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Derivation Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
Euler equation proof make expr power
1. 4923339482; locally 3784785:
$$i x = \log(y)$$
$$pdg_{1464} pdg_{4621} = \frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}}$$
1. 0006656532:
$$e$$
$$pdg_{2718}$$
1. 9482923849; locally 9587572:
$$\exp(i x) = y$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{1452}$$
LHS diff is -pdg2718**(pdg1464*pdg4621) + exp(pdg1464*pdg4621) RHS diff is pdg1452 - pdg2718**(log(pdg1452)/log(10)) 4923339482:
9482923849:
4923339482:
9482923849:
Euler equation proof indefinite integrate RHS over
1. 9482113948; locally 8883737:
$$\frac{dy}{y} = i dx$$
$$pdg_{4621}$$
1. 0004264724:
$$y$$
$$pdg_{1452}$$
1. 9482943948; locally 9984877:
$$\log(y) = i dx$$
$$\frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}} = pdg_{4621} pdg_{9199}$$
Nothing to split 9482113948:
9482943948:
9482113948:
9482943948:
Euler equation proof multiply both sides by
1. 9848294829; locally 9038289:
$$\frac{d}{dx} y = y i$$
$$\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{1452} pdg_{4621}$$
1. 0003954314:
$$dx$$
$$pdg_{9199}$$
1. 9848292229; locally 1111289:
$$dy = y i dx$$
$$pdg_{5842} = pdg_{1452} pdg_{4621} pdg_{9199}$$
LHS diff is -pdg5842 RHS diff is 0 9848294829:
9848292229:
9848294829:
9848292229:
Euler equation proof factor out X from RHS
1. 9429829482; locally 1838300:
$$\frac{d}{dx} y = -\sin(x) + i\cos(x)$$
$$\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{4621} \cos{\left(pdg_{1464} \right)} - \sin{\left(pdg_{1464} \right)}$$
1. 0007563791:
$$i$$
$$pdg_{4621}$$
1. 9482984922; locally 2948271:
$$\frac{d}{dx} y = (i\sin(x) + \cos(x)) i$$
$$\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{4621} \left(pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\right)$$
LHS diff is 0 RHS diff is -(pdg4621**2 + 1)*sin(pdg1464) 9429829482:
9482984922:
9429829482:
9482984922:
Euler equation proof indefinite integrate RHS over
1. 9482943948; locally 9984877:
$$\log(y) = i dx$$
$$\frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}} = pdg_{4621} pdg_{9199}$$
1. 0006563727:
$$x$$
$$pdg_{1464}$$
1. 4928239482; locally 3747585:
$$\log(y) = i x$$
$$\frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}} = pdg_{1464} pdg_{4621}$$
no check performed 9482943948:
4928239482:
9482943948:
4928239482:
Euler equation proof differentiate with respect to
1. 9492920340; locally 1029383:
$$y = \cos(x)+i \sin(x)$$
$$pdg_{1452} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
1. 0007636749:
$$x$$
$$pdg_{1464}$$
1. 9429829482; locally 1838300:
$$\frac{d}{dx} y = -\sin(x) + i\cos(x)$$
$$\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{4621} \cos{\left(pdg_{1464} \right)} - \sin{\left(pdg_{1464} \right)}$$
no check performed 9492920340:
9429829482:
9492920340:
9429829482:
Euler equation proof divide both sides by
1. 9848292229; locally 1111289:
$$dy = y i dx$$
$$pdg_{5842} = pdg_{1452} pdg_{4621} pdg_{9199}$$
1. 0009877781:
$$y$$
$$pdg_{1452}$$
1. 9482113948; locally 8883737:
$$\frac{dy}{y} = i dx$$
$$pdg_{4621}$$
Nothing to split 9848292229:
9482113948:
9848292229:
9482113948:
Euler equation proof declare initial expr
1. 9492920340; locally 1029383:
$$y = \cos(x)+i \sin(x)$$
$$pdg_{1452} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
no validation is available for declarations 9492920340:
9492920340:
Euler equation proof substitute RHS of expr 1 into expr 2
1. 9492920340; locally 1029383:
$$y = \cos(x)+i \sin(x)$$
$$pdg_{1452} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
2. 9482984922; locally 2948271:
$$\frac{d}{dx} y = (i\sin(x) + \cos(x)) i$$
$$\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{4621} \left(pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\right)$$
1. 9848294829; locally 9038289:
$$\frac{d}{dx} y = y i$$
$$\frac{d}{d pdg_{1464}} pdg_{1452} = pdg_{1452} pdg_{4621}$$
valid 9492920340:
9482984922:
9848294829:
9492920340:
9482984922:
9848294829:
Euler equation proof declare final expr
1. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
no validation is available for declarations 4938429483:
4938429483:
Euler equation proof swap LHS with RHS
1. 4928239482; locally 3747585:
$$\log(y) = i x$$
$$\frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}} = pdg_{1464} pdg_{4621}$$
1. 4923339482; locally 3784785:
$$i x = \log(y)$$
$$pdg_{1464} pdg_{4621} = \frac{\log{\left(pdg_{1452} \right)}}{\log{\left(10 \right)}}$$
valid 4928239482:
4923339482:
4928239482:
4923339482:
Euler equation proof substitute RHS of expr 1 into expr 2
1. 9482923849; locally 9587572:
$$\exp(i x) = y$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{1452}$$
2. 9492920340; locally 1029383:
$$y = \cos(x)+i \sin(x)$$
$$pdg_{1452} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
1. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
valid 9482923849:
9492920340:
4938429483:
9482923849:
9492920340:
4938429483:
Euler equation: trig square root declare final expr
1. 9988949211; locally 1231131:
$$(\sin(x))^2 = \frac{1 - \cos(2 x)}{2}$$
$$\sin^{2}{\left(pdg_{1464} \right)} = \frac{1}{2} - \frac{\cos{\left(2 pdg_{1464} \right)}}{2}$$
no validation is available for declarations 9988949211:
9988949211:
Euler equation: trig square root declare identity
1. 5832984291; locally 9385720:
$$(\sin(x))^2 + (\cos(x))^2 = 1$$
$$\sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)} = 1$$
no validation is available for declarations 5832984291:
5832984291:
Euler equation: trig square root divide both sides by
1. 9889984281; locally 7472666:
$$2 (\sin(x))^2 = 1 - \cos(2 x)$$
$$2 \sin^{2}{\left(pdg_{1464} \right)} = 1 - \cos{\left(2 pdg_{1464} \right)}$$
1. 0003838111:
$$2$$
$$2$$
1. 9988949211; locally 1231131:
$$(\sin(x))^2 = \frac{1 - \cos(2 x)}{2}$$
$$\sin^{2}{\left(pdg_{1464} \right)} = \frac{1}{2} - \frac{\cos{\left(2 pdg_{1464} \right)}}{2}$$
valid 9889984281:
9988949211:
9889984281:
9988949211:
Euler equation: trig square root swap LHS with RHS
1. 9482928243; locally 4890284:
$$\cos(2 x) + (\sin(x))^2 = (\cos(x))^2$$
$$\sin^{2}{\left(pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)} = \cos^{2}{\left(pdg_{1464} \right)}$$
1. 9482438243; locally 2936550:
$$(\cos(x))^2 = \cos(2 x) + (\sin(x))^2$$
$$\cos^{2}{\left(pdg_{1464} \right)} = \sin^{2}{\left(pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)}$$
valid 9482928243:
9482438243:
9482928243:
9482438243:
Euler equation: trig square root expand RHS
1. 4638429483; locally 3333333:
$$\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))$$
$$e^{2 pdg_{1464} pdg_{4621}} = \left(pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\right)^{2}$$
1. 4598294821; locally 4444444:
$$\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2$$
$$e^{2 pdg_{1464} pdg_{4621}} = 2 pdg_{4621} \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} - \sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)}$$
LHS diff is 0 RHS diff is (pdg4621**2 + 1)*sin(pdg1464)**2 4638429483:
4598294821:
4638429483:
4598294821:
Euler equation: trig square root change variable X to Y
1. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
1. 0004089571:
$$2 x$$
$$2 pdg_{1464}$$
2. 0004582412:
$$x$$
$$pdg_{1464}$$
1. 4838429483; locally 9999999:
$$\exp(2 i x) = \cos(2 x)+i \sin(2 x)$$
$$e^{2 pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(2 pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)}$$
LHS diff is (1 - exp(pdg1464*pdg4621))*exp(pdg1464*pdg4621) RHS diff is pdg4621*sin(pdg1464) - pdg4621*sin(2*pdg1464) + cos(pdg1464) - cos(2*pdg1464) 4938429483:
4838429483:
4938429483:
4838429483:
Euler equation: trig square root subtract X from both sides
1. 4827492911; locally 9481000:
$$\cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2$$
$$\sin^{2}{\left(pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)} = 1 - \sin^{2}{\left(pdg_{1464} \right)}$$
1. 0006466214:
$$(\sin(x))^2$$
$$\sin^{2}{\left(pdg_{1464} \right)}$$
1. 1248277773; locally 7472641:
$$\cos(2 x) = 1 - 2 (\sin(x))^2$$
$$\cos{\left(2 pdg_{1464} \right)} = 1 - 2 \sin^{2}{\left(pdg_{1464} \right)}$$
valid 4827492911:
1248277773:
4827492911:
1248277773:
Euler equation: trig square root subtract X from both sides
1. 7572664728; locally 1029911:
$$\cos(2 x) + 2 (\sin(x))^2 = 1$$
$$2 \sin^{2}{\left(pdg_{4037} \right)} + \cos{\left(2 pdg_{4037} \right)} = 1$$
1. 0008842811:
$$\cos(2 x)$$
$$\cos{\left(2 pdg_{1464} \right)}$$
1. 9889984281; locally 7472666:
$$2 (\sin(x))^2 = 1 - \cos(2 x)$$
$$2 \sin^{2}{\left(pdg_{1464} \right)} = 1 - \cos{\left(2 pdg_{1464} \right)}$$
valid 7572664728:
9889984281:
7572664728:
9889984281:
Euler equation: trig square root add X to both sides
1. 1248277773; locally 7472641:
$$\cos(2 x) = 1 - 2 (\sin(x))^2$$
$$\cos{\left(2 pdg_{1464} \right)} = 1 - 2 \sin^{2}{\left(pdg_{1464} \right)}$$
1. 0007471778:
$$2(\sin(x))^2$$
$$2 \sin^{2}{\left(pdg_{1464} \right)}$$
1. 7572664728; locally 1029911:
$$\cos(2 x) + 2 (\sin(x))^2 = 1$$
$$2 \sin^{2}{\left(pdg_{4037} \right)} + \cos{\left(2 pdg_{4037} \right)} = 1$$
valid 1248277773:
7572664728:
1248277773:
7572664728:
Euler equation: trig square root declare initial expr
1. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
no validation is available for declarations 4938429483:
4938429483:
Euler equation: trig square root select real parts
1. 9483928192; locally 2222222:
$$\cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2$$
$$pdg_{4621} \sin{\left(2 pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)} = 2 pdg_{4621} \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} - \sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)}$$
1. 9482928242; locally 5828294:
$$\cos(2 x) = (\cos(x))^2 - (\sin(x))^2$$
$$\cos{\left(2 pdg_{1464} \right)} = - \sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)}$$
LHS diff is -cos(2*pdg1464) + cos(2*re(pdg1464))*cosh(2*im(pdg1464)) + re(pdg4621*sin(2*pdg1464)) RHS diff is -cos(2*pdg1464) + 2*cos(2*re(pdg1464))*sinh(im(pdg1464))**2 + cos(2*re(pdg1464)) + re(pdg4621*sin(2*pdg1464)) 9483928192:
9482928242:
9483928192:
9482928242:
Euler equation: trig square root add X to both sides
1. 9482928242; locally 5828294:
$$\cos(2 x) = (\cos(x))^2 - (\sin(x))^2$$
$$\cos{\left(2 pdg_{1464} \right)} = - \sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)}$$
1. 0007894942:
$$(\sin(x))^2$$
$$\sin^{2}{\left(pdg_{1464} \right)}$$
1. 9482928243; locally 4890284:
$$\cos(2 x) + (\sin(x))^2 = (\cos(x))^2$$
$$\sin^{2}{\left(pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)} = \cos^{2}{\left(pdg_{1464} \right)}$$
valid 9482928242:
9482928243:
9482928242:
9482928243:
Euler equation: trig square root multiply expr 1 by expr 2
1. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
2. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
1. 4638429483; locally 3333333:
$$\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))$$
$$e^{2 pdg_{1464} pdg_{4621}} = \left(pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}\right)^{2}$$
valid 4938429483:
4938429483:
4638429483:
4938429483:
4938429483:
4638429483:
Euler equation: trig square root LHS of expr 1 equals LHS of expr 2
1. 9482438243; locally 2936550:
$$(\cos(x))^2 = \cos(2 x) + (\sin(x))^2$$
$$\cos^{2}{\left(pdg_{1464} \right)} = \sin^{2}{\left(pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)}$$
2. 3285732911; locally 9123670:
$$(\cos(x))^2 = 1-(\sin(x))^2$$
$$\cos^{2}{\left(pdg_{1464} \right)} = 1 - \sin^{2}{\left(pdg_{1464} \right)}$$
1. 4827492911; locally 9481000:
$$\cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2$$
$$\sin^{2}{\left(pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)} = 1 - \sin^{2}{\left(pdg_{1464} \right)}$$
valid 9482438243:
3285732911:
4827492911:
9482438243:
3285732911:
4827492911:
Euler equation: trig square root LHS of expr 1 equals LHS of expr 2
1. 4598294821; locally 4444444:
$$\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2$$
$$e^{2 pdg_{1464} pdg_{4621}} = 2 pdg_{4621} \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} - \sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)}$$
2. 4838429483; locally 9999999:
$$\exp(2 i x) = \cos(2 x)+i \sin(2 x)$$
$$e^{2 pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(2 pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)}$$
1. 9483928192; locally 2222222:
$$\cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2$$
$$pdg_{4621} \sin{\left(2 pdg_{1464} \right)} + \cos{\left(2 pdg_{1464} \right)} = 2 pdg_{4621} \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} - \sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)}$$
valid 4598294821:
4838429483:
9483928192:
4598294821:
4838429483:
9483928192:
Euler equation: trig square root subtract X from both sides
1. 5832984291; locally 9385720:
$$(\sin(x))^2 + (\cos(x))^2 = 1$$
$$\sin^{2}{\left(pdg_{1464} \right)} + \cos^{2}{\left(pdg_{1464} \right)} = 1$$
1. 0001111111:
$$(\sin(x))^2$$
$$\sin^{2}{\left(pdg_{1464} \right)}$$
1. 3285732911; locally 9123670:
$$(\cos(x))^2 = 1-(\sin(x))^2$$
$$\cos^{2}{\left(pdg_{1464} \right)} = 1 - \sin^{2}{\left(pdg_{1464} \right)}$$
valid 5832984291:
3285732911:
5832984291:
3285732911:
Euler equation: trigonometric relations divide both sides by
1. 4742644828; locally 2939484:
$$\exp(i x)+\exp(-i x) = 2 \cos(x)$$
$$e^{pdg_{1464} pdg_{4621}} + e^{- pdg_{1464} pdg_{4621}} = 2 \cos{\left(pdg_{1464} \right)}$$
1. 0004829194:
$$2$$
$$2$$
1. 3829492824; locally 4383592:
$$\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x)$$
$$\frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2} = \cos{\left(pdg_{1464} \right)}$$
valid 4742644828:
3829492824:
4742644828:
3829492824:
Euler equation: trigonometric relations declare final expr
1. 4585932229; locally 4849888:
$$\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$$
$$\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}$$
2. 2103023049; locally 4849959:
$$\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)$$
$$\sin{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}$$
no validation is available for declarations 4585932229:
2103023049:
4585932229:
2103023049:
Euler equation: trigonometric relations divide both sides by
1. 3942849294; locally 4825483:
$$\exp(i x)-\exp(-i x) = 2 i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}} = 2 pdg_{4621} \sin{\left(pdg_{1464} \right)}$$
1. 0001921933:
$$2 i$$
$$2 pdg_{4621}$$
1. 4843995999; locally 1133483:
$$\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x)$$
$$\frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}} = \sin{\left(pdg_{1464} \right)}$$
valid 3942849294:
4843995999:
3942849294:
4843995999:
Euler equation: trigonometric relations add expr 1 to expr 2
1. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
2. 2123139121; locally 3194924:
$$-\exp(-i x) = -\cos(x)+i \sin(x)$$
$$- e^{- pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} - \cos{\left(pdg_{1464} \right)}$$
1. 3942849294; locally 4825483:
$$\exp(i x)-\exp(-i x) = 2 i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}} = 2 pdg_{4621} \sin{\left(pdg_{1464} \right)}$$
valid 4938429483:
2123139121:
3942849294:
4938429483:
2123139121:
3942849294:
Euler equation: trigonometric relations swap LHS with RHS
1. 3829492824; locally 4383592:
$$\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x)$$
$$\frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2} = \cos{\left(pdg_{1464} \right)}$$
1. 4585932229; locally 4849888:
$$\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$$
$$\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}$$
valid 3829492824:
4585932229:
3829492824:
4585932229:
Euler equation: trigonometric relations add expr 1 to expr 2
1. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
2. 4938429484; locally 8888883:
$$\exp(-i x) = \cos(x)-i \sin(x)$$
$$e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
1. 4742644828; locally 2939484:
$$\exp(i x)+\exp(-i x) = 2 \cos(x)$$
$$e^{pdg_{1464} pdg_{4621}} + e^{- pdg_{1464} pdg_{4621}} = 2 \cos{\left(pdg_{1464} \right)}$$
valid 4938429483:
4938429484:
4742644828:
4938429483:
4938429484:
4742644828:
Euler equation: trigonometric relations multiply both sides by
1. 4938429484; locally 8888883:
$$\exp(-i x) = \cos(x)-i \sin(x)$$
$$e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
1. 0003747849:
$$-1$$
$$-1$$
1. 2123139121; locally 3194924:
$$-\exp(-i x) = -\cos(x)+i \sin(x)$$
$$- e^{- pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} - \cos{\left(pdg_{1464} \right)}$$
valid 4938429484:
2123139121:
4938429484:
2123139121:
Euler equation: trigonometric relations change variable X to Y
1. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
1. 0002393922:
$$x$$
$$pdg_{1464}$$
2. 0003949052:
$$-x$$
$$- pdg_{1464}$$
1. 2394853829; locally 8888881:
$$\exp(-i x) = \cos(-x)+i \sin(-x)$$
$$e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
valid 4938429483:
2394853829:
4938429483:
2394853829:
Euler equation: trigonometric relations swap LHS with RHS
1. 4843995999; locally 1133483:
$$\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x)$$
$$\frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}} = \sin{\left(pdg_{1464} \right)}$$
1. 2103023049; locally 4849959:
$$\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)$$
$$\sin{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}$$
valid 4843995999:
2103023049:
4843995999:
2103023049:
Euler equation: trigonometric relations function is odd
1. 4938429482; locally 8888882:
$$\exp(-i x) = \cos(x)+i \sin(-x)$$
$$e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
1. 0003919391:
$$x$$
$$pdg_{1464}$$
2. 0003981813:
$$-\sin(x)$$
$$- \sin{\left(pdg_{1464} \right)}$$
3. 0002919191:
$$\sin(-x)$$
$$- \sin{\left(pdg_{1464} \right)}$$
1. 4938429484; locally 8888883:
$$\exp(-i x) = \cos(x)-i \sin(x)$$
$$e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
no check performed 4938429482:
4938429484:
4938429482:
4938429484:
Euler equation: trigonometric relations declare initial expr
1. 4938429483; locally 8888888:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
no validation is available for declarations 4938429483:
4938429483:
Euler equation: trigonometric relations function is even
1. 2394853829; locally 8888881:
$$\exp(-i x) = \cos(-x)+i \sin(-x)$$
$$e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
1. 0004849392:
$$x$$
$$pdg_{1464}$$
2. 0001030901:
$$\cos(x)$$
$$\cos{\left(pdg_{1464} \right)}$$
3. 0003413423:
$$\cos(-x)$$
$$\cos{\left(pdg_{1464} \right)}$$
1. 4938429482; locally 8888882:
$$\exp(-i x) = \cos(x)+i \sin(-x)$$
$$e^{- pdg_{1464} pdg_{4621}} = - pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
no check performed 2394853829:
4938429482:
2394853829:
4938429482:
Maxwell equations to electric field wave equation partially differentiate with respect to
1. 1314864131; locally 1199299:
$$\vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}$$
$$nabla \times pdg_{2069} = pdg_{7940} \frac{d}{d pdg_{1467}} pdg_{4326}$$
1. 0005626421:
$$t$$
$$pdg_{1467}$$
1. 1314464131; locally 4642245:
$$\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$$
$$pdg_{1467}$$
Nothing to split 1314864131:
1314464131:
1314864131:
1314464131:
Maxwell equations to electric field wave equation substitute LHS of expr 1 into expr 2
1. 9291999979; locally 2392932:
$$\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}$$
$$nabla^{2} pdg_{4326} = - nabla pdg_{6197} \frac{d}{d pdg_{1467}} pdg_{2069}$$
2. 1314464131; locally 4642245:
$$\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$$
$$pdg_{1467}$$
1. 3947269979; locally 2962831:
$$\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$$
$$pdg_{1467}$$
Nothing to split 9291999979:
1314464131:
3947269979:
9291999979:
1314464131:
3947269979:
Maxwell equations to electric field wave equation declare initial expr
1. 9991999979; locally 4757562:
$$\vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}$$
$$- nabla \times pdg_{6238} = - pdg_{6197} \frac{d}{d pdg_{1467}} pdg_{2069}$$
no validation is available for declarations 9991999979:
9991999979:
Maxwell equations to electric field wave equation declare initial expr
1. 1314864131; locally 1199299:
$$\vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}$$
$$nabla \times pdg_{2069} = pdg_{7940} \frac{d}{d pdg_{1467}} pdg_{4326}$$
no validation is available for declarations 1314864131:
1314864131:
Maxwell equations to electric field wave equation declare initial expr
1. 9999999981; locally 4857731:
$$\vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0$$
$$nabla pdg_{4326} = \frac{pdg_{3935}}{pdg_{7940}}$$
no validation is available for declarations 9999999981:
9999999981:
Maxwell equations to electric field wave equation substitute LHS of expr 1 into expr 2
1. 1636453295; locally 3738373:
$$\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E}$$
$$nabla \times \left(nabla \times pdg_{4326}\right) = - nabla^{2} pdg_{4326}$$
2. 3947269979; locally 2962831:
$$\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$$
$$pdg_{1467}$$
1. 8494839423; locally 4758592:
$$\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$$
$$nabla^{2} pdg_{4326} = \frac{partial pdg_{4326} pdg_{6197} pdg_{7940}}{pdg_{1467}^{2}}$$
Nothing to split 1636453295:
3947269979:
8494839423:
1636453295:
3947269979:
8494839423:
Maxwell equations to electric field wave equation declare assumption
1. 9919999981; locally 3984852:
$$\rho = 0$$
$$pdg_{3935} = 0$$
no validation is available for declarations 9919999981:
9919999981:
Maxwell equations to electric field wave equation declare identity
1. 7575859295; locally 1939485:
$$\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$
$$pdg_{6238} \times \left(nabla \times nabla\right) = \operatorname{nabla}{\left(- nabla^{2} pdg_{6238} + nabla \cdot pdg_{6238} \right)}$$
no validation is available for declarations 7575859295:
7575859295:
Maxwell equations to electric field wave equation declare final expr
1. 8494839423; locally 4758592:
$$\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$$
$$nabla^{2} pdg_{4326} = \frac{partial pdg_{4326} pdg_{6197} pdg_{7940}}{pdg_{1467}^{2}}$$
no validation is available for declarations 8494839423:
8494839423:
Maxwell equations to electric field wave equation take curl of both sides
1. 9991999979; locally 4757562:
$$\vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}$$
$$- nabla \times pdg_{6238} = - pdg_{6197} \frac{d}{d pdg_{1467}} pdg_{2069}$$
1. 9291999979; locally 2392932:
$$\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}$$
$$nabla^{2} pdg_{4326} = - nabla pdg_{6197} \frac{d}{d pdg_{1467}} pdg_{2069}$$
no check performed 9991999979:
9291999979:
9991999979:
9291999979:
Maxwell equations to electric field wave equation substitute LHS of expr 1 into expr 2
1. 7575859295; locally 1939485:
$$\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$
$$pdg_{6238} \times \left(nabla \times nabla\right) = \operatorname{nabla}{\left(- nabla^{2} pdg_{6238} + nabla \cdot pdg_{6238} \right)}$$
2. 7466829492; locally 2837471:
$$\vec{ \nabla} \cdot \vec{E} = 0$$
$$nabla \cdot pdg_{6238} = 0$$
1. 1636453295; locally 3738373:
$$\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E}$$
$$nabla \times \left(nabla \times pdg_{4326}\right) = - nabla^{2} pdg_{4326}$$
failed 7575859295:
7466829492:
1636453295:
7575859295:
7466829492:
1636453295:
Maxwell equations to electric field wave equation substitute LHS of expr 1 into expr 2
1. 9999999981; locally 4857731:
$$\vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0$$
$$nabla pdg_{4326} = \frac{pdg_{3935}}{pdg_{7940}}$$
2. 9919999981; locally 3984852:
$$\rho = 0$$
$$pdg_{3935} = 0$$
1. 7466829492; locally 2837471:
$$\vec{ \nabla} \cdot \vec{E} = 0$$
$$nabla \cdot pdg_{6238} = 0$$
LHS diff is pdg3935 - Dot(nabla, pdg6238) RHS diff is 0 9999999981:
9919999981:
7466829492:
9999999981:
9919999981:
7466829492:
curl curl identity replace summation notation with vector notation
1. 7575859310; locally 3948472:
$$\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$

1. 7575859312; locally 2109231:
$$\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$

Nothing to split 7575859310:
7575859312:
7575859310:
7575859312:
curl curl identity replace curl with LeviCevita summation contravariant
1. 7575859295; locally 1939485:
$$\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$
$$pdg_{6238} \times \left(nabla \times nabla\right) = \operatorname{nabla}{\left(- nabla^{2} pdg_{6238} + nabla \cdot pdg_{6238} \right)}$$
1. 7575859300; locally 9485482:
$$\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$

Nothing to split 7575859295:
7575859300:
7575859295:
7575859300:
curl curl identity substitute RHS of expr 1 into expr 2
1. 7575859304; locally 2934842:
$$\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}$$

2. 7575859302; locally 2941319:
$$\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$

1. 7575859306; locally 3949292:
$$\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$

failed 7575859304:
7575859302:
7575859306:
7575859304:
7575859302:
7575859306:
curl curl identity simplify
1. 7575859308; locally 3844221:
$$\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$

1. 7575859310; locally 3948472:
$$\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$

Nothing to split 7575859308:
7575859310:
7575859308:
7575859310:
curl curl identity simplify
1. 7575859306; locally 3949292:
$$\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$

1. 7575859308; locally 3844221:
$$\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$

failed 7575859306:
7575859308:
7575859306:
7575859308:
curl curl identity replace curl with LeviCevita summation contravariant
1. 7575859300; locally 9485482:
$$\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$

1. 7575859302; locally 2941319:
$$\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$

Nothing to split 7575859300:
7575859302:
7575859300:
7575859302:
curl curl identity declare identity
1. 7575859295; locally 1939485:
$$\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$
$$pdg_{6238} \times \left(nabla \times nabla\right) = \operatorname{nabla}{\left(- nabla^{2} pdg_{6238} + nabla \cdot pdg_{6238} \right)}$$
no validation is available for declarations 7575859295:
7575859295:
curl curl identity claim LHS equals RHS
1. 7575859312; locally 2109231:
$$\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$$

Nothing to split 7575859312:
7575859312:
curl curl identity declare identity
1. 7575859304; locally 2934842:
$$\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}$$

no validation is available for declarations 7575859304:
7575859304:
derivation of Schrodinger Equation replace scalar with vector
1. 9999999870; locally 4948325:
$$\frac{p}{\hbar} = k$$

1. 9999998870; locally 2948487:
$$\frac{ \vec{p}}{\hbar} = \vec{k}$$

no check performed 9999999870:
9999998870:
9999999870:
9999998870:
derivation of Schrodinger Equation substitute RHS of expr 1 into expr 2
1. 3948574226; locally 2100421:
$$\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)$$

2. 9999999961; locally 4499582:
$$\frac{E}{\hbar} = \omega$$

1. 3948574228; locally 1291313:
$$\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)$$

LHS diff is -pdg9489(pdg9472, pdg1467) + pdg4931/pdg1054 RHS diff is pdg2321 - pdg8330*pdg2718(pdg4621((pdg1134*pdg9472 - pdg1467*pdg6238)/pdg1054)) 3948574226:
9999999961:
3948574228:
3948574226:
9999999961:
3948574228:
derivation of Schrodinger Equation declare initial expr
1. 3121513111; locally 2934848:
$$k = \frac{2 \pi}{\lambda}$$
$$pdg_{5321} = \frac{2 pdg_{3141}}{pdg_{1115}}$$
no validation is available for declarations 3121513111:
3121513111:
derivation of Schrodinger Equation declare initial expr
1. 1029039903; locally 1039948:
$$p = m v$$

no validation is available for declarations 1029039903:
1029039903:
derivation of Schrodinger Equation substitute RHS of expr 1 into expr 2
1. 1020394900; locally 1203491:
$$p = h/\lambda$$

2. 3121234211; locally 1039485:
$$\frac{k}{2\pi} = \lambda$$

1. 3121234212; locally 2901049:
$$p = \frac{h k}{2\pi}$$

LHS diff is -pdg1134 + pdg5321/(2*pdg3141) RHS diff is pdg1115 - pdg4413*pdg5321/(2*pdg3141) 1020394900:
3121234211:
3121234212:
1020394900:
3121234211:
3121234212:
derivation of Schrodinger Equation declare initial expr
1. 9999999960; locally 2949002:
$$\hbar = h/(2 \pi)$$

no validation is available for declarations 9999999960:
9999999960:
derivation of Schrodinger Equation substitute RHS of expr 1 into expr 2
1. 3147472131; locally 2939402:
$$\frac{\omega}{2 \pi} = f$$

2. 1020394902; locally 3499522:
$$E = h f$$

1. 4147472132; locally 2949821:
$$E = \frac{h \omega}{2 \pi}$$

valid 3147472131:
1020394902:
4147472132:
3147472131:
1020394902:
4147472132:
derivation of Schrodinger Equation simplify
1. 3948574228; locally 1291313:
$$\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)$$

1. 3948574230; locally 1305534:
$$\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$$

LHS diff is 0 RHS diff is pdg8330*(-pdg2718(pdg4621*(pdg1134*pdg9472 - pdg1467*pdg6238)/pdg1054) + pdg2718(pdg4621((pdg1134*pdg9472 - pdg1467*pdg6238)/pdg1054))) 3948574228:
3948574230:
3948574228:
3948574230:
derivation of Schrodinger Equation partially differentiate with respect to
1. 3948574230; locally 1305534:
$$\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$$

1. 0006544644:
$$t$$

1. 3948574233; locally 2364546:
$$\frac{\partial}{\partial t} \psi( \vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)$$

no check performed 3948574230:
3948574233:
3948574230:
3948574233:
derivation of Schrodinger Equation declare initial expr
1. 4298359835; locally 1353583:
$$E = \frac{1}{2}m v^2$$

no validation is available for declarations 4298359835:
4298359835:
derivation of Schrodinger Equation declare initial expr
1. 3948574224; locally 3940505:
$$\psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)$$

no validation is available for declarations 3948574224:
3948574224:
derivation of Schrodinger Equation divide both sides by
1. 9999999962; locally 1039013:
$$p = \hbar k$$

1. 0001304952:
$$\hbar$$

1. 9999999870; locally 4948325:
$$\frac{p}{\hbar} = k$$

valid 9999999962:
9999999870:
9999999962:
9999999870:
derivation of Schrodinger Equation multiply RHS by unity
1. 4298359835; locally 1353583:
$$E = \frac{1}{2}m v^2$$

1. 0002342425:
$$m/m$$

1. 4298359845; locally 2326309:
$$E = \frac{1}{2m}m^2 v^2$$

valid 4298359835:
4298359845:
4298359835:
4298359845:
derivation of Schrodinger Equation simplify
1. 4394958389; locally 4938589:
$$\vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) = \frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right)$$

1. 1648958381; locally 1495034:
$$\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)$$

failed 4394958389:
1648958381:
4394958389:
1648958381:
derivation of Schrodinger Equation raise both sides to power
1. 1029039903; locally 1039948:
$$p = m v$$

1. 0002239424:
$$2$$

1. 1029039904; locally 1432042:
$$p^2 = m^2 v^2$$

no check is performed 1029039903:
1029039904:
1029039903:
1029039904:
derivation of Schrodinger Equation substitute RHS of expr 1 into expr 2
1. 1029039904; locally 1432042:
$$p^2 = m^2 v^2$$

2. 4298359845; locally 2326309:
$$E = \frac{1}{2m}m^2 v^2$$

1. 4298359851; locally 3576787:
$$E = \frac{p^2}{2m}$$

LHS diff is 0 RHS diff is (-pdg1134**2 + pdg1357**2*pdg5156**2)/(2*pdg5156) 1029039904:
4298359845:
4298359851:
1029039904:
4298359845:
4298359851:
derivation of Schrodinger Equation apply gradient to scalar function
1. 3948574230; locally 1305534:
$$\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$$

1. 3948574230; locally 5577584:
$$\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$$

no check performed 3948574230:
3948574230:
3948574230:
3948574230:
derivation of Schrodinger Equation declare initial expr
1. 1158485859; locally 2344324:
$$\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}$$

no validation is available for declarations 1158485859:
1158485859:
derivation of Schrodinger Equation substitute RHS of expr 1 into expr 2
1. 3948574233; locally 2364546:
$$\frac{\partial}{\partial t} \psi( \vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)$$

2. 3948574230; locally 1305534:
$$\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$$

1. 3948571256; locally 5345567:
$$\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t)$$

LHS diff is pdg9489(pdg9472, pdg1467) - Derivative(pdg9489(pdg9472, pdg1467), pdg1467) RHS diff is (pdg1054*pdg8330*pdg2718(pdg4621*(pdg1134*pdg9472 - pdg1467*pdg6238)/pdg1054) + pdg4621*pdg6238*pdg9489(pdg9472, pdg1467))/pdg1054 3948574233:
3948574230:
3948571256:
3948574233:
3948574230:
3948571256:
derivation of Schrodinger Equation divide both sides by
1. 3121513111; locally 2934848:
$$k = \frac{2 \pi}{\lambda}$$
$$pdg_{5321} = \frac{2 pdg_{3141}}{pdg_{1115}}$$
1. 0001209482:
$$2 \pi$$

1. 3121234211; locally 1039485:
$$\frac{k}{2\pi} = \lambda$$

LHS diff is 0 RHS diff is -pdg1115 + 1/pdg1115 3121513111:
3121234211:
3121513111:
3121234211:
derivation of Schrodinger Equation declare initial expr
1. 3131211131; locally 9214650:
$$\omega = 2 \pi f$$
$$pdg_{2321} = 2 pdg_{3141} pdg_{4201}$$
no validation is available for declarations 3131211131:
3131211131:
derivation of Schrodinger Equation substitute RHS of expr 1 into expr 2
1. 9999999960; locally 2949002:
$$\hbar = h/(2 \pi)$$

2. 4147472132; locally 2949821:
$$E = \frac{h \omega}{2 \pi}$$

1. 9999999965; locally 3741728:
$$E = \omega \hbar$$

valid 9999999960:
4147472132:
9999999965:
9999999960:
4147472132:
9999999965:
derivation of Schrodinger Equation substitute RHS of expr 1 into expr 2
1. 3948574224; locally 3940505:
$$\psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)$$

2. 9999998870; locally 2948487:
$$\frac{ \vec{p}}{\hbar} = \vec{k}$$

1. 3948574226; locally 2100421:
$$\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)$$

LHS diff is -pdg9489(pdg9472, pdg1467) + pdg2046/pdg1054 RHS diff is pdg7394 - pdg8330*pdg2718(pdg4621(-pdg1467*pdg2321 + pdg1134*pdg9472/pdg1054)) 3948574224:
9999998870:
3948574226:
3948574224:
9999998870:
3948574226:
derivation of Schrodinger Equation substitute RHS of expr 1 into expr 2
1. 3121234212; locally 2901049:
$$p = \frac{h k}{2\pi}$$

2. 9999999960; locally 2949002:
$$\hbar = h/(2 \pi)$$

1. 9999999962; locally 1039013:
$$p = \hbar k$$

LHS diff is pdg1054 - pdg1134 RHS diff is -pdg1054*pdg5321 + pdg4413/(2*pdg3141) 3121234212:
9999999960:
9999999962:
3121234212:
9999999960:
9999999962:
derivation of Schrodinger Equation substitute LHS of expr 1 into expr 2
1. 1158485859; locally 2344324:
$$\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}$$

2. 9958485859; locally 1304924:
$$\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)$$

1. 2258485859; locally 2456546:
$${\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)$$

Nothing to split 1158485859:
9958485859:
2258485859:
1158485859:
9958485859:
2258485859:
derivation of Schrodinger Equation apply divergence
1. 5985371230; locally 5535257:
$$\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)$$

1. 4394958389; locally 4938589:
$$\vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) = \frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right)$$

failed 5985371230:
4394958389:
5985371230:
4394958389:
derivation of Schrodinger Equation substitute RHS of expr 1 into expr 2
1. 4298359851; locally 3576787:
$$E = \frac{p^2}{2m}$$

2. 3948571256; locally 5345567:
$$\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t)$$

1. 4348571256; locally 2495835:
$$\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)$$

LHS diff is 0 RHS diff is pdg4621*(pdg1134**2 - 2*pdg5156*pdg6238)*pdg9489(pdg9472, pdg1467)/(2*pdg1054*pdg5156) 4298359851:
3948571256:
4348571256:
4298359851:
3948571256:
4348571256:
derivation of Schrodinger Equation divide both sides by
1. 9999999965; locally 3741728:
$$E = \omega \hbar$$

1. 0003949921:
$$\hbar$$

1. 9999999961; locally 4499582:
$$\frac{E}{\hbar} = \omega$$

valid 9999999965:
9999999961:
9999999965:
9999999961:
derivation of Schrodinger Equation declare initial expr
1. 1020394902; locally 3499522:
$$E = h f$$

no validation is available for declarations 1020394902:
1020394902:
derivation of Schrodinger Equation divide both sides by
1. 3131211131; locally 9214650:
$$\omega = 2 \pi f$$
$$pdg_{2321} = 2 pdg_{3141} pdg_{4201}$$
1. 0002940021:
$$2 \pi$$

1. 3147472131; locally 2939402:
$$\frac{\omega}{2 \pi} = f$$

valid 3131211131:
3147472131:
3131211131:
3147472131:
derivation of Schrodinger Equation substitute RHS of expr 1 into expr 2
1. 5985371230; locally 5535257:
$$\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)$$

2. 1648958381; locally 1495034:
$$\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)$$

1. 2648958382; locally 1049553:
$$\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)$$

Nothing to split 5985371230:
1648958381:
2648958382:
5985371230:
1648958381:
2648958382:
derivation of Schrodinger Equation simplify
1. 2648958382; locally 1049553:
$$\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)$$

1. 2395958385; locally 4959593:
$$\nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t)$$

Nothing to split 2648958382:
2395958385:
2648958382:
2395958385:
derivation of Schrodinger Equation multiply both sides by
1. 4348571256; locally 2495835:
$$\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)$$

1. 0002436656:
$$i \hbar$$

1. 4341171256; locally 3429538:
$$i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t)$$

LHS diff is 0 RHS diff is -pdg1134**2*(pdg4621**2 + 1)*pdg9489(pdg9472, pdg1467)/(2*pdg5156) 4348571256:
4341171256:
4348571256:
4341171256:
derivation of Schrodinger Equation multiply both sides by
1. 2395958385; locally 4959593:
$$\nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t)$$

1. 0005938585:
$$\frac{-\hbar^2}{2m}$$

1. 5868688585; locally 4349493:
$$\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t)$$

LHS diff is 0 RHS diff is pdg1134**2*(pdg1054 - 1)*pdg9489(pdg9472, pdg1467)/(2*pdg5156) 2395958385:
5868688585:
2395958385:
5868688585:
derivation of Schrodinger Equation substitute RHS of expr 1 into expr 2
1. 4943571230; locally 3454565:
$$\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$$

2. 3948574230; locally 1305534:
$$\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$$

1. 5985371230; locally 5535257:
$$\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)$$

failed 4943571230:
3948574230:
5985371230:
4943571230:
3948574230:
5985371230:
derivation of Schrodinger Equation simplify
1. 3948574230; locally 5577584:
$$\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$$

1. 4943571230; locally 3454565:
$$\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$$

failed 3948574230:
4943571230:
3948574230:
4943571230:
derivation of Schrodinger Equation declare final expr
1. 2258485859; locally 2456546:
$${\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)$$

no validation is available for declarations 2258485859:
2258485859:
derivation of Schrodinger Equation LHS of expr 1 equals LHS of expr 2
1. 4341171256; locally 3429538:
$$i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t)$$

2. 5868688585; locally 4349493:
$$\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t)$$

1. 9958485859; locally 1304924:
$$\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)$$

Nothing to split 4341171256:
5868688585:
9958485859:
4341171256:
5868688585:
9958485859:
derivation of Schrodinger Equation declare initial expr
1. 1020394900; locally 1203491:
$$p = h/\lambda$$

no validation is available for declarations 1020394900:
1020394900:
electric field wave equation: from time dependent to time independent substitute LHS of expr 1 into expr 2
1. 9499428242; locally 3994928:
$$E( \vec{r},t) = E( \vec{r})\exp(i \omega t)$$
$$\operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)} = \operatorname{pdg_{2718}}{\left(pdg_{1467} pdg_{2321} pdg_{4621} \right)} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)}$$
2. 9394939493; locally 3839493:
$$\nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t)$$
$$nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)} = \frac{partial pdg_{6197} pdg_{7940} \operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)}}{pdg_{1467}^{2}}$$
1. 2029293929; locally 1029393:
$$\nabla^2 E( \vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t)$$
$$nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}} = \frac{partial pdg_{6197} pdg_{7940} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}}}{pdg_{1467}^{2}}$$
LHS diff is nabla**2*(pdg2718(pdg1467*pdg2321*pdg4621) - exp(pdg1467*pdg2321*pdg4621))*pdg6238(pdg9472) RHS diff is partial*pdg6197*pdg7940*(pdg2718(pdg1467*pdg2321*pdg4621) - exp(pdg1467*pdg2321*pdg4621))*pdg6238(pdg9472)/pdg1467**2 9499428242:
9394939493:
2029293929:
9499428242:
9394939493:
2029293929:
electric field wave equation: from time dependent to time independent differentiate with respect to
1. 2029293929; locally 1029393:
$$\nabla^2 E( \vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t)$$
$$nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}} = \frac{partial pdg_{6197} pdg_{7940} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}}}{pdg_{1467}^{2}}$$
1. 0003232242:
$$t$$
$$pdg_{1467}$$
1. 4985825552; locally 2939392:
$$\nabla^2 E( \vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t)$$
$$nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}} = pdg_{2321} pdg_{4621} pdg_{6197} pdg_{7940} \frac{\partial}{\partial pdg_{1467}} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}}$$
no check performed 2029293929:
4985825552:
2029293929:
4985825552:
electric field wave equation: from time dependent to time independent declare initial expr
1. 8572852424; locally 9393848:
$$\vec{E} = E( \vec{r},t)$$
$$pdg_{4326} = \operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)}$$
no validation is available for declarations 8572852424:
8572852424:
electric field wave equation: from time dependent to time independent declare guess solution
1. 8494839423; locally 4758592:
$$\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$$
$$nabla^{2} pdg_{4326} = \frac{partial pdg_{4326} pdg_{6197} pdg_{7940}}{pdg_{1467}^{2}}$$
1. 9499428242; locally 3994928:
$$E( \vec{r},t) = E( \vec{r})\exp(i \omega t)$$
$$\operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)} = \operatorname{pdg_{2718}}{\left(pdg_{1467} pdg_{2321} pdg_{4621} \right)} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)}$$
no validation is available for declarations 8494839423:
9499428242:
8494839423:
9499428242:
electric field wave equation: from time dependent to time independent substitute LHS of expr 1 into expr 2
1. 8572852424; locally 9393848:
$$\vec{E} = E( \vec{r},t)$$
$$pdg_{4326} = \operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)}$$
2. 8494839423; locally 4758592:
$$\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$$
$$nabla^{2} pdg_{4326} = \frac{partial pdg_{4326} pdg_{6197} pdg_{7940}}{pdg_{1467}^{2}}$$
1. 9394939493; locally 3839493:
$$\nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t)$$
$$nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)} = \frac{partial pdg_{6197} pdg_{7940} \operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)}}{pdg_{1467}^{2}}$$
valid 8572852424:
8494839423:
9394939493:
8572852424:
8494839423:
9394939493:
electric field wave equation: from time dependent to time independent differentiate with respect to
1. 4985825552; locally 2939392:
$$\nabla^2 E( \vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t)$$
$$nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}} = pdg_{2321} pdg_{4621} pdg_{6197} pdg_{7940} \frac{\partial}{\partial pdg_{1467}} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}}$$
1. 0003232242:
$$t$$
$$pdg_{1467}$$
1. 1858578388; locally 4958573:
$$\nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t)$$
$$nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}} = - pdg_{2321}^{2} pdg_{6197} pdg_{7940} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}}$$
no check performed 4985825552:
1858578388:
4985825552:
1858578388:
electric field wave equation: from time dependent to time independent simplify
1. 9485384858; locally 9495903:
$$\nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t)$$
$$nabla^{2} \operatorname{pdg_{2718}}{\left(pdg_{1467} pdg_{2321} pdg_{4621} \right)} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} = - \frac{pdg_{2321}^{2} \operatorname{pdg_{2718}}{\left(pdg_{1467} pdg_{2321} pdg_{4621} \right)} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)}}{pdg_{4567}^{2}}$$
1. 3485475729; locally 3949492:
$$\nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r})$$
$$nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} = - \frac{pdg_{2321}^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)}}{pdg_{4567}^{2}}$$
LHS diff is nabla**2*(pdg2718(pdg1467*pdg2321*pdg4621) - 1)*pdg6238(pdg9472) RHS diff is pdg2321**2*(1 - pdg2718(pdg1467*pdg2321*pdg4621))*pdg6238(pdg9472)/pdg4567**2 9485384858:
3485475729:
9485384858:
3485475729:
electric field wave equation: from time dependent to time independent declare final expr
1. 3485475729; locally 3949492:
$$\nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r})$$
$$nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} = - \frac{pdg_{2321}^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)}}{pdg_{4567}^{2}}$$
no validation is available for declarations 3485475729:
3485475729:
electric field wave equation: from time dependent to time independent substitute LHS of expr 1 into expr 2
1. 1858578388; locally 4958573:
$$\nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t)$$
$$nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}} = - pdg_{2321}^{2} pdg_{6197} pdg_{7940} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}}$$
2. 4585828572; locally 4949582:
$$\epsilon_0 \mu_0 = \frac{1}{c^2}$$
$$pdg_{6197} pdg_{7940} = \frac{1}{pdg_{4567}^{2}}$$
1. 9485384858; locally 9495903:
$$\nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t)$$
$$nabla^{2} \operatorname{pdg_{2718}}{\left(pdg_{1467} pdg_{2321} pdg_{4621} \right)} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} = - \frac{pdg_{2321}^{2} \operatorname{pdg_{2718}}{\left(pdg_{1467} pdg_{2321} pdg_{4621} \right)} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)}}{pdg_{4567}^{2}}$$
LHS diff is -nabla**2*pdg2718(pdg1467*pdg2321*pdg4621)*pdg6238(pdg9472) + pdg6197*pdg7940 RHS diff is (pdg2321**2*pdg2718(pdg1467*pdg2321*pdg4621)*pdg6238(pdg9472) + 1)/pdg4567**2 1858578388:
4585828572:
9485384858:
1858578388:
4585828572:
9485384858:
electric field wave equation: from time dependent to time independent declare initial expr
1. 8494839423; locally 4758592:
$$\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$$
$$nabla^{2} pdg_{4326} = \frac{partial pdg_{4326} pdg_{6197} pdg_{7940}}{pdg_{1467}^{2}}$$
no validation is available for declarations 8494839423:
8494839423:
electric field wave equation: from time dependent to time independent declare initial expr
1. 4585828572; locally 4949582:
$$\epsilon_0 \mu_0 = \frac{1}{c^2}$$
$$pdg_{6197} pdg_{7940} = \frac{1}{pdg_{4567}^{2}}$$
no validation is available for declarations 4585828572:
4585828572:
frequency relations declare initial expr
1. 5900595848; locally 3293094:
$$k = \frac{\omega}{v}$$
$$pdg_{5321} = \frac{pdg_{2321}}{pdg_{1357}}$$
no validation is available for declarations 5900595848:
5900595848:
frequency relations substitute RHS of expr 1 into expr 2
1. 3131111133; locally 8482459:
$$T = 1 / f$$
$$pdg_{9491} = \frac{1}{pdg_{4201}}$$
2. 0404050504; locally 3294004:
$$\lambda = \frac{v}{f}$$
$$pdg_{1115} = \frac{pdg_{1357}}{pdg_{4201}}$$
1. 1293923844; locally 3993940:
$$\lambda = v T$$
$$pdg_{1115} = pdg_{1357} pdg_{9491}$$
valid 3131111133:
0404050504:
1293923844:
3131111133:
0404050504:
1293923844:
frequency relations declare initial expr
1. 0404050504; locally 3294004:
$$\lambda = \frac{v}{f}$$
$$pdg_{1115} = \frac{pdg_{1357}}{pdg_{4201}}$$
no validation is available for declarations 0404050504:
0404050504:
frequency relations substitute RHS of expr 1 into expr 2
1. 3132131132; locally 8374556:
$$\omega = \frac{2\pi}{T}$$
$$pdg_{2321} = \frac{2 pdg_{3141}}{pdg_{9491}}$$
2. 5900595848; locally 3293094:
$$k = \frac{\omega}{v}$$
$$pdg_{5321} = \frac{pdg_{2321}}{pdg_{1357}}$$
1. 0934990943; locally 8394853:
$$k = \frac{2 \pi}{v T}$$
$$pdg_{5321} = \frac{2 pdg_{3141}}{pdg_{1357} pdg_{9491}}$$
LHS diff is 0 RHS diff is (pdg2321*pdg9491 - 2*pdg3141)/(pdg1357*pdg9491) 3132131132:
5900595848:
0934990943:
3132131132:
5900595848:
0934990943:
frequency relations multiply both sides by
1. 3131111133; locally 8482459:
$$T = 1 / f$$
$$pdg_{9491} = \frac{1}{pdg_{4201}}$$
1. 0005749291:
$$f$$
$$pdg_{6235}$$
1. 2131616531; locally 8341200:
$$T f = 1$$
$$pdg_{4201} pdg_{9491} = 1$$
LHS diff is pdg9491*(-pdg4201 + pdg6235) RHS diff is (-pdg4201 + pdg6235)/pdg4201 3131111133:
2131616531:
3131111133:
2131616531:
frequency relations declare initial expr
1. 3131211131; locally 9214650:
$$\omega = 2 \pi f$$
$$pdg_{2321} = 2 pdg_{3141} pdg_{4201}$$
no validation is available for declarations 3131211131:
3131211131:
frequency relations declare final expr
1. 3121513111; locally 2934848:
$$k = \frac{2 \pi}{\lambda}$$
$$pdg_{5321} = \frac{2 pdg_{3141}}{pdg_{1115}}$$
no validation is available for declarations 3121513111:
3121513111:
frequency relations substitute RHS of expr 1 into expr 2
1. 1293923844; locally 3993940:
$$\lambda = v T$$
$$pdg_{1115} = pdg_{1357} pdg_{9491}$$
2. 0934990943; locally 8394853:
$$k = \frac{2 \pi}{v T}$$
$$pdg_{5321} = \frac{2 pdg_{3141}}{pdg_{1357} pdg_{9491}}$$
1. 3121513111; locally 2934848:
$$k = \frac{2 \pi}{\lambda}$$
$$pdg_{5321} = \frac{2 pdg_{3141}}{pdg_{1115}}$$
valid 1293923844:
0934990943:
3121513111:
1293923844:
0934990943:
3121513111:
frequency relations declare initial expr
1. 3131111133; locally 8482459:
$$T = 1 / f$$
$$pdg_{9491} = \frac{1}{pdg_{4201}}$$
no validation is available for declarations 3131111133:
3131111133:
frequency relations divide both sides by
1. 2131616531; locally 8341200:
$$T f = 1$$
$$pdg_{4201} pdg_{9491} = 1$$
1. 0008837284:
$$T$$
$$pdg_{9491}$$
1. 2113211456; locally 9380032:
$$f = 1/T$$
$$pdg_{4201} = \frac{1}{pdg_{9491}}$$
valid 2131616531:
2113211456:
2131616531:
2113211456:
frequency relations substitute RHS of expr 1 into expr 2
1. 2113211456; locally 9380032:
$$f = 1/T$$
$$pdg_{4201} = \frac{1}{pdg_{9491}}$$
2. 3131211131; locally 9214650:
$$\omega = 2 \pi f$$
$$pdg_{2321} = 2 pdg_{3141} pdg_{4201}$$
1. 3132131132; locally 8374556:
$$\omega = \frac{2\pi}{T}$$
$$pdg_{2321} = \frac{2 pdg_{3141}}{pdg_{9491}}$$
LHS diff is 0 RHS diff is 2*pdg3141*(pdg4201*pdg9491 - 1)/pdg9491 2113211456:
3131211131:
3132131132:
2113211456:
3131211131:
3132131132:
integration by parts declare final expr
1. 8489593964; locally 3848329:
$$\int u dv = u v - \int v du$$
$$\int pdg_{4221}\, dpdg_{5177} = pdg_{4221} pdg_{5177} - \int pdg_{5177}\, dpdg_{4221}$$
no validation is available for declarations 8489593964:
8489593964:
integration by parts subtract X from both sides
1. 8489593958; locally 3494854:
$$d(u v) = u dv + v du$$
$$pdg_{4221}$$
1. 0009492929:
$$v du$$
$$pdg_{4221} pdg_{5177}$$
1. 8489593960; locally 2938188:
$$d(u v) - v du = u dv$$
$$pdg_{4221}$$
Nothing to split 8489593958:
8489593960:
8489593958:
8489593960:
integration by parts swap LHS with RHS
1. 8489593960; locally 2938188:
$$d(u v) - v du = u dv$$
$$pdg_{4221}$$
1. 8489593962; locally 2938190:
$$u dv = d(u v) - v du$$
$$pdg_{4221}$$
Nothing to split 8489593960:
8489593962:
8489593960:
8489593962:
integration by parts declare identity
1. 8489593958; locally 3494854:
$$d(u v) = u dv + v du$$
$$pdg_{4221}$$
no validation is available for declarations 8489593958:
8489593958:
integration by parts indefinite integration
1. 8489593962; locally 2938190:
$$u dv = d(u v) - v du$$
$$pdg_{4221}$$
1. 8489593964; locally 3848329:
$$\int u dv = u v - \int v du$$
$$\int pdg_{4221}\, dpdg_{5177} = pdg_{4221} pdg_{5177} - \int pdg_{5177}\, dpdg_{4221}$$
Nothing to split 8489593962:
8489593964:
8489593962:
8489593964:
particle in a 1D box change variable X to Y
1. 9988949211; locally 1231131:
$$(\sin(x))^2 = \frac{1 - \cos(2 x)}{2}$$
$$\sin^{2}{\left(pdg_{1464} \right)} = \frac{1}{2} - \frac{\cos{\left(2 pdg_{1464} \right)}}{2}$$
1. 0009484724:
$$\frac{n \pi}{W}x$$
$$\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}}$$
2. 0004934845:
$$x$$
$$pdg_{1464}$$
1. 7575738420; locally 0100404:
$$\left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}$$
$$\sin^{2}{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)} = \frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}$$
LHS diff is sin(pdg1464)**2 - sin(pdg1592*pdg3141*pdg4037/pdg2523)**2 RHS diff is -cos(2*pdg1464)/2 + cos(2*pdg1464*pdg1592*pdg3141/pdg2523)/2 9988949211:
7575738420:
9988949211:
7575738420:
particle in a 1D box square root both sides
1. 8485867742; locally 1029384:
$$\frac{2}{W} = a^2$$
$$\frac{2}{pdg_{2523}} = pdg_{9139}^{2}$$
1. 9485747245; locally 9394857:
$$\sqrt{\frac{2}{W}} = a$$
$$\sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} = pdg_{9139}$$
2. 9485747246; locally 9394858:
$$-\sqrt{\frac{2}{W}} = a$$
$$- \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} = pdg_{9139}$$
no check performed 8485867742:
9485747245:
9485747246:
8485867742:
9485747245:
9485747246:
particle in a 1D box declare guess solution
1. 5727578862; locally 7572748:
$$\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)$$
$$pdg_{9199}$$
1. 8582885111; locally 7572118:
$$\psi(x) = a \sin(kx) + b \cos(kx)$$
$$\operatorname{pdg_{9489}}{\left(pdg_{4037} \right)} = pdg_{1939} \cos{\left(pdg_{4037} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{4037} pdg_{5321} \right)}$$
no validation is available for declarations 5727578862:
8582885111:
5727578862:
8582885111:
particle in a 1D box substitute LHS of expr 1 into expr 2
1. 9485747245; locally 9394857:
$$\sqrt{\frac{2}{W}} = a$$
$$\sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} = pdg_{9139}$$
2. 2944838499; locally 3452131:
$$\psi(x) = a \sin(\frac{n \pi}{W} x)$$
$$\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}$$
1. 9393939991; locally 8474766:
$$\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)$$
$$\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = - \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}$$
LHS diff is 0 RHS diff is pdg9139*sin(pdg1592*pdg3141*pdg4037/pdg2523) + sqrt(2)*sqrt(1/pdg2523)*sin(pdg1464*pdg1592*pdg3141/pdg2523) 9485747245:
2944838499:
9393939991:
9485747245:
2944838499:
9393939991:
particle in a 1D box change variable X to Y
1. 5857434758; locally 0021030:
$$\int a dx = a x$$
$$\int pdg_{9139}\, dpdg_{1464} = pdg_{1464} pdg_{9139}$$
1. 0002929944:
$$1/2$$
$$\frac{1}{2}$$
2. 0004948585:
$$a$$
$$pdg_{9139}$$
1. 8575746378; locally 9339495:
$$\int \frac{1}{2} dx = \frac{1}{2} x$$
$$\int \frac{1}{2}\, dpdg_{1464} = \frac{pdg_{1464}}{2}$$
LHS diff is pdg1464*(pdg9139 - 1/2) RHS diff is pdg1464*(pdg9139 - 1/2) 5857434758:
8575746378:
5857434758:
8575746378:
particle in a 1D box simplify
1. 8575748999; locally 2838288:
$$\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right) = -k^2 \left(a \sin(kx) + b \cos(kx) \right)$$
$$\frac{d^{2} \left(pdg_{1939} \cos{\left(pdg_{1464} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{1464} pdg_{5321} \right)}\right)}{pdg_{9199}^{2}} = - pdg_{5321}^{2} \left(pdg_{1939} \cos{\left(pdg_{1464} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{1464} pdg_{5321} \right)}\right)$$
1. 8485757728; locally 8474762:
$$a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)$$
$$pdg_{9199}$$
Nothing to split 8575748999:
8485757728:
8575748999:
8485757728:
particle in a 1D box declare identity
1. 5857434758; locally 0021030:
$$\int a dx = a x$$
$$\int pdg_{9139}\, dpdg_{1464} = pdg_{1464} pdg_{9139}$$
no validation is available for declarations 5857434758:
5857434758:
particle in a 1D box declare identity
1. 0948572140; locally 3992939:
$$\int \cos(a x) dx = \frac{1}{a}\sin(a x)$$
$$\int \cos{\left(pdg_{1464} pdg_{9139} \right)}\, dpdg_{9199} = \frac{\sin{\left(pdg_{1464} pdg_{9139} \right)}}{pdg_{9139}}$$
no validation is available for declarations 0948572140:
0948572140:
particle in a 1D box substitute LHS of expr 1 into expr 2
1. 8575746378; locally 9339495:
$$\int \frac{1}{2} dx = \frac{1}{2} x$$
$$\int \frac{1}{2}\, dpdg_{1464} = \frac{pdg_{1464}}{2}$$
2. 1202310110; locally 0203020:
$$\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx$$
$$\frac{1}{pdg_{9139}^{2}} = \int\limits_{0}^{pdg_{2523}} \left(\frac{pdg_{9199}}{2} - \frac{\int\limits_{0}^{pdg_{2523}} \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037}}{2}\right)\, dpdg_{4037}$$
1. 1202312210; locally 8584733:
$$\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx$$
$$\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2} - \frac{\int\limits_{0}^{pdg_{2523}} \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037}}{2}$$
LHS diff is 0 RHS diff is Piecewise((pdg2523*(2*pdg1592*pdg3141*pdg9199 - 2*pdg1592*pdg3141 - pdg2523*sin(2*pdg1592*pdg3141) + sin(2*pdg1592*pdg3141))/(4*pdg1592*pdg3141), Ne(2*pdg1592*pdg3141/pdg2523, 0)), (pdg2523*(-pdg2523 + pdg9199)/2, True)) 8575746378:
1202310110:
1202312210:
8575746378:
1202310110:
1202312210:
particle in a 1D box substitute LHS of expr 1 into expr 2
1. 9485747246; locally 9394858:
$$-\sqrt{\frac{2}{W}} = a$$
$$- \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} = pdg_{9139}$$
2. 2944838499; locally 3452131:
$$\psi(x) = a \sin(\frac{n \pi}{W} x)$$
$$\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}$$
1. 9393939992; locally 8474765:
$$\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)$$
$$\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}$$
LHS diff is 0 RHS diff is pdg9139*sin(pdg1592*pdg3141*pdg4037/pdg2523) - sqrt(2)*sqrt(1/pdg2523)*sin(pdg1464*pdg1592*pdg3141/pdg2523) 9485747246:
2944838499:
9393939992:
9485747246:
2944838499:
9393939992:
particle in a 1D box expand integrand
1. 9858028950; locally 0495054:
$$\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx$$
$$\frac{1}{pdg_{9139}^{2}} = \int\limits_{0}^{pdg_{2523}} \left(\frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}\right)\, dpdg_{1464}$$
1. 1202310110; locally 0203020:
$$\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx$$
$$\frac{1}{pdg_{9139}^{2}} = \int\limits_{0}^{pdg_{2523}} \left(\frac{pdg_{9199}}{2} - \frac{\int\limits_{0}^{pdg_{2523}} \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037}}{2}\right)\, dpdg_{4037}$$
no check performed 9858028950:
1202310110:
9858028950:
1202310110:
particle in a 1D box normalization condition
1. 1934748140; locally 7575626:
$$\int |\psi(x)|^2 dx = 1$$
$$\int \left|{\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)}}\right|^{2}\, dpdg_{9199} = 1$$
no validation is available for assumptions 1934748140:
1934748140:
particle in a 1D box substitute LHS of expr 1 into expr 2
1. 2944838499; locally 3452131:
$$\psi(x) = a \sin(\frac{n \pi}{W} x)$$
$$\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}$$
2. 4857472413; locally 0595847:
$$1 = \int \psi(x)\psi(x)^* dx$$
$$pdg_{9199}$$
1. 0203024440; locally 0495950:
$$1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx$$
$$1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)} \overline{\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)}}\, dpdg_{1464}$$
Nothing to split 2944838499:
4857472413:
0203024440:
2944838499:
4857472413:
0203024440:
particle in a 1D box change variable X to Y
1. 0948572140; locally 3992939:
$$\int \cos(a x) dx = \frac{1}{a}\sin(a x)$$
$$\int \cos{\left(pdg_{1464} pdg_{9139} \right)}\, dpdg_{9199} = \frac{\sin{\left(pdg_{1464} pdg_{9139} \right)}}{pdg_{9139}}$$
1. 0009485858:
$$\frac{2n\pi}{W}$$
$$\frac{2 pdg_{1592} pdg_{3141}}{pdg_{2523}}$$
2. 0004831494:
$$a$$
$$pdg_{9139}$$
1. 7564894985; locally 4948377:
$$\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)$$
$$\int \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037} = \frac{pdg_{2523} \sin{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}}{2 pdg_{1592} pdg_{3141}}$$
LHS diff is pdg9199*cos(pdg1464*pdg9139) - Piecewise((pdg2523*sin(2*pdg1592*pdg3141*pdg4037/pdg2523)/(2*pdg1592*pdg3141), Ne(2*pdg1592*pdg3141/pdg2523, 0)), (pdg4037, True)) RHS diff is sin(pdg1464*pdg9139)/pdg9139 - pdg2523*sin(2*pdg1592*pdg3141*pdg4037/pdg2523)/(2*pdg1592*pdg3141) 0948572140:
7564894985:
0948572140:
7564894985:
particle in a 1D box swap LHS with RHS
1. 1934748140; locally 7575626:
$$\int |\psi(x)|^2 dx = 1$$
$$\int \left|{\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)}}\right|^{2}\, dpdg_{9199} = 1$$
1. 8572657110; locally 5577567:
$$1 = \int |\psi(x)|^2 dx$$
$$1 = \int \left|{\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)}}\right|^{2}\, dpdg_{1464}$$
LHS diff is pdg9199*Abs(pdg9489(pdg1464))**2 - Integral(Abs(pdg9489(pdg1464))**2, pdg1464) RHS diff is pdg9199*Abs(pdg9489(pdg1464))**2 - Integral(Abs(pdg9489(pdg1464))**2, pdg1464) 1934748140:
8572657110:
1934748140:
8572657110:
particle in a 1D box substitute RHS of expr 1 into expr 2
1. 5727578862; locally 7572748:
$$\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)$$
$$pdg_{9199}$$
2. 8582885111; locally 7572118:
$$\psi(x) = a \sin(kx) + b \cos(kx)$$
$$\operatorname{pdg_{9489}}{\left(pdg_{4037} \right)} = pdg_{1939} \cos{\left(pdg_{4037} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{4037} pdg_{5321} \right)}$$
1. 8575748999; locally 2838288:
$$\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right) = -k^2 \left(a \sin(kx) + b \cos(kx) \right)$$
$$\frac{d^{2} \left(pdg_{1939} \cos{\left(pdg_{1464} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{1464} pdg_{5321} \right)}\right)}{pdg_{9199}^{2}} = - pdg_{5321}^{2} \left(pdg_{1939} \cos{\left(pdg_{1464} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{1464} pdg_{5321} \right)}\right)$$
Nothing to split 5727578862:
8582885111:
8575748999:
5727578862:
8582885111:
8575748999:
particle in a 1D box substitute LHS of expr 1 into expr 2
1. 8849289982; locally 3452132:
$$\psi(x)^* = a \sin(\frac{n \pi}{W} x)$$
$$\overline{\operatorname{pdg_{9489}}{\left(pdg_{4037} \right)}} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}$$
2. 0203024440; locally 0495950:
$$1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx$$
$$1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)} \overline{\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)}}\, dpdg_{1464}$$
1. 8889444440; locally 8478550:
$$1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx$$
$$1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139}^{2} \sin^{2}{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}\, dpdg_{1464}$$
LHS diff is 0 RHS diff is pdg9139*(-pdg9139*Piecewise((pdg2523*(pdg1592*pdg3141/2 - sin(pdg1592*pdg3141)*cos(pdg1592*pdg3141)/2)/(pdg1592*pdg3141), Ne(pdg1592*pdg3141/pdg2523, 0)), (0, True)) + Integral(sin(pdg1464*pdg1592*pdg3141/pdg2523)*conjugate(pdg9489(pdg1464)), (pdg1464, 0, pdg2523))) 8849289982:
0203024440:
8889444440:
8849289982:
0203024440:
8889444440:
particle in a 1D box conjugate function X
1. 2944838499; locally 3452131:
$$\psi(x) = a \sin(\frac{n \pi}{W} x)$$
$$\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}$$
1. 0009587738:
$$\psi$$
$$pdg_{9489}$$
1. 8849289982; locally 3452132:
$$\psi(x)^* = a \sin(\frac{n \pi}{W} x)$$
$$\overline{\operatorname{pdg_{9489}}{\left(pdg_{4037} \right)}} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}$$
no check performed 2944838499:
8849289982:
2944838499:
8849289982:
particle in a 1D box multiply both sides by
1. 4857475848; locally 9493949:
$$\frac{1}{a^2} = \frac{W}{2}$$
$$\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2}$$
1. 0009485857:
$$a^2\frac{2}{W}$$
$$\frac{2 pdg_{9139}^{2}}{pdg_{2523}}$$
1. 8485867742; locally 1029384:
$$\frac{2}{W} = a^2$$
$$\frac{2}{pdg_{2523}} = pdg_{9139}^{2}$$
valid 4857475848:
8485867742:
4857475848:
8485867742:
particle in a 1D box divide both sides by
1. 8576785890; locally 9485800:
$$1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx$$
$$1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139}^{2} \left(\frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}\right)\, dpdg_{1464}$$
1. 0000040490:
$$a^2$$
$$pdg_{9139}^{2}$$
1. 9858028950; locally 0495054:
$$\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx$$
$$\frac{1}{pdg_{9139}^{2}} = \int\limits_{0}^{pdg_{2523}} \left(\frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}\right)\, dpdg_{1464}$$
valid 8576785890:
9858028950:
8576785890:
9858028950:
particle in a 1D box claim LHS equals RHS
1. 8484544728; locally 1214762:
$$-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)$$
$$pdg_{4037}$$
Nothing to split 8484544728:
8484544728:
particle in a 1D box expand magnitude to conjugate
1. 8572657110; locally 5577567:
$$1 = \int |\psi(x)|^2 dx$$
$$1 = \int \left|{\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)}}\right|^{2}\, dpdg_{1464}$$
1. 0009458842:
$$\psi(x)$$
$$\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)}$$
1. 4857472413; locally 0595847:
$$1 = \int \psi(x)\psi(x)^* dx$$
$$pdg_{9199}$$
Nothing to split 8572657110:
4857472413:
8572657110:
4857472413:
particle in a 1D box declare final expr
1. 9393939992; locally 8474765:
$$\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)$$
$$\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}$$
no validation is available for declarations 9393939992:
9393939992:
particle in a 1D box simplify
1. 0439492440; locally 0405049:
$$\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W$$
$$\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2} - \frac{pdg_{2523} \sin{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}}{4 pdg_{1592} pdg_{3141}}$$
1. 4857475848; locally 9493949:
$$\frac{1}{a^2} = \frac{W}{2}$$
$$\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2}$$
LHS diff is 0 RHS diff is -pdg2523*sin(2*pdg1592*pdg3141*pdg4037/pdg2523)/(4*pdg1592*pdg3141) 0439492440:
4857475848:
0439492440:
4857475848:
particle in a 1D box declare identity
1. 9988949211; locally 1231131:
$$(\sin(x))^2 = \frac{1 - \cos(2 x)}{2}$$
$$\sin^{2}{\left(pdg_{1464} \right)} = \frac{1}{2} - \frac{\cos{\left(2 pdg_{1464} \right)}}{2}$$
no validation is available for declarations 9988949211:
9988949211:
particle in a 1D box substitute RHS of expr 1 into expr 2
1. 7564894985; locally 4948377:
$$\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)$$
$$\int \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037} = \frac{pdg_{2523} \sin{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}}{2 pdg_{1592} pdg_{3141}}$$
2. 1202312210; locally 8584733:
$$\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx$$
$$\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2} - \frac{\int\limits_{0}^{pdg_{2523}} \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037}}{2}$$
1. 0439492440; locally 0405049:
$$\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W$$
$$\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2} - \frac{pdg_{2523} \sin{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}}{4 pdg_{1592} pdg_{3141}}$$
LHS diff is 0 RHS diff is -Piecewise((pdg2523*sin(2*pdg1592*pdg3141)/(2*pdg1592*pdg3141), Ne(2*pdg1592*pdg3141/pdg2523, 0)), (pdg2523, True))/2 + pdg2523*sin(2*pdg1592*pdg3141*pdg4037/pdg2523)/(4*pdg1592*pdg3141) 7564894985:
1202312210:
0439492440:
7564894985:
1202312210:
0439492440:
particle in a 1D box LHS of expr 1 equals LHS of expr 2
1. 9585727710; locally 8577781:
$$\psi(x=0) = 0$$
$$\operatorname{pdg_{9489}}{\left(pdg_{1464} = 0 \right)} = 0$$
2. 8582885111; locally 7572118:
$$\psi(x) = a \sin(kx) + b \cos(kx)$$
$$\operatorname{pdg_{9489}}{\left(pdg_{4037} \right)} = pdg_{1939} \cos{\left(pdg_{4037} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{4037} pdg_{5321} \right)}$$
1. 8577275751; locally 7547581:
$$0 = a \sin(0) + b\cos(0)$$
$$0 = pdg_{1939}$$
input diff is -pdg9489(pdg4037) + pdg9489(Eq(pdg1464, 0)) diff is 0 diff is -pdg1939*cos(pdg4037*pdg5321) + pdg1939 - pdg9139*sin(pdg4037*pdg5321) 9585727710:
8582885111:
8577275751:
9585727710:
8582885111:
8577275751:
particle in a 1D box substitute RHS of expr 1 into expr 2
1. 1293913110; locally 7572859:
$$0 = b$$
$$0 = pdg_{1939}$$
2. 8582885111; locally 7572118:
$$\psi(x) = a \sin(kx) + b \cos(kx)$$
$$\operatorname{pdg_{9489}}{\left(pdg_{4037} \right)} = pdg_{1939} \cos{\left(pdg_{4037} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{4037} pdg_{5321} \right)}$$
1. 9059289981; locally 7562671:
$$\psi(x) = a \sin(k x)$$
$$pdg_{1464}$$
Nothing to split 1293913110:
8582885111:
9059289981:
1293913110:
8582885111:
9059289981:
particle in a 1D box boundary condition for expr
1. 5727578862; locally 7572748:
$$\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)$$
$$pdg_{9199}$$
1. 9585727710; locally 8577781:
$$\psi(x=0) = 0$$
$$\operatorname{pdg_{9489}}{\left(pdg_{1464} = 0 \right)} = 0$$
no validation is available for assumptions 5727578862:
9585727710:
5727578862:
9585727710:
particle in a 1D box boundary condition for expr
1. 5727578862; locally 7572748:
$$\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)$$
$$pdg_{9199}$$
1. 9495857278; locally 8585727:
$$\psi(x=W) = 0$$
$$pdg_{2523}$$
no validation is available for assumptions 5727578862:
9495857278:
5727578862:
9495857278:
particle in a 1D box expr 1 is true under condition expr 2
1. 1020010291; locally 8577672:
$$0 = a \sin(k W)$$
$$0 = pdg_{9139} \sin{\left(pdg_{2523} pdg_{5321} \right)}$$
2. 1857710291; locally 8577711:
$$0 = a \sin(n \pi)$$
$$0 = pdg_{9139} \sin{\left(pdg_{1592} pdg_{3141} \right)}$$
1. 1010923823; locally 9847600:
$$k W = n \pi$$
$$pdg_{2523} pdg_{5321} = pdg_{1592} pdg_{3141}$$
no check performed 1020010291:
1857710291:
1010923823:
1020010291:
1857710291:
1010923823:
particle in a 1D box declare identity
1. 1857710291; locally 8577711:
$$0 = a \sin(n \pi)$$
$$0 = pdg_{9139} \sin{\left(pdg_{1592} pdg_{3141} \right)}$$
no validation is available for declarations 1857710291:
1857710291:
particle in a 1D box substitute RHS of expr 1 into expr 2
1. 1858772113; locally 9495882:
$$k = \frac{n \pi}{W}$$
$$pdg_{5321} = \frac{pdg_{1592} pdg_{3141}}{pdg_{2523}}$$
2. 9059289981; locally 7562671:
$$\psi(x) = a \sin(k x)$$
$$pdg_{1464}$$
1. 2944838499; locally 3452131:
$$\psi(x) = a \sin(\frac{n \pi}{W} x)$$
$$\operatorname{pdg_{9489}}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}$$
Nothing to split 1858772113:
9059289981:
2944838499:
1858772113:
9059289981:
2944838499:
particle in a 1D box simplify
1. 8577275751; locally 7547581:
$$0 = a \sin(0) + b\cos(0)$$
$$0 = pdg_{1939}$$
1. 1293913110; locally 7572859:
$$0 = b$$
$$0 = pdg_{1939}$$
valid 8577275751:
1293913110:
8577275751:
1293913110:
particle in a 1D box declare initial expr
1. 5727578862; locally 7572748:
$$\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)$$
$$pdg_{9199}$$
no validation is available for declarations 5727578862:
5727578862:
particle in a 1D box substitute RHS of expr 1 into expr 2
1. 7575738420; locally 0100404:
$$\left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}$$
$$\sin^{2}{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)} = \frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}$$
2. 8889444440; locally 8478550:
$$1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx$$
$$1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139}^{2} \sin^{2}{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}\, dpdg_{1464}$$
1. 8576785890; locally 9485800:
$$1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx$$
$$1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139}^{2} \left(\frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}\right)\, dpdg_{1464}$$
valid 7575738420:
8889444440:
8576785890:
7575738420:
8889444440:
8576785890:
particle in a 1D box divide both sides by
1. 1010923823; locally 9847600:
$$k W = n \pi$$
$$pdg_{2523} pdg_{5321} = pdg_{1592} pdg_{3141}$$
1. 0001334112:
$$W$$
$$pdg_{2523}$$
1. 1858772113; locally 9495882:
$$k = \frac{n \pi}{W}$$
$$pdg_{5321} = \frac{pdg_{1592} pdg_{3141}}{pdg_{2523}}$$
valid 1010923823:
1858772113:
1010923823:
1858772113:
particle in a 1D box simplify
1. 8485757728; locally 8474762:
$$a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)$$
$$pdg_{9199}$$
1. 8484544728; locally 1214762:
$$-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)$$
$$pdg_{4037}$$
Nothing to split 8485757728:
8484544728:
8485757728:
8484544728:
particle in a 1D box LHS of expr 1 equals LHS of expr 2
1. 9495857278; locally 8585727:
$$\psi(x=W) = 0$$
$$pdg_{2523}$$
2. 9059289981; locally 7562671:
$$\psi(x) = a \sin(k x)$$
$$pdg_{1464}$$
1. 1020010291; locally 8577672:
$$0 = a \sin(k W)$$
$$0 = pdg_{9139} \sin{\left(pdg_{2523} pdg_{5321} \right)}$$
Nothing to split 9495857278:
9059289981:
1020010291:
9495857278:
9059289981:
1020010291:
quadratic equation derivation subtract X from both sides
1. 5982958249; locally 6608123:
$$x+(b/(2 a)) = -\sqrt{(b/(2 a))^2 - (c/a)}$$
$$pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}} = - \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}$$
1. 0002838490:
$$b/(2 a)$$
$$\frac{pdg_{1939}}{2 pdg_{9139}}$$
1. 9582958293; locally 4433112:
$$x = \sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))$$
$$pdg_{1464} = - \frac{pdg_{1939}}{2 pdg_{9139}} + \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}$$
LHS diff is 0 RHS diff is -sqrt((pdg1939**2 - 4*pdg4231*pdg9139)/pdg9139**2) 5982958249:
9582958293:
5982958249:
9582958293:
quadratic equation derivation subtract X from both sides
1. 9582958294; locally 6608102:
$$x+(b/(2 a)) = \sqrt{(b/(2 a))^2 - (c/a)}$$
$$pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}} = \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}$$
1. 0002449291:
$$b/(2 a)$$
$$\frac{pdg_{1939}}{2 pdg_{9139}}$$
1. 5982958248; locally 2657355:
$$x = -\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))$$
$$pdg_{1464} = - \frac{pdg_{1939}}{2 pdg_{9139}} - \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}$$
LHS diff is 0 RHS diff is sqrt((pdg1939**2 - 4*pdg4231*pdg9139)/pdg9139**2) 9582958294:
5982958248:
9582958294:
5982958248:
1. 5982958248; locally 2657355:
$$x = -\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))$$
$$pdg_{1464} = - \frac{pdg_{1939}}{2 pdg_{9139}} - \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}$$
1. 9999999968; locally 8811221:
$$x = \frac{-b-\sqrt{b^2-4ac}}{2 a}$$
$$pdg_{1464} = \frac{- pdg_{1939} - \sqrt{pdg_{1939}^{2} - 4 pdg_{4231} pdg_{9139}}}{2 pdg_{9139}}$$
LHS diff is 0 RHS diff is (-pdg9139*sqrt((pdg1939**2 - 4*pdg4231*pdg9139)/pdg9139**2) + sqrt(pdg1939**2 - 4*pdg4231*pdg9139))/(2*pdg9139) 5982958248:
9999999968:
5982958248:
9999999968:
1. 9582958293; locally 4433112:
$$x = \sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a))$$
$$pdg_{1464} = - \frac{pdg_{1939}}{2 pdg_{9139}} + \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}$$
1. 9999999969; locally 8761200:
$$x = \frac{-b+\sqrt{b^2-4ac}}{2 a}$$
$$pdg_{1464} = \frac{- pdg_{1939} + \sqrt{pdg_{1939}^{2} - 4 pdg_{4231} pdg_{9139}}}{2 pdg_{9139}}$$
LHS diff is 0 RHS diff is (pdg9139*sqrt((pdg1939**2 - 4*pdg4231*pdg9139)/pdg9139**2) - sqrt(pdg1939**2 - 4*pdg4231*pdg9139))/(2*pdg9139) 9582958293:
9999999969:
9582958293:
9999999969:
1. 5938459282; locally 1212129:
$$x^2 + (b/a)x = -c/a$$
$$pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} = - \frac{pdg_{4231}}{pdg_{9139}}$$
1. 0004307451:
$$(b/(2 a))^2$$
$$\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}}$$
1. 5928292841; locally 1120000:
$$x^2 + (b/a)x + (b/(2 a))^2 = -c/a + (b/(2 a))^2$$
$$pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} + \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} = \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}$$
valid 5938459282:
5928292841:
5938459282:
5928292841:
quadratic equation derivation subtract X from both sides
1. 5958392859; locally 7777621:
$$x^2 + (b/a)x+(c/a) = 0$$
$$pdg_{1464}^{2} + pdg_{1464} + \frac{pdg_{4231}}{pdg_{9139}} = 0$$
1. 0006644853:
$$c/a$$
$$\frac{pdg_{4231}}{pdg_{9139}}$$
1. 5938459282; locally 1212129:
$$x^2 + (b/a)x = -c/a$$
$$pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} = - \frac{pdg_{4231}}{pdg_{9139}}$$
LHS diff is pdg1464*(-pdg1939 + pdg9139)/pdg9139 RHS diff is 0 5958392859:
5938459282:
5958392859:
5938459282:
quadratic equation derivation square root both sides
1. 9385938295; locally 2985412:
$$(x+(b/(2 a)))^2 = -(c/a) + (b/(2 a))^2$$
$$\left(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}}\right)^{2} = \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}$$
1. 5982958249; locally 6608123:
$$x+(b/(2 a)) = -\sqrt{(b/(2 a))^2 - (c/a)}$$
$$pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}} = - \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}$$
2. 9582958294; locally 6608102:
$$x+(b/(2 a)) = \sqrt{(b/(2 a))^2 - (c/a)}$$
$$pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}} = \sqrt{\frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}}$$
no check performed 9385938295:
5982958249:
9582958294:
9385938295:
5982958249:
9582958294:
quadratic equation derivation LHS of expr 1 equals LHS of expr 2
1. 5928292841; locally 1120000:
$$x^2 + (b/a)x + (b/(2 a))^2 = -c/a + (b/(2 a))^2$$
$$pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} + \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} = \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}$$
2. 5959282914; locally 1734000:
$$x^2 + x(b/a) + (b/(2 a))^2 = (x+(b/(2 a)))^2$$
$$pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} + \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} = \left(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}}\right)^{2}$$
1. 9385938295; locally 2985412:
$$(x+(b/(2 a)))^2 = -(c/a) + (b/(2 a))^2$$
$$\left(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}}\right)^{2} = \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} - \frac{pdg_{4231}}{pdg_{9139}}$$
input diff is 0 diff is (pdg1464**2*pdg9139 + pdg1464*pdg1939 + pdg4231)/pdg9139 diff is -(pdg1464**2*pdg9139 + pdg1464*pdg1939 + pdg4231)/pdg9139 5928292841:
5959282914:
9385938295:
5928292841:
5959282914:
9385938295:
1. 5928285821; locally 1239010:
$$x^2 + 2 x (b/(2 a)) + (b/(2 a))^2 = (x + (b/(2 a)))^2$$
$$pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} + \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} = \left(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}}\right)^{2}$$
1. 5959282914; locally 1734000:
$$x^2 + x(b/a) + (b/(2 a))^2 = (x+(b/(2 a)))^2$$
$$pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} + \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} = \left(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}}\right)^{2}$$
valid 5928285821:
5959282914:
5928285821:
5959282914:
quadratic equation derivation change variable X to Y
1. 8582954722; locally 9091270:
$$x^2 + 2 x h + h^2 = (x + h)^2$$
$$pdg_{1464}^{2} + 2 pdg_{1464} pdg_{3410} + pdg_{3410}^{2} = \left(pdg_{1464} + pdg_{3410}\right)^{2}$$
1. 0004858592:
$$h$$
$$pdg_{3410}$$
2. 0000999900:
$$b/(2 a)$$
$$\frac{pdg_{1939}}{2 pdg_{9139}}$$
1. 5928285821; locally 1239010:
$$x^2 + 2 x (b/(2 a)) + (b/(2 a))^2 = (x + (b/(2 a)))^2$$
$$pdg_{1464}^{2} + \frac{pdg_{1464} pdg_{1939}}{pdg_{9139}} + \frac{pdg_{1939}^{2}}{4 pdg_{9139}^{2}} = \left(pdg_{1464} + \frac{pdg_{1939}}{2 pdg_{9139}}\right)^{2}$$
valid 8582954722:
5928285821:
8582954722:
5928285821:
quadratic equation derivation declare final expr
1. 9999999968; locally 8811221:
$$x = \frac{-b-\sqrt{b^2-4ac}}{2 a}$$
$$pdg_{1464} = \frac{- pdg_{1939} - \sqrt{pdg_{1939}^{2} - 4 pdg_{4231} pdg_{9139}}}{2 pdg_{9139}}$$
no validation is available for declarations 9999999968:
9999999968:
quadratic equation derivation declare final expr
1. 9999999969; locally 8761200:
$$x = \frac{-b+\sqrt{b^2-4ac}}{2 a}$$
$$pdg_{1464} = \frac{- pdg_{1939} + \sqrt{pdg_{1939}^{2} - 4 pdg_{4231} pdg_{9139}}}{2 pdg_{9139}}$$
no validation is available for declarations 9999999969:
9999999969:
quadratic equation derivation divide both sides by
1. 9285928292; locally 8882098:
$$ax^2 + bx + c = 0$$
$$pdg_{1464}^{2} pdg_{9139} + pdg_{1464} pdg_{1939} + pdg_{4231} = 0$$
1. 0002424922:
$$a$$
$$pdg_{9139}$$
1. 5958392859; locally 7777621:
$$x^2 + (b/a)x+(c/a) = 0$$
$$pdg_{1464}^{2} + pdg_{1464} + \frac{pdg_{4231}}{pdg_{9139}} = 0$$
LHS diff is pdg1464*(pdg1939 - pdg9139)/pdg9139 RHS diff is 0 9285928292:
5958392859:
9285928292:
5958392859:
quadratic equation derivation declare initial expr
1. 9285928292; locally 8882098:
$$ax^2 + bx + c = 0$$
$$pdg_{1464}^{2} pdg_{9139} + pdg_{1464} pdg_{1939} + pdg_{4231} = 0$$
no validation is available for declarations 9285928292:
9285928292:
quadratic equation derivation declare initial expr
1. 8582954722; locally 9091270:
$$x^2 + 2 x h + h^2 = (x + h)^2$$
$$pdg_{1464}^{2} + 2 pdg_{1464} pdg_{3410} + pdg_{3410}^{2} = \left(pdg_{1464} + pdg_{3410}\right)^{2}$$
no validation is available for declarations 8582954722:
8582954722:
quantum basics Hermitian operators have realvalued observables declare assumption
1. 9294858532; locally 2484892:
$$\hat{A}^+ = \hat{A}$$

no validation is available for declarations 9294858532:
9294858532:
quantum basics Hermitian operators have realvalued observables distribute conjugate transpose to factors
1. 2394935835; locally 2495954:
$$\left(\langle\psi| \hat{A} |\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+$$

1. 1010393913; locally 2390094:
$$\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*$$

Nothing to split 2394935835:
1010393913:
2394935835:
1010393913:
quantum basics Hermitian operators have realvalued observables substitute RHS of expr 1 into expr 2
1. 9294858532; locally 2484892:
$$\hat{A}^+ = \hat{A}$$

2. 1010393913; locally 2390094:
$$\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*$$

1. 4948934890; locally 2494040:
$$\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle^*$$

failed 9294858532:
1010393913:
4948934890:
9294858532:
1010393913:
4948934890:
quantum basics Hermitian operators have realvalued observables substitute RHS of expr 1 into expr 2
1. 4948934890; locally 2494040:
$$\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle^*$$

2. 9999999975; locally 3402919:
$$\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle$$

1. 2848934890; locally 4930585:
$$\langle a \rangle^* = \langle a \rangle$$

Nothing to split 4948934890:
9999999975:
2848934890:
4948934890:
9999999975:
2848934890:
quantum basics Hermitian operators have realvalued observables declare final expr
1. 2848934890; locally 4930585:
$$\langle a \rangle^* = \langle a \rangle$$

no validation is available for declarations 2848934890:
2848934890:
quantum basics Hermitian operators have realvalued observables declare initial expr
1. 9999999975; locally 3402919:
$$\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle$$

no validation is available for declarations 9999999975:
9999999975:
quantum basics Hermitian operators have realvalued observables conjugate transpose both sides
1. 9999999975; locally 3402919:
$$\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle$$

1. 2394935835; locally 2495954:
$$\left(\langle\psi| \hat{A} |\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+$$

Nothing to split 9999999975:
2394935835:
9999999975:
2394935835:
quantum basics orthogonality apply operator to bra
1. 9596004948; locally 3849595:
$$x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle$$
$$pdg_{1464} = pdg_{5598} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }$$
1. 1395858355; locally 4349300:
$$x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle$$
$$pdg_{1464} = pdg_{2427} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }$$
no check performed 9596004948:
1395858355:
9596004948:
1395858355:
quantum basics orthogonality simplify
1. 1010393944; locally 4940359:
$$x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle$$
$$pdg_{1464} = pdg_{7752} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }$$
1. 2394240499; locally 2409402:
$$x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle$$
$$pdg_{1464} = pdg_{7752} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }$$
valid 1010393944:
2394240499:
1010393944:
2394240499:
quantum basics orthogonality declare final expr
1. 2394935831; locally 3494855:
$$( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0$$
$$\left(- pdg_{2427} + pdg_{7752}\right) {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle } = 0$$
no validation is available for declarations 2394935831:
2394935831:
quantum basics orthogonality declare initial expr
1. 9596004948; locally 3849595:
$$x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle$$
$$pdg_{1464} = pdg_{5598} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }$$
no validation is available for declarations 9596004948:
9596004948:
quantum basics orthogonality apply operator to ket
1. 9596004948; locally 3849595:
$$x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle$$
$$pdg_{1464} = pdg_{5598} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }$$
1. 1010393944; locally 4940359:
$$x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle$$
$$pdg_{1464} = pdg_{7752} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }$$
no check performed 9596004948:
1010393944:
9596004948:
1010393944:
quantum basics orthogonality subtract X from both sides
1. 1203938249; locally 3495045:
$$a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle$$
$$\text{True}$$
1. 0005395034:
$$a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle$$
$$pdg_{2427} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }$$
1. 3924948349; locally 4939583:
$$a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0$$
$$pdg_{7752}$$
failed 1203938249:
3924948349:
1203938249:
3924948349:
quantum basics orthogonality simplify
1. 1395858355; locally 4349300:
$$x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle$$
$$pdg_{1464} = pdg_{2427} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }$$
1. 3943939590; locally 4934893:
$$x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle$$
$$pdg_{2427}$$
Nothing to split 1395858355:
3943939590:
1395858355:
3943939590:
quantum basics orthogonality LHS of expr 1 equals LHS of expr 2
1. 2394240499; locally 2409402:
$$x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle$$
$$pdg_{1464} = pdg_{7752} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }$$
2. 3943939590; locally 4934893:
$$x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle$$
$$pdg_{2427}$$
1. 1203938249; locally 3495045:
$$a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle$$
$$\text{True}$$
Nothing to split 2394240499:
3943939590:
1203938249:
2394240499:
3943939590:
1203938249:
quantum basics orthogonality combine like terms
1. 3924948349; locally 4939583:
$$a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0$$
$$pdg_{7752}$$
1. 2394935831; locally 3494855:
$$( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0$$
$$\left(- pdg_{2427} + pdg_{7752}\right) {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle } = 0$$
Nothing to split 3924948349:
2394935831:
3924948349:
2394935831:
variance relation declare identity
1. 3585845894; locally 3493498:
$$\langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2$$
$$pdg_{1464}$$
no validation is available for declarations 3585845894:
3585845894:
variance relation simplify
1. 3585845894; locally 3493498:
$$\langle \left(x-\langle x \rangle\right)^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2$$
$$pdg_{1464}$$
1. 8399484849; locally 5049530:
$$\langle x^2 - 2 x \langle x \rangle + \langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2$$
$$pdg_{1464}$$
Nothing to split 3585845894:
8399484849:
3585845894:
8399484849:
variance relation simplify
1. 8399484849; locally 5049530:
$$\langle x^2 - 2 x \langle x \rangle + \langle x \rangle^2 \rangle = \langle x^2 \rangle-\langle x \rangle^2$$
$$pdg_{1464}$$
1. 2404934990; locally 6757584:
$$\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2$$
$$pdg_{1464}$$
Nothing to split 8399484849:
2404934990:
8399484849:
2404934990:
variance relation simplify
1. 2404934990; locally 6757584:
$$\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2$$
$$pdg_{1464}$$
1. 4949359835; locally 3294824:
$$\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2$$
$$pdg_{1464}$$
Nothing to split 2404934990:
4949359835:
2404934990:
4949359835:
variance relation simplify
1. 4949359835; locally 3294824:
$$\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2$$
$$pdg_{1464}$$
1. 2494533900; locally 5949484:
$$\langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2$$
$$pdg_{1464}$$
Nothing to split 4949359835:
2494533900:
4949359835:
2494533900:
variance relation claim LHS equals RHS
1. 2494533900; locally 5949484:
$$\langle x^2\rangle -\langle x \rangle^2 = \langle x^2 \rangle-\langle x \rangle^2$$
$$pdg_{1464}$$
Nothing to split 2494533900:
2494533900:
Compton's equation for scattering declare initial expr
1. 8257621077; locally 2840008:
$$\vec{p}_{\rm before} = \vec{p}_{1}$$
$$pdg_{1302} = pdg_{6029}$$
no validation is available for declarations 8257621077:
8257621077:
Compton's equation for scattering substitute LHS of expr 1 into expr 2
1. 3951205425; locally 2491904:
$$\vec{p}_{\rm after} = \vec{p}_{1}$$
$$pdg_{5493} = pdg_{6029}$$
2. 8311458118; locally 1209604:
$$\vec{p}_{\rm after} = \vec{p}_{2}+\vec{p}_{electron}$$
$$pdg_{5493} = pdg_{2097} + pdg_{4299}$$
1. 8139187332; locally 5610925:
$$\vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}$$
$$pdg_{6029} = pdg_{2097} + pdg_{4299}$$
valid 3951205425:
8311458118:
8139187332:
3951205425:
8311458118:
8139187332:
Compton's equation for scattering swap LHS with RHS
1. 5530148480; locally 4068150:
$$\vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}$$
$$- pdg_{2097} + pdg_{6029} = pdg_{4299}$$
1. 7917051060; locally 4200334:
$$\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}$$
$$pdg_{4299} = - pdg_{2097} + pdg_{6029}$$
valid 5530148480:
7917051060:
5530148480:
7917051060:
Compton's equation for scattering declare initial expr
1. 8311458118; locally 1209604:
$$\vec{p}_{\rm after} = \vec{p}_{2}+\vec{p}_{electron}$$
$$pdg_{5493} = pdg_{2097} + pdg_{4299}$$
no validation is available for declarations 8311458118:
8311458118:
Compton's equation for scattering multiply expr 1 by expr 2
1. 7917051060; locally 4200334:
$$\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}$$
$$pdg_{4299} = - pdg_{2097} + pdg_{6029}$$
2. 7917051060; locally 4200334:
$$\vec{p}_{electron} = \vec{p}_{1}-\vec{p}_{2}$$
$$pdg_{4299} = - pdg_{2097} + pdg_{6029}$$
1. 6742123016; locally 4218805:
$$\vec{p}_{electron}\cdot\vec{p}_{electron} = ( \vec{p}_{1}\cdot\vec{p}_{1})+( \vec{p}_{2}\cdot\vec{p}_{2})-2( \vec{p}_{1}\cdot\vec{p}_{2})$$
$$pdg_{4299}$$
Nothing to split 7917051060:
7917051060:
6742123016:
7917051060:
7917051060:
6742123016:
Compton's equation for scattering subtract X from both sides
1. 8139187332; locally 5610925:
$$\vec{p}_{1} = \vec{p}_{2}+\vec{p}_{electron}$$
$$pdg_{6029} = pdg_{2097} + pdg_{4299}$$
1. 0002338514:
$$\vec{p}_{2}$$
$$pdg_{2097}$$
1. 5530148480; locally 4068150:
$$\vec{p}_{1}-\vec{p}_{2} = \vec{p}_{electron}$$
$$- pdg_{2097} + pdg_{6029} = pdg_{4299}$$
valid 8139187332:
5530148480:
8139187332:
5530148480:
Compton's equation for scattering substitute LHS of expr 1 into expr 2
1. 8257621077; locally 2840008:
$$\vec{p}_{\rm before} = \vec{p}_{1}$$
$$pdg_{1302} = pdg_{6029}$$
2. 1638282134; locally 4978059:
$$\vec{p}_{\rm before} = \vec{p}_{\rm after}$$
$$pdg_{1302} = pdg_{5493}$$
1. 3951205425; locally 2491904:
$$\vec{p}_{\rm after} = \vec{p}_{1}$$
$$pdg_{5493} = pdg_{6029}$$
LHS diff is -pdg5493 + pdg6029 RHS diff is pdg5493 - pdg6029 8257621077:
1638282134:
3951205425:
8257621077:
1638282134:
3951205425:
Compton's equation for scattering declare initial expr
1. 1638282134; locally 4978059:
$$\vec{p}_{\rm before} = \vec{p}_{\rm after}$$
$$pdg_{1302} = pdg_{5493}$$
no validation is available for declarations 1638282134:
1638282134:
identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation multiply expr 1 by expr 2
1. 2103023049; locally 6060683:
$$\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)$$
$$\sin{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}$$
2. 4585932229; locally 5011637:
$$\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$$
$$\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}$$
1. 3470587782; locally 6350246:
$$\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$$
$$\sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{\left(\frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}\right) \left(e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}\right)}{2 pdg_{4621}}$$
valid 2103023049:
4585932229:
3470587782:
2103023049:
4585932229:
3470587782:
identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation RHS of expr 1 equals RHS of expr 2
1. 9180861128; locally 6229292:
$$2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)$$
$$2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{e^{2 pdg_{1464} pdg_{4621}} - e^{- 2 pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}$$
2. 8483686863; locally 1414263:
$$\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)$$
$$\sin{\left(2 pdg_{1464} \right)} = \frac{e^{2 pdg_{1464} pdg_{4621}} - e^{- 2 pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}$$
1. 2405307372; locally 7647794:
$$\sin(2 x) = 2 \sin(x) \cos(x)$$
$$\sin{\left(2 pdg_{1464} \right)} = 2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)}$$
valid 9180861128:
8483686863:
2405307372:
9180861128:
8483686863:
2405307372:
identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation declare initial expr
1. 2103023049; locally 6060683:
$$\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)$$
$$\sin{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}$$
no validation is available for declarations 2103023049:
2103023049:
identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation declare initial expr
1. 4585932229; locally 5011637:
$$\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$$
$$\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}$$
no validation is available for declarations 4585932229:
4585932229:
identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation change variable X to Y
1. 2103023049; locally 6060683:
$$\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)$$
$$\sin{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}$$
1. 4961662865:
$$x$$
$$pdg_{1464}$$
2. 9110536742:
$$2 x$$
$$2 pdg_{1464}$$
1. 8483686863; locally 1414263:
$$\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)$$
$$\sin{\left(2 pdg_{1464} \right)} = \frac{e^{2 pdg_{1464} pdg_{4621}} - e^{- 2 pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}$$
valid 2103023049:
8483686863:
2103023049:
8483686863:
identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation simplify
1. 8699789241; locally 5714636:
$$2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)$$
$$2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{e^{2 pdg_{1464} pdg_{4621}} - e^{- 2 pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}$$
1. 9180861128; locally 6229292:
$$2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)$$
$$2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{e^{2 pdg_{1464} pdg_{4621}} - e^{- 2 pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}$$
valid 8699789241:
9180861128:
8699789241:
9180861128:
identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation declare final expr
1. 2405307372; locally 7647794:
$$\sin(2 x) = 2 \sin(x) \cos(x)$$
$$\sin{\left(2 pdg_{1464} \right)} = 2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)}$$
no validation is available for declarations 2405307372:
2405307372:
identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation multiply both sides by
1. 3470587782; locally 6350246:
$$\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$$
$$\sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{\left(\frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}\right) \left(e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}\right)}{2 pdg_{4621}}$$
1. 8642992037:
$$2$$
$$2$$
1. 9894826550; locally 7867574:
$$2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)$$
$$2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{\left(e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}\right) \left(e^{pdg_{1464} pdg_{4621}} + e^{- pdg_{1464} pdg_{4621}}\right)}{2 pdg_{4621}}$$
valid 3470587782:
9894826550:
3470587782:
9894826550:
identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation simplify
1. 9894826550; locally 7867574:
$$2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)$$
$$2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{\left(e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}\right) \left(e^{pdg_{1464} pdg_{4621}} + e^{- pdg_{1464} pdg_{4621}}\right)}{2 pdg_{4621}}$$
1. 8699789241; locally 5714636:
$$2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)$$
$$2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{e^{2 pdg_{1464} pdg_{4621}} - e^{- 2 pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}$$
valid 9894826550:
8699789241:
9894826550:
8699789241:
Euler equation to e^(i pi) + 1 = 0 simplify
1. 8332931442; locally 1148677:
$$\exp(i \pi) = \cos(\pi)+i \sin(\pi)$$
$$e^{pdg_{3141} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{3141} \right)} + \cos{\left(pdg_{3141} \right)}$$
1. 6885625907; locally 8524301:
$$\exp(i \pi) = -1 + i 0$$
$$e^{pdg_{3141} pdg_{4621}} = -1$$
LHS diff is 0 RHS diff is pdg4621*sin(pdg3141) + cos(pdg3141) + 1 8332931442:
6885625907:
8332931442:
6885625907:
Euler equation to e^(i pi) + 1 = 0 change variable X to Y
1. 4938429483; locally 1580045:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
1. 3268645065:
$$x$$
$$pdg_{1464}$$
2. 9350663581:
$$\pi$$
$$pdg_{3141}$$
1. 8332931442; locally 1148677:
$$\exp(i \pi) = \cos(\pi)+i \sin(\pi)$$
$$e^{pdg_{3141} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{3141} \right)} + \cos{\left(pdg_{3141} \right)}$$
valid 4938429483:
8332931442:
4938429483:
8332931442:
Euler equation to e^(i pi) + 1 = 0 declare initial expr
1. 4938429483; locally 1580045:
$$\exp(i x) = \cos(x)+i \sin(x)$$
$$e^{pdg_{1464} pdg_{4621}} = pdg_{4621} \sin{\left(pdg_{1464} \right)} + \cos{\left(pdg_{1464} \right)}$$
no validation is available for declarations 4938429483:
4938429483:
Euler equation to e^(i pi) + 1 = 0 add X to both sides
1. 3331824625; locally 9610540:
$$\exp(i \pi) = -1$$
$$e^{pdg_{3141} pdg_{4621}} = -1$$
1. 4901237716:
$$1$$
$$1$$
1. 2501591100; locally 9472905:
$$\exp(i \pi) + 1 = 0$$
$$e^{pdg_{3141} pdg_{4621}} + 1 = 0$$
valid 3331824625:
2501591100:
3331824625:
2501591100:
Euler equation to e^(i pi) + 1 = 0 simplify
1. 6885625907; locally 8524301:
$$\exp(i \pi) = -1 + i 0$$
$$e^{pdg_{3141} pdg_{4621}} = -1$$
1. 3331824625; locally 9610540:
$$\exp(i \pi) = -1$$
$$e^{pdg_{3141} pdg_{4621}} = -1$$
valid 6885625907:
3331824625:
6885625907:
3331824625:
Euler equation to e^(i pi) + 1 = 0 declare final expr
1. 2501591100; locally 9472905:
$$\exp(i \pi) + 1 = 0$$
$$e^{pdg_{3141} pdg_{4621}} + 1 = 0$$
no validation is available for declarations 2501591100:
2501591100:
time invariant force conserves energy substitute RHS of expr 1 into expr 2
1. 9337785146; locally 6154610:
$$v = \frac{x_2 - x_1}{t}$$
$$pdg_{1357} = \frac{- pdg_{3852} + pdg_{5467}}{pdg_{1467}}$$
2. 7267155233; locally 7539016:
$$\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)$$
$$\frac{- pdg_{4093} + pdg_{8849}}{pdg_{1467}} = - \frac{pdg_{4202} \left(- pdg_{3852} + pdg_{5467}\right)}{pdg_{1467}}$$
1. 4872970974; locally 9383749:
$$\frac{PE_2 - PE_1}{t} = -F v$$
$$\frac{- pdg_{4093} + pdg_{8849}}{pdg_{1467}} = - pdg_{1357} pdg_{4202}$$
valid 9337785146:
7267155233:
4872970974:
9337785146:
7267155233:
4872970974:
time invariant force conserves energy substitute RHS of expr 1 into expr 2
1. 4648451961; locally 8696678:
$$v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)$$
$$- pdg_{2473}^{2} + pdg_{4770}^{2} = \left(- pdg_{2473} + pdg_{4770}\right) \left(pdg_{2473} + pdg_{4770}\right)$$
2. 4270680309; locally 3040361:
$$\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = \frac{pdg_{5156} \left(- pdg_{2473}^{2} + pdg_{4770}^{2}\right)}{2 pdg_{1467}}$$
1. 9356924046; locally 6246951:
$$\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = \frac{pdg_{5156} \left(- pdg_{2473} + pdg_{4770}\right) \left(\frac{pdg_{2473}}{2} + \frac{pdg_{4770}}{2}\right)}{pdg_{1467}}$$
valid 4648451961:
4270680309:
9356924046:
4648451961:
4270680309:
9356924046:
time invariant force conserves energy substitute RHS of expr 1 into expr 2
1. 2857430695; locally 6973462:
$$a = \frac{v_2 - v_1}{t}$$
$$pdg_{9140} = \frac{- pdg_{2473} + pdg_{4770}}{pdg_{1467}}$$
2. 7735737409; locally 6733685:
$$\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = \frac{pdg_{1357} pdg_{5156} \left(- pdg_{2473} + pdg_{4770}\right)}{pdg_{1467}}$$
1. 4784793837; locally 4876963:
$$\frac{KE_2 - KE_1}{t} = m v a$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = pdg_{1357} pdg_{5156} pdg_{9140}$$
valid 2857430695:
7735737409:
4784793837:
2857430695:
7735737409:
4784793837:
time invariant force conserves energy simplify
1. 1772416655; locally 5300304:
$$\frac{E_2 - E_1}{t} = v F - F v$$
$$\frac{pdg_{4550} - pdg_{5579}}{pdg_{1467}} = 0$$
1. 1809909100; locally 6495233:
$$\frac{E_2 - E_1}{t} = 0$$
$$\frac{pdg_{4550} - pdg_{5579}}{pdg_{1467}} = 0$$
valid 1772416655:
1809909100:
1772416655:
1809909100:
time invariant force conserves energy multiply both sides by
1. 1809909100; locally 6495233:
$$\frac{E_2 - E_1}{t} = 0$$
$$\frac{pdg_{4550} - pdg_{5579}}{pdg_{1467}} = 0$$
1. 5778176146:
$$t$$
$$pdg_{1467}$$
1. 3806977900; locally 2075807:
$$E_2 - E_1 = 0$$
$$pdg_{4550} - pdg_{5579} = 0$$
valid 1809909100:
3806977900:
1809909100:
3806977900:
time invariant force conserves energy declare initial expr
1. 8357234146; locally 6559987:
$$KE = \frac{1}{2} m v^2$$
$$pdg_{4929} = \frac{pdg_{1357}^{2} pdg_{5156}}{2}$$
no validation is available for declarations 8357234146:
8357234146:
time invariant force conserves energy substitute RHS of expr 1 into expr 2
1. 9397152918; locally 3484339:
$$v = \frac{v_1 + v_2}{2}$$
$$pdg_{1357} = \frac{pdg_{2473}}{2} + \frac{pdg_{4770}}{2}$$
2. 9356924046; locally 6246951:
$$\frac{KE_2 - KE_1}{t} = m \frac{v_2 + v_1}{2} \frac{ v_2 - v_1 }{t}$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = \frac{pdg_{5156} \left(- pdg_{2473} + pdg_{4770}\right) \left(\frac{pdg_{2473}}{2} + \frac{pdg_{4770}}{2}\right)}{pdg_{1467}}$$
1. 7735737409; locally 6733685:
$$\frac{KE_2 - KE_1}{t} = m v \frac{ v_2 - v_1 }{t}$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = \frac{pdg_{1357} pdg_{5156} \left(- pdg_{2473} + pdg_{4770}\right)}{pdg_{1467}}$$
valid 9397152918:
9356924046:
7735737409:
9397152918:
9356924046:
7735737409:
time invariant force conserves energy substitute RHS of expr 1 into expr 2
1. 4872970974; locally 9383749:
$$\frac{PE_2 - PE_1}{t} = -F v$$
$$\frac{- pdg_{4093} + pdg_{8849}}{pdg_{1467}} = - pdg_{1357} pdg_{4202}$$
2. 2770069250; locally 2692856:
$$\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}$$
$$\frac{pdg_{4550} - pdg_{5579}}{pdg_{1467}} = \frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} + \frac{- pdg_{4093} + pdg_{8849}}{pdg_{1467}}$$
1. 3591237106; locally 9714818:
$$\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v$$
$$\frac{pdg_{4550} - pg_{5579}}{pdg_{1467}} = - pdg_{1357} pdg_{4202} + \frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}}$$
LHS diff is (-pdg5579 + pg5579)/pdg1467 RHS diff is (pdg1357*pdg1467*pdg4202 - pdg4093 + pdg8849)/pdg1467 4872970974:
2770069250:
3591237106:
4872970974:
2770069250:
3591237106:
time invariant force conserves energy change two variables in expr
1. 8357234146; locally 6559987:
$$KE = \frac{1}{2} m v^2$$
$$pdg_{4929} = \frac{pdg_{1357}^{2} pdg_{5156}}{2}$$
1. 6383056612:
$$KE$$
$$pdg_{4929}$$
2. 6838659900:
$$KE_2$$
$$pdg_{1352}$$
3. 9305761407:
$$v$$
$$pdg_{1357}$$
4. 5011888122:
$$v_2$$
$$pdg_{4770}$$
1. 7676652285; locally 6632540:
$$KE_2 = \frac{1}{2} m v_2^2$$
$$pdg_{1352} = \frac{pdg_{4770}^{2} pdg_{5156}}{2}$$
valid 8357234146:
7676652285:
8357234146:
7676652285:
time invariant force conserves energy divide both sides by
1. 5733146966; locally 9602854:
$$KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)$$
$$pdg_{1352} - pdg_{1955} = \frac{pdg_{5156} \left(- pdg_{2473}^{2} + pdg_{4770}^{2}\right)}{2}$$
1. 6554292307:
$$t$$
$$pdg_{1467}$$
1. 4270680309; locally 3040361:
$$\frac{KE_2 - KE_1}{t} = \frac{1}{2} m \frac{\left( v_2^2 - v_1^2 \right)}{t}$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = \frac{pdg_{5156} \left(- pdg_{2473}^{2} + pdg_{4770}^{2}\right)}{2 pdg_{1467}}$$
valid 5733146966:
4270680309:
5733146966:
4270680309:
time invariant force conserves energy declare initial expr
1. 5136652623; locally 8844119:
$$E = KE + PE$$
$$pdg_{4931} = pdg_{4929} + pdg_{4930}$$
no validation is available for declarations 5136652623:
5136652623:
time invariant force conserves energy change three variables in expr
1. 5136652623; locally 8844119:
$$E = KE + PE$$
$$pdg_{4931} = pdg_{4929} + pdg_{4930}$$
1. 1258245373:
$$E$$
$$pdg_{4931}$$
2. 2344320475:
$$E_2$$
$$pdg_{4550}$$
3. 6383056612:
$$KE$$
$$pdg_{4929}$$
4. 7939947931:
$$KE_2$$
$$pdg_{1352}$$
5. 5075406409:
$$PE$$
$$pdg_{4930}$$
6. 5803210729:
$$PE_2$$
$$pdg_{8849}$$
1. 7875206161; locally 5642407:
$$E_2 = KE_2 + PE_2$$
$$pdg_{4550} = pdg_{1352} + pdg_{8849}$$
valid 5136652623:
7875206161:
5136652623:
7875206161:
time invariant force conserves energy divide both sides by
1. 5514556106; locally 2443387:
$$E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)$$
$$pdg_{4550} - pdg_{5579} = pdg_{1352} - pdg_{1955} - pdg_{4093} + pdg_{8849}$$
1. 2081689540:
$$t$$
$$pdg_{1467}$$
1. 2770069250; locally 2692856:
$$\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} + \frac{(PE_2 - PE_1)}{t}$$
$$\frac{pdg_{4550} - pdg_{5579}}{pdg_{1467}} = \frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} + \frac{- pdg_{4093} + pdg_{8849}}{pdg_{1467}}$$
valid 5514556106:
2770069250:
5514556106:
2770069250:
time invariant force conserves energy subtract expr 1 from expr 2
1. 4928007622; locally 4208138:
$$KE_1 = \frac{1}{2} m v_1^2$$
$$pdg_{1955} = \frac{pdg_{2473}^{2} pdg_{5156}}{2}$$
2. 7676652285; locally 6632540:
$$KE_2 = \frac{1}{2} m v_2^2$$
$$pdg_{1352} = \frac{pdg_{4770}^{2} pdg_{5156}}{2}$$
1. 5733146966; locally 9602854:
$$KE_2 - KE_1 = \frac{1}{2} m \left(v_2^2 - v_1^2\right)$$
$$pdg_{1352} - pdg_{1955} = \frac{pdg_{5156} \left(- pdg_{2473}^{2} + pdg_{4770}^{2}\right)}{2}$$
valid 4928007622:
7676652285:
5733146966:
4928007622:
7676652285:
5733146966:
time invariant force conserves energy declare initial expr
1. 2857430695; locally 6973462:
$$a = \frac{v_2 - v_1}{t}$$
$$pdg_{9140} = \frac{- pdg_{2473} + pdg_{4770}}{pdg_{1467}}$$
no validation is available for declarations 2857430695:
2857430695:
time invariant force conserves energy divide both sides by
1. 7734996511; locally 1550851:
$$PE_2 - PE_1 = -F ( x_2 - x_1 )$$
$$- pdg_{4093} + pdg_{8849} = - pdg_{4202} \left(- pdg_{3852} + pdg_{5467}\right)$$
1. 2016063530:
$$t$$
$$pdg_{1467}$$
1. 7267155233; locally 7539016:
$$\frac{PE_2 - PE_1}{t} = -F \left( \frac{x_2 - x_1}{t} \right)$$
$$\frac{- pdg_{4093} + pdg_{8849}}{pdg_{1467}} = - \frac{pdg_{4202} \left(- pdg_{3852} + pdg_{5467}\right)}{pdg_{1467}}$$
valid 7734996511:
7267155233:
7734996511:
7267155233:
time invariant force conserves energy change two variables in expr
1. 6715248283; locally 8497204:
$$PE = -F x$$
$$pdg_{4930} = - pdg_{4037} pdg_{4202}$$
1. 3809726424:
$$PE$$
$$pdg_{4930}$$
2. 6749533119:
$$PE_1$$
$$pdg_{4093}$$
3. 4218009993:
$$x$$
$$pdg_{4037}$$
4. 1552869972:
$$x_1$$
$$pdg_{3852}$$
1. 4669290568; locally 9081932:
$$PE_1 = -F x_1$$
$$pdg_{4093} = - pdg_{3852} pdg_{4202}$$
valid 6715248283:
4669290568:
6715248283:
4669290568:
time invariant force conserves energy declare initial expr
1. 5345738321; locally 8447573:
$$F = m a$$
$$pdg_{4202} = pdg_{5156} pdg_{9140}$$
no validation is available for declarations 5345738321:
5345738321:
time invariant force conserves energy change two variables in expr
1. 6715248283; locally 8497204:
$$PE = -F x$$
$$pdg_{4930} = - pdg_{4037} pdg_{4202}$$
1. 5075406409:
$$PE$$
$$pdg_{4930}$$
2. 4522137851:
$$PE_2$$
$$pdg_{8849}$$
3. 4188639044:
$$x$$
$$pdg_{4037}$$
4. 4755369593:
$$x_2$$
$$pdg_{5467}$$
1. 2431507955; locally 3988671:
$$PE_2 = -F x_2$$
$$pdg_{8849} = - pdg_{4202} pdg_{5467}$$
valid 6715248283:
2431507955:
6715248283:
2431507955:
assumes constant force
time invariant force conserves energy subtract expr 1 from expr 2
1. 4303372136; locally 1298003:
$$E_1 = KE_1 + PE_1$$
$$pdg_{5579} = pdg_{1955} + pdg_{4093}$$
2. 7875206161; locally 5642407:
$$E_2 = KE_2 + PE_2$$
$$pdg_{4550} = pdg_{1352} + pdg_{8849}$$
1. 5514556106; locally 2443387:
$$E_2 - E_1 = (KE_2 - KE_1) + (PE_2 - PE_1)$$
$$pdg_{4550} - pdg_{5579} = pdg_{1352} - pdg_{1955} - pdg_{4093} + pdg_{8849}$$
valid 4303372136:
7875206161:
5514556106:
4303372136:
7875206161:
5514556106:
time invariant force conserves energy declare final expr
1. 8558338742; locally 1781127:
$$E_2 = E_1$$
$$pdg_{4550} = pdg_{5579}$$
no validation is available for declarations 8558338742:
8558338742:
time invariant force conserves energy substitute RHS of expr 1 into expr 2
1. 5345738321; locally 8447573:
$$F = m a$$
$$pdg_{4202} = pdg_{5156} pdg_{9140}$$
2. 4784793837; locally 4876963:
$$\frac{KE_2 - KE_1}{t} = m v a$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = pdg_{1357} pdg_{5156} pdg_{9140}$$
1. 2186083170; locally 7034924:
$$\frac{KE_2 - KE_1}{t} = v F$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = pdg_{1357} pdg_{4202}$$
valid 5345738321:
4784793837:
2186083170:
5345738321:
4784793837:
2186083170:
time invariant force conserves energy declare initial expr
1. 5781981178; locally 2776565:
$$x^2 - y^2 = (x+y)(x-y)$$
$$- pdg_{1452}^{2} + pdg_{1464}^{2} = \left(- pdg_{1452} + pdg_{1464}\right) \left(pdg_{1452} + pdg_{1464}\right)$$
no validation is available for declarations 5781981178:
5781981178:
time invariant force conserves energy change two variables in expr
1. 8357234146; locally 6559987:
$$KE = \frac{1}{2} m v^2$$
$$pdg_{4929} = \frac{pdg_{1357}^{2} pdg_{5156}}{2}$$
1. 4147101187:
$$KE$$
$$pdg_{4929}$$
2. 6964468708:
$$KE_1$$
$$pdg_{1955}$$
3. 5398681503:
$$v$$
$$pdg_{1357}$$
4. 3105350101:
$$v_1$$
$$pdg_{2473}$$
1. 4928007622; locally 4208138:
$$KE_1 = \frac{1}{2} m v_1^2$$
$$pdg_{1955} = \frac{pdg_{2473}^{2} pdg_{5156}}{2}$$
valid 8357234146:
4928007622:
8357234146:
4928007622:
time invariant force conserves energy declare initial expr
1. 6715248283; locally 8497204:
$$PE = -F x$$
$$pdg_{4930} = - pdg_{4037} pdg_{4202}$$
no validation is available for declarations 6715248283:
6715248283:
time invariant force conserves energy subtract expr 1 from expr 2
1. 4669290568; locally 9081932:
$$PE_1 = -F x_1$$
$$pdg_{4093} = - pdg_{3852} pdg_{4202}$$
2. 2431507955; locally 3988671:
$$PE_2 = -F x_2$$
$$pdg_{8849} = - pdg_{4202} pdg_{5467}$$
1. 7734996511; locally 1550851:
$$PE_2 - PE_1 = -F ( x_2 - x_1 )$$
$$- pdg_{4093} + pdg_{8849} = - pdg_{4202} \left(- pdg_{3852} + pdg_{5467}\right)$$
valid 4669290568:
2431507955:
7734996511:
4669290568:
2431507955:
7734996511:
time invariant force conserves energy add X to both sides
1. 3806977900; locally 2075807:
$$E_2 - E_1 = 0$$
$$pdg_{4550} - pdg_{5579} = 0$$
1. 5960438249:
$$E_1$$
$$pdg_{5579}$$
1. 8558338742; locally 1781127:
$$E_2 = E_1$$
$$pdg_{4550} = pdg_{5579}$$
valid 3806977900:
8558338742:
3806977900:
8558338742:
time invariant force conserves energy change two variables in expr
1. 5781981178; locally 2776565:
$$x^2 - y^2 = (x+y)(x-y)$$
$$- pdg_{1452}^{2} + pdg_{1464}^{2} = \left(- pdg_{1452} + pdg_{1464}\right) \left(pdg_{1452} + pdg_{1464}\right)$$
1. 1025759423:
$$y$$
$$pdg_{1452}$$
2. 5239755033:
$$v_1$$
$$pdg_{2473}$$
3. 8173074178:
$$x$$
$$pdg_{1464}$$
4. 4319470443:
$$v_2$$
$$pdg_{4770}$$
1. 4648451961; locally 8696678:
$$v_2^2 - v_1^2 = (v_2 + v_1)(v_2 - v_1)$$
$$- pdg_{2473}^{2} + pdg_{4770}^{2} = \left(- pdg_{2473} + pdg_{4770}\right) \left(pdg_{2473} + pdg_{4770}\right)$$
valid 5781981178:
4648451961:
5781981178:
4648451961:
time invariant force conserves energy declare initial expr
1. 9337785146; locally 6154610:
$$v = \frac{x_2 - x_1}{t}$$
$$pdg_{1357} = \frac{- pdg_{3852} + pdg_{5467}}{pdg_{1467}}$$
no validation is available for declarations 9337785146:
9337785146:
time invariant force conserves energy declare initial expr
1. 9397152918; locally 3484339:
$$v = \frac{v_1 + v_2}{2}$$
$$pdg_{1357} = \frac{pdg_{2473}}{2} + \frac{pdg_{4770}}{2}$$
no validation is available for declarations 9397152918:
9397152918:
time invariant force conserves energy change three variables in expr
1. 5136652623; locally 8844119:
$$E = KE + PE$$
$$pdg_{4931} = pdg_{4929} + pdg_{4930}$$
1. 3749492596:
$$E$$
$$pdg_{4931}$$
2. 4213426349:
$$E_1$$
$$pdg_{5579}$$
3. 4147101187:
$$KE$$
$$pdg_{4929}$$
4. 1092872200:
$$KE_1$$
$$pdg_{1955}$$
5. 3809726424:
$$PE$$
$$pdg_{4930}$$
6. 8061701434:
$$PE_1$$
$$pdg_{4093}$$
1. 4303372136; locally 1298003:
$$E_1 = KE_1 + PE_1$$
$$pdg_{5579} = pdg_{1955} + pdg_{4093}$$
valid 5136652623:
4303372136:
5136652623:
4303372136:
time invariant force conserves energy substitute RHS of expr 1 into expr 2
1. 2186083170; locally 7034924:
$$\frac{KE_2 - KE_1}{t} = v F$$
$$\frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}} = pdg_{1357} pdg_{4202}$$
2. 3591237106; locally 9714818:
$$\frac{E_2 - E_1}{t} = \frac{(KE_2 - KE_1)}{t} - F v$$
$$\frac{pdg_{4550} - pg_{5579}}{pdg_{1467}} = - pdg_{1357} pdg_{4202} + \frac{pdg_{1352} - pdg_{1955}}{pdg_{1467}}$$
1. 1772416655; locally 5300304:
$$\frac{E_2 - E_1}{t} = v F - F v$$
$$\frac{pdg_{4550} - pdg_{5579}}{pdg_{1467}} = 0$$
LHS diff is (pdg5579 - pg5579)/pdg1467 RHS diff is 0 2186083170:
3591237106:
1772416655:
2186083170:
3591237106:
1772416655:
escape velocity swap LHS with RHS
1. 2977457786; locally 3358651:
$$2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2$$
$$\frac{2 pdg_{5458} pdg_{6277}}{pdg_{3236}} = pdg_{8656}^{2}$$
1. 9412953728; locally 3908344:
$$v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}$$
$$pdg_{8656}^{2} = \frac{2 pdg_{5458} pdg_{6277}}{pdg_{3236}}$$
valid 2977457786:
9412953728:
2977457786:
9412953728:
escape velocity substitute LHS of two expressions into expr
1. 4303372136; locally 6310702:
$$E_1 = KE_1 + PE_1$$
$$pdg_{5579} = pdg_{1955} + pdg_{4093}$$
2. 7875206161; locally 5160388:
$$E_2 = KE_2 + PE_2$$
$$pdg_{4550} = pdg_{1352} + pdg_{8849}$$
3. 8558338742; locally 6330719:
$$E_2 = E_1$$
$$pdg_{4550} = pdg_{5579}$$
1. 8960645192; locally 4840471:
$$KE_2 + PE_2 = KE_1 + PE_1$$
$$pdg_{1552} + pdg_{8849} = pdg_{1955} + pdg_{4093}$$
failed 4303372136:
7875206161:
8558338742:
8960645192:
4303372136:
7875206161:
8558338742:
8960645192:
escape velocity declare assumption
1. 2267521164; locally 7682341:
$$PE_2 = 0$$
$$pdg_{8849} = 0$$
no validation is available for declarations 2267521164:
2267521164:
escape velocity declare initial expr
1. 7573835180; locally 6773616:
$$PE_{\rm Earth\ surface} = -W$$
$$pdg_{6431} = - pdg_{6789}$$
no validation is available for declarations 7573835180:
7573835180:
escape velocity multiply both sides by
1. 1143343287; locally 7567097:
$$G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2$$
$$\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}} = \frac{pdg_{8656}^{2}}{2}$$
1. 5775658332:
$$2$$
$$2$$
1. 2977457786; locally 3358651:
$$2 G \frac{m_{\rm Earth}}{r_{\rm Earth}} = v_{\rm escape}^2$$
$$\frac{2 pdg_{5458} pdg_{6277}}{pdg_{3236}} = pdg_{8656}^{2}$$
valid 1143343287:
2977457786:
1143343287:
2977457786:
escape velocity evaluate definite integral
1. 4447113478; locally 4803506:
$$\int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx$$
$$\int 1\, dpdg_{6789} = pdg_{4851} pdg_{5022} pdg_{6277} \int\limits_{pdg_{3236}}^{infty} \frac{1}{pdg_{4037}^{2}}\, dpdg_{4037}$$
1. 5732331610; locally 1089445:
$$W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)$$
$$pdg_{6277}$$
Nothing to split 4447113478:
5732331610:
4447113478:
5732331610:
escape velocity simplify
1. 5978756813; locally 2190752:
$$W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)$$
$$pdg_{6789} = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}$$
1. 7749253510; locally 2238158:
$$W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}$$
$$pdg_{6789} = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}$$
valid 5978756813:
7749253510:
5978756813:
7749253510:
escape velocity declare final expr
1. 1330874553; locally 6389964:
$$v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}$$
$$pdg_{8656} = \sqrt{2} \sqrt{\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}}}$$
no validation is available for declarations 1330874553:
1330874553:
escape velocity change two variables in expr
1. 1330874553; locally 6389964:
$$v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}$$
$$pdg_{8656} = \sqrt{2} \sqrt{\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}}}$$
1. 2674546234:
$$m_{\rm Earth}$$
$$pdg_{5458}$$
2. 2135482543:
$$m$$
$$pdg_{5156}$$
3. 2396787389:
$$r_{\rm Earth}$$
$$pdg_{3236}$$
4. 8020058613:
$$r$$
$$pdg_{2530}$$
1. 5404822208; locally 1619188:
$$v_{\rm escape} = \sqrt{2 G \frac{m}{r}}$$
$$pdg_{8656} = \sqrt{2} \sqrt{\frac{pdg_{5156} pdg_{6277}}{pdg_{2530}}}$$
valid 1330874553:
5404822208:
1330874553:
5404822208:
replaced Earth-specific variables
escape velocity square root both sides
1. 9412953728; locally 3908344:
$$v_{\rm escape}^2 = 2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}$$
$$pdg_{8656}^{2} = \frac{2 pdg_{5458} pdg_{6277}}{pdg_{3236}}$$
1. 1330874553; locally 6389964:
$$v_{\rm escape} = \sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}$$
$$pdg_{8656} = \sqrt{2} \sqrt{\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}}}$$
2. 2750380042; locally 8779043:
$$v_{\rm escape} = -\sqrt{2 G \frac{m_{\rm Earth}}{r_{\rm Earth}}}$$
$$pdg_{8656} = - \sqrt{2} \sqrt{\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}}}$$
no check performed 9412953728:
1330874553:
2750380042:
9412953728:
1330874553:
2750380042:
escape velocity substitute LHS of expr 1 into expr 2
1. 6935745841; locally 3279838:
$$F = G \frac{m_1 m_2}{x^2}$$
$$pdg_{4202} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{4037}^{2}}$$
2. 1590774089; locally 2123766:
$$dW = F dx$$
$$pdg_{9398} = pdg_{4202} pdg_{9199}$$
1. 8604483515; locally 3686928:
$$dW = G \frac{m_1 m_2}{x^2} dx$$
$$pdg_{9398} = \frac{pdg_{4851} pdg_{5022} pdg_{6277} pdg_{9199}}{pdg_{4037}^{2}}$$
valid 6935745841:
1590774089:
8604483515:
6935745841:
1590774089:
8604483515:
escape velocity simplify
1. 9703482302; locally 6523887:
$$G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2$$
$$\frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}} = \frac{pdg_{5156} pdg_{8656}^{2}}{2}$$
1. 1143343287; locally 7567097:
$$G \frac{m_{\rm Earth}}{r_{\rm Earth}} = \frac{1}{2} v_{\rm escape}^2$$
$$\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}} = \frac{pdg_{8656}^{2}}{2}$$
LHS diff is pdg5458*pdg6277*(pdg5156 - 1)/pdg3236 RHS diff is pdg8656**2*(pdg5156 - 1)/2 9703482302:
1143343287:
9703482302:
1143343287:
escape velocity integrate
1. 8604483515; locally 3686928:
$$dW = G \frac{m_1 m_2}{x^2} dx$$
$$pdg_{9398} = \frac{pdg_{4851} pdg_{5022} pdg_{6277} pdg_{9199}}{pdg_{4037}^{2}}$$
1. 4447113478; locally 4803506:
$$\int dW = G m_1 m_2 \int_{ r_{\rm Earth} }^{\infty} \frac{1}{x^2} dx$$
$$\int 1\, dpdg_{6789} = pdg_{4851} pdg_{5022} pdg_{6277} \int\limits_{pdg_{3236}}^{infty} \frac{1}{pdg_{4037}^{2}}\, dpdg_{4037}$$
no check performed 8604483515:
4447113478:
8604483515:
4447113478:
escape velocity substitute LHS of expr 1 into expr 2
1. 7749253510; locally 2238158:
$$W = G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}$$
$$pdg_{6789} = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}$$
2. 7573835180; locally 6773616:
$$PE_{\rm Earth\ surface} = -W$$
$$pdg_{6431} = - pdg_{6789}$$
1. 3846041519; locally 9437784:
$$PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}}$$
$$pdg_{6431} = - \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}$$
valid 7749253510:
7573835180:
3846041519:
7749253510:
7573835180:
3846041519:
escape velocity declare initial expr
1. 6935745841; locally 3279838:
$$F = G \frac{m_1 m_2}{x^2}$$
$$pdg_{4202} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{4037}^{2}}$$
no validation is available for declarations 6935745841:
6935745841:
escape velocity declare assumption
1. 1840080113; locally 9324316:
$$KE_2 = 0$$
$$pdg_{1552} = 0$$
no validation is available for declarations 1840080113:
1840080113:
escape velocity substitute LHS of two expressions into expr
1. 2267521164; locally 7682341:
$$PE_2 = 0$$
$$pdg_{8849} = 0$$
2. 1840080113; locally 9324316:
$$KE_2 = 0$$
$$pdg_{1552} = 0$$
3. 8960645192; locally 4840471:
$$KE_2 + PE_2 = KE_1 + PE_1$$
$$pdg_{1552} + pdg_{8849} = pdg_{1955} + pdg_{4093}$$
1. 9749777192; locally 8369684:
$$0 = KE_1 + PE_1$$
$$0 = pdg_{1955} + pdg_{4093}$$
failed 2267521164:
1840080113:
8960645192:
9749777192:
2267521164:
1840080113:
8960645192:
9749777192:
escape velocity change two variables in expr
1. 5732331610; locally 1089445:
$$W = G m_1 m_2 \left( \frac{1}{x} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)$$
$$pdg_{6277}$$
1. 1413137236:
$$m_1$$
$$pdg_{5022}$$
2. 9072369552:
$$m_{\rm Earth}$$
$$pdg_{5458}$$
3. 2764966428:
$$m_2$$
$$pdg_{4851}$$
4. 7140470627:
$$m$$
$$pdg_{5156}$$
1. 6131764194; locally 2341415:
$$W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)$$
$$W = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{4037}^{2}}$$
Nothing to split 5732331610:
6131764194:
5732331610:
6131764194:
escape velocity substitute LHS of two expressions into expr
1. 6870322215; locally 5106827:
$$KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2$$
$$pdg_{5332} = \frac{pdg_{5156} pdg_{8656}^{2}}{2}$$
2. 3846041519; locally 9437784:
$$PE_{\rm Earth\ surface} = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}}$$
$$pdg_{6431} = - \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}$$
3. 2503972039; locally 9967559:
$$0 = KE_{\rm escape} + PE_{\rm Earth\ surface}$$
$$0 = pdg_{5332} + pdg_{6431}$$
1. 2042298788; locally 3493665:
$$0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2$$
$$0 = \frac{pdg_{5156} pdg_{8656}^{2}}{2} - \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}$$
failed 6870322215:
3846041519:
2503972039:
2042298788:
6870322215:
3846041519:
2503972039:
2042298788:
escape velocity change two variables in expr
1. 8357234146; locally 3778087:
$$KE = \frac{1}{2} m v^2$$
$$pdg_{4929} = \frac{pdg_{1357}^{2} pdg_{5156}}{2}$$
1. 5021965469:
$$KE$$
$$pdg_{4929}$$
2. 9370882921:
$$KE_{\rm escape}$$
$$pdg_{5332}$$
3. 6681646197:
$$v$$
$$pdg_{1357}$$
4. 6498985149:
$$v_{\rm escape}$$
$$pdg_{8656}$$
1. 6870322215; locally 5106827:
$$KE_{\rm escape} = \frac{1}{2} m v_{\rm escape}^2$$
$$pdg_{5332} = \frac{pdg_{5156} pdg_{8656}^{2}}{2}$$
valid 8357234146:
6870322215:
8357234146:
6870322215:
escape velocity add X to both sides
1. 2042298788; locally 3493665:
$$0 = -G \frac{m_{\rm Earth} m}{r_{\rm Earth}} + \frac{1}{2} m v_{\rm escape}^2$$
$$0 = \frac{pdg_{5156} pdg_{8656}^{2}}{2} - \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}$$
1. 5050429607:
$$G \frac{m_{\rm Earth} m}{r_{\rm Earth}}$$
$$\frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}$$
1. 9703482302; locally 6523887:
$$G \frac{m_{\rm Earth} m}{r_{\rm Earth}} = \frac{1}{2} m v_{\rm escape}^2$$
$$\frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}} = \frac{pdg_{5156} pdg_{8656}^{2}}{2}$$
valid 2042298788:
9703482302:
2042298788:
9703482302:
escape velocity simplify
1. 6131764194; locally 2341415:
$$W = G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)$$
$$W = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{4037}^{2}}$$
1. 5978756813; locally 2190752:
$$W = G m_{\rm Earth} m \left( 0 - \frac{-1}{ r_{\rm Earth}} \right)$$
$$pdg_{6789} = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}}$$
LHS diff is W - pdg6789 RHS diff is pdg5156*pdg5458*pdg6277*(pdg3236 - pdg4037**2)/(pdg3236*pdg4037**2) 6131764194:
5978756813:
6131764194:
5978756813:
escape velocity change two variables in expr
1. 9749777192; locally 8369684:
$$0 = KE_1 + PE_1$$
$$0 = pdg_{1955} + pdg_{4093}$$
1. 5591692598:
$$KE_1$$
$$pdg_{1955}$$
2. 8416464049:
$$KE_{\rm escape}$$
$$pdg_{5332}$$
3. 6158970683:
$$PE_1$$
$$pdg_{4093}$$
4. 8871333437:
$$PE_{\rm Earth\ surface}$$
$$pdg_{6431}$$
1. 2503972039; locally 9967559:
$$0 = KE_{\rm escape} + PE_{\rm Earth\ surface}$$
$$0 = pdg_{5332} + pdg_{6431}$$
valid 9749777192:
2503972039:
9749777192:
2503972039:
Schwarzschild radius for non-rotating black hole change two variables in expr
1. 8946383937; locally 2478510:
$$v_{\rm escape}^2 = 2 G \frac{m}{r}$$
$$pdg_{8656}^{2} = \frac{2 pdg_{5156} pdg_{6277}}{pdg_{2530}}$$
1. 8362338572:
$$v_{\rm escape}$$
$$pdg_{8656}$$
2. 1238593037:
$$c$$
$$pdg_{4567}$$
3. 2660368546:
$$r$$
$$pdg_{2530}$$
4. 9933742680:
$$r_{\rm Schwarzschild}$$
$$pdg_{4518}$$
1. 4275004561; locally 5459812:
$$c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}}$$
$$pdg_{4567}^{2} = \frac{2 pdg_{5156} pdg_{6277}}{pdg_{4518}}$$
valid 8946383937:
4275004561:
8946383937:
4275004561:
Schwarzschild radius for non-rotating black hole divide both sides by
1. 2883079365; locally 9932666:
$$r_{\rm Schwarzschild} c^2 = 2 G m$$
$$pdg_{4518} pdg_{4567}^{2} = 2 pdg_{5156} pdg_{6277}$$
1. 7263534144:
$$c^2$$
$$pdg_{4567}^{2}$$
1. 6800170830; locally 9994959:
$$r_{\rm Schwarzschild} = \frac{2 G m}{c^2}$$
$$pdg_{4518} = \frac{2 pdg_{5156} pdg_{6277}}{pdg_{4567}^{2}}$$
valid 2883079365:
6800170830:
2883079365:
6800170830:
Schwarzschild radius for non-rotating black hole raise both sides to power
1. 5404822208; locally 1044984:
$$v_{\rm escape} = \sqrt{2 G \frac{m}{r}}$$
$$pdg_{8656} = \sqrt{2} \sqrt{\frac{pdg_{5156} pdg_{6277}}{pdg_{2530}}}$$
1. 3663007361:
$$2$$
$$2$$
1. 8946383937; locally 2478510:
$$v_{\rm escape}^2 = 2 G \frac{m}{r}$$
$$pdg_{8656}^{2} = \frac{2 pdg_{5156} pdg_{6277}}{pdg_{2530}}$$
no check is performed 5404822208:
8946383937:
5404822208:
8946383937:
Schwarzschild radius for non-rotating black hole multiply both sides by
1. 4275004561; locally 5459812:
$$c^2 = 2 G \frac{m}{r_{\rm Schwarzschild}}$$
$$pdg_{4567}^{2} = \frac{2 pdg_{5156} pdg_{6277}}{pdg_{4518}}$$
1. 7194432406:
$$r_{\rm Schwarzschild}$$
$$pdg_{4518}$$
1. 2883079365; locally 9932666:
$$r_{\rm Schwarzschild} c^2 = 2 G m$$
$$pdg_{4518} pdg_{4567}^{2} = 2 pdg_{5156} pdg_{6277}$$
valid 4275004561:
2883079365:
4275004561:
2883079365:
coefficient of thermal expansion using the equation of state for an ideal gas simplify
1. 6925244346; locally 7845152:
$$\alpha = \frac{PV}{T} \frac{1}{VP}$$
$$pdg_{4686} = \frac{pdg_{7586} pdg_{8134}}{pdg_{7343}}$$
1. 2472653783; locally 2491768:
$$\alpha = \frac{1}{T}$$
$$pdg_{4686} = \frac{1}{pdg_{7343}}$$
LHS diff is 0 RHS diff is (pdg7586*pdg8134 - 1)/pdg7343 6925244346:
2472653783:
6925244346:
2472653783:
coefficient of thermal expansion using the equation of state for an ideal gas declare initial expr
1. 8435841627; locally 5130250:
$$P V = n R T$$
$$pdg_{7586} pdg_{8134} = pdg_{2834} pdg_{7343} pdg_{8179}$$
no validation is available for declarations 8435841627:
8435841627:
coefficient of thermal expansion using the equation of state for an ideal gas declare initial expr
1. 3497828859; locally 5927974:
$$V = \frac{n R T}{P}$$
$$pdg_{7586} = \frac{pdg_{2834} pdg_{7343} pdg_{8179}}{pdg_{8134}}$$
no validation is available for declarations 3497828859:
3497828859:
coefficient of thermal expansion using the equation of state for an ideal gas substitute RHS of expr 1 into expr 2
1. 2613006036; locally 8283443:
$$\frac{PV}{T} = nR$$
$$\frac{pdg_{7586} pdg_{8134}}{pdg_{7343}} = pdg_{2834} pdg_{8179}$$
2. 5962145508; locally 4600503:
$$\alpha = \frac{nR}{VP}$$
$$pdg_{4686} = \frac{pdg_{2834} pdg_{8179}}{pdg_{7586} pdg_{8134}}$$
1. 6925244346; locally 7845152:
$$\alpha = \frac{PV}{T} \frac{1}{VP}$$
$$pdg_{4686} = \frac{pdg_{7586} pdg_{8134}}{pdg_{7343}}$$
LHS diff is 0 RHS diff is (-pdg7586*pdg8134 + 1)/pdg7343 2613006036:
5962145508:
6925244346:
2613006036:
5962145508:
6925244346:
coefficient of thermal expansion using the equation of state for an ideal gas declare initial expr
1. 3464107376; locally 5888046:
$$\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p$$
$$pdg_{4686} = \frac{\frac{d}{d pdg_{7343}} pdg_{7586}}{pdg_{7586}}$$
no validation is available for declarations 3464107376:
3464107376:
coefficient of thermal expansion using the equation of state for an ideal gas simplify
1. 1311403394; locally 7236464:
$$\alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P$$
$$pdg_{4686} = \frac{pdg_{2834} pdg_{8179} \frac{d}{d pdg_{7343}} pdg_{7343}}{pdg_{7586} pdg_{8134}}$$
1. 5962145508; locally 4600503:
$$\alpha = \frac{nR}{VP}$$
$$pdg_{4686} = \frac{pdg_{2834} pdg_{8179}}{pdg_{7586} pdg_{8134}}$$
valid 1311403394:
5962145508:
1311403394:
5962145508:
coefficient of thermal expansion using the equation of state for an ideal gas divide both sides by
1. 8435841627; locally 5130250:
$$P V = n R T$$
$$pdg_{7586} pdg_{8134} = pdg_{2834} pdg_{7343} pdg_{8179}$$
1. 7924842770:
$$T$$
$$pdg_{7343}$$
1. 2613006036; locally 8283443:
$$\frac{PV}{T} = nR$$
$$\frac{pdg_{7586} pdg_{8134}}{pdg_{7343}} = pdg_{2834} pdg_{8179}$$
valid 8435841627:
2613006036:
8435841627:
2613006036:
coefficient of thermal expansion using the equation of state for an ideal gas declare final expr
1. 2472653783; locally 2491768:
$$\alpha = \frac{1}{T}$$
$$pdg_{4686} = \frac{1}{pdg_{7343}}$$
no validation is available for declarations 2472653783:
2472653783:
coefficient of thermal expansion using the equation of state for an ideal gas substitute LHS of expr 1 into expr 2
1. 3497828859; locally 5927974:
$$V = \frac{n R T}{P}$$
$$pdg_{7586} = \frac{pdg_{2834} pdg_{7343} pdg_{8179}}{pdg_{8134}}$$
2. 3464107376; locally 5888046:
$$\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p$$
$$pdg_{4686} = \frac{\frac{d}{d pdg_{7343}} pdg_{7586}}{pdg_{7586}}$$
1. 1311403394; locally 7236464:
$$\alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P$$
$$pdg_{4686} = \frac{pdg_{2834} pdg_{8179} \frac{d}{d pdg_{7343}} pdg_{7343}}{pdg_{7586} pdg_{8134}}$$
LHS diff is 0 RHS diff is -pdg2834*pdg8179/(pdg7586*pdg8134) + 1/pdg7343 3497828859:
3464107376:
1311403394:
3497828859:
3464107376:
1311403394:
equations of motion in 2D (calculus) substitute LHS of expr 1 into expr 2
1. 6134836751; locally 8435615:
$$v_{0, x} = v_x$$
$$pdg_{2958} = pdg_{5505}$$
2. 8460820419; locally 4895553:
$$v_x = \frac{dx}{dt}$$
$$pdg_{5505} = \frac{d}{d pdg_{1467}} pdg_{9199}$$
1. 7455581657; locally 5123314:
$$v_{0, x} = \frac{dx}{dt}$$
$$pdg_{2958} = \frac{d}{d pdg_{1467}} pdg_{9199}$$
LHS diff is -pdg2958 + pdg5505 RHS diff is 0 6134836751:
8460820419:
7455581657:
6134836751:
8460820419:
7455581657:
equations of motion in 2D (calculus) declare initial expr
1. 7252338326; locally 3936380:
$$v_y = \frac{dy}{dt}$$
$$pdg_{9107} = \frac{d}{d pdg_{1467}} pdg_{5647}$$
no validation is available for declarations 7252338326:
7252338326:
equations of motion in 2D (calculus) multiply both sides by
1. 8750379055; locally 8742281:
$$0 = \frac{d}{dt} v_x$$
$$0 = \frac{d}{d pdg_{1467}} pdg_{5505}$$
1. 8717193282:
$$dt$$
$$pdg_{4711}$$
1. 1166310428; locally 5239397:
$$0 dt = d v_x$$
$$0 = pdg_{5005}$$
LHS diff is 0 RHS diff is -pdg5005 8750379055:
1166310428:
8750379055:
1166310428:
equations of motion in 2D (calculus) assume N dimensions
1. 8880467139:
$$2$$
$$2$$
1. 5349866551; locally 5359560:
$$\vec{v} = v_x \hat{x} + v_y \hat{y}$$
$$pdg_{6373} = pdg_{1700} pdg_{9107} + pdg_{5505} pdg_{8339}$$
no validation is available for assumptions 5349866551:
5349866551:
equations of motion in 2D (calculus) multiply both sides by
1. 7455581657; locally 5123314:
$$v_{0, x} = \frac{dx}{dt}$$
$$pdg_{2958} = \frac{d}{d pdg_{1467}} pdg_{9199}$$
1. 8607458157:
$$dt$$
$$pdg_{4711}$$
1. 1963253044; locally 8062944:
$$v_{0, x} dt = dx$$
$$pdg_{2958} pdg_{4711} = pdg_{9199}$$
LHS diff is 0 RHS diff is -pdg9199 7455581657:
1963253044:
7455581657:
1963253044:
equations of motion in 2D (calculus) add X to both sides
1. 9973952056; locally 1321587:
$$-g t = v_y - v_{0, y}$$
$$- pdg_{1467} pdg_{1649} = - pdg_{5153} + pdg_{9431}$$
1. 4167526462:
$$v_{0, y}$$
$$pdg_{9431}$$
1. 6572039835; locally 2682139:
$$-g t + v_{0, y} = v_y$$
$$- pdg_{1467} pdg_{1649} + pdg_{9431} = pdg_{9107}$$
LHS diff is 0 RHS diff is -pdg5153 - pdg9107 + 2*pdg9431 9973952056:
6572039835:
9973952056:
6572039835:
equations of motion in 2D (calculus) substitute LHS of expr 1 into expr 2
1. 9707028061; locally 2060958:
$$a_x = 0$$
$$pdg_{7159} = 0$$
2. 1819663717; locally 5765841:
$$a_x = \frac{d}{dt} v_x$$
$$pdg_{7159} = \frac{d}{d pdg_{1467}} pdg_{5505}$$
1. 8750379055; locally 8742281:
$$0 = \frac{d}{dt} v_x$$
$$0 = \frac{d}{d pdg_{1467}} pdg_{5505}$$
valid 9707028061:
1819663717:
8750379055:
9707028061:
1819663717:
8750379055:
equations of motion in 2D (calculus) indefinite integration
1. 1963253044; locally 8062944:
$$v_{0, x} dt = dx$$
$$pdg_{2958} pdg_{4711} = pdg_{9199}$$
1. 3676159007; locally 2732393:
$$v_{0, x} \int dt = \int dx$$
$$pdg_{2958} \int 1\, dpdg_{1467} = \int 1\, dpdg_{1464}$$
no check performed 1963253044:
3676159007:
1963253044:
3676159007:
equations of motion in 2D (calculus) multiply both sides by
1. 7376526845; locally 2378061:
$$\sin(\theta) = \frac{v_{0, y}}{v_0}$$
$$\sin{\left(pdg_{1575} \right)} = \frac{pdg_{5153}}{pdg_{9431}}$$
1. 5620558729:
$$v_0$$
$$pdg_{5153}$$
1. 8949329361; locally 3041148:
$$v_0 \sin(\theta) = v_{0, y}$$
$$pdg_{5153} \sin{\left(pdg_{1575} \right)} = pdg_{9431}$$
LHS diff is 0 RHS diff is pdg5153**2/pdg9431 - pdg9431 7376526845:
8949329361:
7376526845:
8949329361:
equations of motion in 2D (calculus) swap LHS with RHS
1. 8486706976; locally 6277762:
$$v_{0, x} t + x_0 = x$$
$$pdg_{1467} pdg_{2958} + pdg_{1572} = pdg_{4037}$$
1. 1306360899; locally 3011802:
$$x = v_{0, x} t + x_0$$
$$pdg_{4037} = pdg_{1467} pdg_{2958} + pdg_{1572}$$
valid 8486706976:
1306360899:
8486706976:
1306360899:
equations of motion in 2D (calculus) substitute LHS of expr 1 into expr 2
1. 2741489181; locally 1439312:
$$a_y = -g$$
$$pdg_{7055} = - pdg_{1649}$$
2. 8228733125; locally 2080932:
$$a_y = \frac{d}{dt} v_y$$
$$pdg_{7055} = \frac{d}{d pdg_{1467}} pdg_{9107}$$
1. 1977955751; locally 3939933:
$$-g = \frac{d}{dt} v_y$$
$$- pdg_{1649} = \frac{d}{d pdg_{1467}} pdg_{9107}$$
valid 2741489181:
8228733125:
1977955751:
2741489181:
8228733125:
1977955751:
equations of motion in 2D (calculus) separate two vector components
1. 7729413831; locally 4904941:
$$a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)$$
$$pdg_{1700} pdg_{7055} + pdg_{7159} pdg_{8339} = \frac{\partial}{\partial pdg_{1467}} \left(pdg_{1700} pdg_{9107} + pdg_{5505} pdg_{8339}\right)$$
1. 1819663717; locally 5765841:
$$a_x = \frac{d}{dt} v_x$$
$$pdg_{7159} = \frac{d}{d pdg_{1467}} pdg_{5505}$$
2. 8228733125; locally 2080932:
$$a_y = \frac{d}{dt} v_y$$
$$pdg_{7055} = \frac{d}{d pdg_{1467}} pdg_{9107}$$
no check performed 7729413831:
1819663717:
8228733125:
7729413831:
1819663717:
8228733125:
equations of motion in 2D (calculus) multiply both sides by
1. 1977955751; locally 3939933:
$$-g = \frac{d}{dt} v_y$$
$$- pdg_{1649} = \frac{d}{d pdg_{1467}} pdg_{9107}$$
1. 6672141531:
$$dt$$
$$pdg_{4711}$$
1. 1702349646; locally 4777195:
$$-g dt = d v_y$$
$$- dt pdg_{1649} = pdg_{5674}$$
LHS diff is pdg1649*(dt - pdg4711) RHS diff is -pdg5674 1977955751:
1702349646:
1977955751:
1702349646:
equations of motion in 2D (calculus) substitute LHS of expr 1 into expr 2
1. 6083821265; locally 6010171:
$$v_0 \cos(\theta) = v_{0, x}$$
$$pdg_{5153} \cos{\left(pdg_{1575} \right)} = pdg_{2958}$$
2. 1306360899; locally 3011802:
$$x = v_{0, x} t + x_0$$
$$pdg_{4037} = pdg_{1467} pdg_{2958} + pdg_{1572}$$
1. 5438722682; locally 6795282:
$$x = v_0 t \cos(\theta) + x_0$$
$$pdg_{4037} = pdg_{1467} pdg_{5153} \cos{\left(pdg_{1575} \right)} + pdg_{1572}$$
LHS diff is 0 RHS diff is pdg1467*(pdg2958 - pdg5153*cos(pdg1575)) 6083821265:
1306360899:
5438722682:
6083821265:
1306360899:
5438722682:
equations of motion in 2D (calculus) indefinite integration
1. 1702349646; locally 4777195:
$$-g dt = d v_y$$
$$- dt pdg_{1649} = pdg_{5674}$$
1. 8584698994; locally 3366698:
$$-g \int dt = \int d v_y$$
$$- dt g = pdg_{5674}$$
no check performed 1702349646:
8584698994:
1702349646:
8584698994:
equations of motion in 2D (calculus) swap LHS with RHS
1. 2461349007; locally 7541692:
$$- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y$$
$$- \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{9431} + pdg_{1469} = pdg_{5647}$$
1. 1405465835; locally 1910429:
$$y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0$$
$$pdg_{5647} = - \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{9107} + pdg_{1469}$$
LHS diff is pdg1467*(-pdg9107 + pdg9431) RHS diff is pdg1467*(-pdg9107 + pdg9431) 2461349007:
1405465835:
2461349007:
1405465835:
equations of motion in 2D (calculus) add X to both sides
1. 2858549874; locally 8638087:
$$- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0$$
$$- \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{9431} = - pdg_{1469} + pdg_{5647}$$
1. 6098638221:
$$y_0$$
$$pdg_{1469}$$
1. 2461349007; locally 7541692:
$$- \frac{1}{2} g t^2 + v_{0, y} t + y_0 = y$$
$$- \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{9431} + pdg_{1469} = pdg_{5647}$$
valid 2858549874:
2461349007:
2858549874:
2461349007:
equations of motion in 2D (calculus) simplify
1. 8584698994; locally 3366698:
$$-g \int dt = \int d v_y$$
$$- dt g = pdg_{5674}$$
1. 9973952056; locally 1321587:
$$-g t = v_y - v_{0, y}$$
$$- pdg_{1467} pdg_{1649} = - pdg_{5153} + pdg_{9431}$$
LHS diff is -dt*g + pdg1467*pdg1649 RHS diff is pdg5153 + pdg5674 - pdg9431 8584698994:
9973952056:
8584698994:
9973952056:
equations of motion in 2D (calculus) declare assumption
1. 9707028061; locally 2060958:
$$a_x = 0$$
$$pdg_{7159} = 0$$
no validation is available for declarations 9707028061:
9707028061:
define the orientation of the coordinate system with respect to the gravitational acceleration such that x axis is perpendicular to gravity
equations of motion in 2D (calculus) declare assumption
1. 2741489181; locally 1439312:
$$a_y = -g$$
$$pdg_{7055} = - pdg_{1649}$$
no validation is available for declarations 2741489181:
2741489181:
define the orientation of the coordinate system with respect to the gravitational acceleration such that y axis is parallel to gravity
equations of motion in 2D (calculus) multiply both sides by
1. 7391837535; locally 5523081:
$$\cos(\theta) = \frac{v_{0, x}}{v_0}$$
$$\cos{\left(pdg_{1575} \right)} = \frac{pdg_{5153}}{pdg_{2958}}$$
1. 5868731041:
$$v_0$$
$$pdg_{5153}$$
1. 6083821265; locally 6010171:
$$v_0 \cos(\theta) = v_{0, x}$$
$$pdg_{5153} \cos{\left(pdg_{1575} \right)} = pdg_{2958}$$
LHS diff is 0 RHS diff is -pdg2958 + pdg5153**2/pdg2958 7391837535:
6083821265:
7391837535:
6083821265:
equations of motion in 2D (calculus) add X to both sides
1. 9882526611; locally 2740672:
$$v_{0, x} t = x - x_0$$
$$pdg_{1467} pdg_{2958} = - pdg_{1572} + pdg_{4037}$$
1. 3182907803:
$$x_0$$
$$pdg_{1572}$$
1. 8486706976; locally 6277762:
$$v_{0, x} t + x_0 = x$$
$$pdg_{1467} pdg_{2958} + pdg_{1572} = pdg_{4037}$$
valid 9882526611:
8486706976:
9882526611:
8486706976:
equations of motion in 2D (calculus) substitute LHS of expr 1 into expr 2
1. 5349866551; locally 5359560:
$$\vec{v} = v_x \hat{x} + v_y \hat{y}$$
$$pdg_{6373} = pdg_{1700} pdg_{9107} + pdg_{5505} pdg_{8339}$$
2. 4158986868; locally 4755350:
$$a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}$$
$$pdg_{1467}$$
1. 7729413831; locally 4904941:
$$a_x \hat{x} + a_y \hat{y} = \frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)$$
$$pdg_{1700} pdg_{7055} + pdg_{7159} pdg_{8339} = \frac{\partial}{\partial pdg_{1467}} \left(pdg_{1700} pdg_{9107} + pdg_{5505} pdg_{8339}\right)$$
Nothing to split 5349866551:
4158986868:
7729413831:
5349866551:
4158986868:
7729413831:
equations of motion in 2D (calculus) indefinite integration
1. 1166310428; locally 5239397:
$$0 dt = d v_x$$
$$0 = pdg_{5005}$$
1. 2366691988; locally 3137944:
$$\int 0 dt = \int d v_x$$
$$\int 0\, dpdg_{1467} = \int 1\, dpdg_{5005}$$
no check performed 1166310428:
2366691988:
1166310428:
2366691988:
equations of motion in 2D (calculus) declare final expr
1. 9862900242; locally 9780510:
$$y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0$$
$$pdg_{5647} = - \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{5153} \sin{\left(pdg_{1575} \right)} + pdg_{1469}$$
no validation is available for declarations 9862900242:
9862900242:
equations of motion in 2D (calculus) declare final expr
1. 5438722682; locally 6795282:
$$x = v_0 t \cos(\theta) + x_0$$
$$pdg_{4037} = pdg_{1467} pdg_{5153} \cos{\left(pdg_{1575} \right)} + pdg_{1572}$$
no validation is available for declarations 5438722682:
5438722682:
equations of motion in 2D (calculus) assume N dimensions
1. 3270039798:
$$2$$
$$2$$
1. 8602512487; locally 4862823:
$$\vec{a} = a_x \hat{x} + a_y \hat{y}$$
$$pdg_{2423} = pdg_{1700} pdg_{7055} + pdg_{7159} pdg_{8339}$$
no validation is available for assumptions 8602512487:
8602512487:
equations of motion in 2D (calculus) substitute LHS of expr 1 into expr 2
1. 7252338326; locally 3936380:
$$v_y = \frac{dy}{dt}$$
$$pdg_{9107} = \frac{d}{d pdg_{1467}} pdg_{5647}$$
2. 6572039835; locally 2682139:
$$-g t + v_{0, y} = v_y$$
$$- pdg_{1467} pdg_{1649} + pdg_{9431} = pdg_{9107}$$
1. 6204539227; locally 5010170:
$$-g t + v_{0, y} = \frac{dy}{dt}$$
$$- pdg_{1467} pdg_{6277} + pdg_{9431} = \frac{d}{d pdg_{1467}} pdg_{5647}$$
LHS diff is pdg1467*(-pdg1649 + pdg6277) RHS diff is 0 7252338326:
6572039835:
6204539227:
7252338326:
6572039835:
6204539227:
equations of motion in 2D (calculus) declare initial expr
1. 8460820419; locally 4895553:
$$v_x = \frac{dx}{dt}$$
$$pdg_{5505} = \frac{d}{d pdg_{1467}} pdg_{9199}$$
no validation is available for declarations 8460820419:
8460820419:
equations of motion in 2D (calculus) simplify
1. 2366691988; locally 3137944:
$$\int 0 dt = \int d v_x$$
$$\int 0\, dpdg_{1467} = \int 1\, dpdg_{5005}$$
1. 1676472948; locally 9737190:
$$0 = v_x - v_{0, x}$$
$$0 = - pdg_{2958} + pdg_{5505}$$
LHS diff is 0 RHS diff is pdg2958 + pdg5005 - pdg5505 2366691988:
1676472948:
2366691988:
1676472948:
equations of motion in 2D (calculus) substitute LHS of expr 1 into expr 2
1. 3169580383; locally 6758737:
$$\vec{a} = \frac{d\vec{v}}{dt}$$
$$pdg_{2423} = \frac{d}{d pdg_{1467}} pdg_{6373}$$
2. 8602512487; locally 4862823:
$$\vec{a} = a_x \hat{x} + a_y \hat{y}$$
$$pdg_{2423} = pdg_{1700} pdg_{7055} + pdg_{7159} pdg_{8339}$$
1. 4158986868; locally 4755350:
$$a_x \hat{x} + a_y \hat{y} = \frac{d\vec{v}}{dt}$$
$$pdg_{1467}$$
Nothing to split 3169580383:
8602512487:
4158986868:
3169580383:
8602512487:
4158986868:
equations of motion in 2D (calculus) indefinite integration
1. 8145337879; locally 5577963:
$$-g t dt + v_{0, y} dt = dy$$
$$- pdg_{1467} pdg_{1649} pdg_{4711} + pdg_{4711} pdg_{9431} = pdg_{5842}$$
1. 8808860551; locally 8020644:
$$-g \int t dt + v_{0, y} \int dt = \int dy$$
$$- pdg_{1649} \int pdg_{1467}\, dpdg_{1467} + pdg_{9431} \int 1\, dpdg_{1467} = \int 1\, dpdg_{5647}$$
no check performed 8145337879:
8808860551:
8145337879:
8808860551:
equations of motion in 2D (calculus) separate vector into two trigonometric ratios
1. 9341391925; locally 1381925:
$$\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}$$
$$pdg_{6091} = pdg_{1700} pdg_{9431} + pdg_{2958} pdg_{8339}$$
1. 6410818363:
$$\theta$$
$$pdg_{1575}$$
1. 7391837535; locally 5523081:
$$\cos(\theta) = \frac{v_{0, x}}{v_0}$$
$$\cos{\left(pdg_{1575} \right)} = \frac{pdg_{5153}}{pdg_{2958}}$$
2. 7376526845; locally 2378061:
$$\sin(\theta) = \frac{v_{0, y}}{v_0}$$
$$\sin{\left(pdg_{1575} \right)} = \frac{pdg_{5153}}{pdg_{9431}}$$
no check performed 9341391925:
7391837535:
7376526845:
9341391925:
7391837535:
7376526845:
equations of motion in 2D (calculus) substitute LHS of expr 1 into expr 2
1. 8949329361; locally 3041148:
$$v_0 \sin(\theta) = v_{0, y}$$
$$pdg_{5153} \sin{\left(pdg_{1575} \right)} = pdg_{9431}$$
2. 1405465835; locally 1910429:
$$y = - \frac{1}{2} g t^2 + v_{0, y} t + y_0$$
$$pdg_{5647} = - \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{9107} + pdg_{1469}$$
1. 9862900242; locally 9780510:
$$y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0$$
$$pdg_{5647} = - \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{5153} \sin{\left(pdg_{1575} \right)} + pdg_{1469}$$
LHS diff is 0 RHS diff is pdg1467*(-pdg5153*sin(pdg1575) + pdg9107) 8949329361:
1405465835:
9862900242:
8949329361:
1405465835:
9862900242:
equations of motion in 2D (calculus) simplify
1. 8808860551; locally 8020644:
$$-g \int t dt + v_{0, y} \int dt = \int dy$$
$$- pdg_{1649} \int pdg_{1467}\, dpdg_{1467} + pdg_{9431} \int 1\, dpdg_{1467} = \int 1\, dpdg_{5647}$$
1. 2858549874; locally 8638087:
$$- \frac{1}{2} g t^2 + v_{0, y} t = y - y_0$$
$$- \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{9431} = - pdg_{1469} + pdg_{5647}$$
LHS diff is 0 RHS diff is pdg1469 8808860551:
2858549874:
8808860551:
2858549874:
equations of motion in 2D (calculus) simplify
1. 3676159007; locally 2732393:
$$v_{0, x} \int dt = \int dx$$
$$pdg_{2958} \int 1\, dpdg_{1467} = \int 1\, dpdg_{1464}$$
1. 9882526611; locally 2740672:
$$v_{0, x} t = x - x_0$$
$$pdg_{1467} pdg_{2958} = - pdg_{1572} + pdg_{4037}$$
LHS diff is 0 RHS diff is pdg1464 + pdg1572 - pdg4037 3676159007:
9882526611:
3676159007:
9882526611:
equations of motion in 2D (calculus) assume N dimensions
1. 7049769409:
$$2$$
$$2$$
1. 9341391925; locally 1381925:
$$\vec{v}_0 = v_{0, x} \hat{x} + v_{0, y} \hat{y}$$
$$pdg_{6091} = pdg_{1700} pdg_{9431} + pdg_{2958} pdg_{8339}$$
no validation is available for assumptions 9341391925:
9341391925:
equations of motion in 2D (calculus) multiply both sides by
1. 6204539227; locally 5010170:
$$-g t + v_{0, y} = \frac{dy}{dt}$$
$$- pdg_{1467} pdg_{6277} + pdg_{9431} = \frac{d}{d pdg_{1467}} pdg_{5647}$$
1. 1614343171:
$$dt$$
$$pdg_{4711}$$
1. 8145337879; locally 5577963:
$$-g t dt + v_{0, y} dt = dy$$
$$- pdg_{1467} pdg_{1649} pdg_{4711} + pdg_{4711} pdg_{9431} = pdg_{5842}$$
LHS diff is pdg1467*pdg4711*(pdg1649 - pdg6277) RHS diff is -pdg5842 6204539227:
8145337879:
6204539227:
8145337879:
equations of motion in 2D (calculus) declare initial expr
1. 3169580383; locally 6758737:
$$\vec{a} = \frac{d\vec{v}}{dt}$$
$$pdg_{2423} = \frac{d}{d pdg_{1467}} pdg_{6373}$$
no validation is available for declarations 3169580383:
3169580383:
equations of motion in 2D (calculus) add X to both sides
1. 1676472948; locally 9737190:
$$0 = v_x - v_{0, x}$$
$$0 = - pdg_{2958} + pdg_{5505}$$
1. 1439089569:
$$v_{0, x}$$
$$pdg_{2958}$$
1. 6134836751; locally 8435615:
$$v_{0, x} = v_x$$
$$pdg_{2958} = pdg_{5505}$$
valid 1676472948:
6134836751:
1676472948:
6134836751:
angle of maximum distance for projectile motion divide both sides by
1. 1087417579; locally 7465542:
$$0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)$$
$$0 = - \frac{pdg_{1649} pdg_{2467}^{2}}{2} + pdg_{2467} pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
1. 4829590294:
$$t_f$$
$$pdg_{2467}$$
1. 2086924031; locally 5115586:
$$0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)$$
$$0 = - \frac{pdg_{1649} pdg_{2467}}{2} + pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
valid 1087417579:
2086924031:
1087417579:
2086924031:
angle of maximum distance for projectile motion LHS of expr 1 equals LHS of expr 2
1. 5379546684; locally 8592617:
$$y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0$$
$$pdg_{7092} = pdg_{1469} - \frac{pdg_{1649} pdg_{2467}^{2}}{2} + pdg_{2467} pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
2. 9112191201; locally 4911015:
$$y_f = 0$$
$$pdg_{7092} = 0$$
1. 8198310977; locally 7336772:
$$0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0$$
$$0 = pdg_{1469} - \frac{pdg_{1649} pdg_{2467}^{2}}{2} + pdg_{2467} pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
input diff is 0 diff is -pdg1469 + pdg1649*pdg2467**2/2 - pdg2467*pdg5153*sin(pdg1575) diff is pdg1469 - pdg1649*pdg2467**2/2 + pdg2467*pdg5153*sin(pdg1575) 5379546684:
9112191201:
8198310977:
5379546684:
9112191201:
8198310977:
angle of maximum distance for projectile motion declare final expr
1. 5353282496; locally 6972103:
$$d = \frac{v_0^2}{g}$$
$$pdg_{1943} = \frac{pdg_{5153}^{2}}{pdg_{1649}}$$
no validation is available for declarations 5353282496:
5353282496:
angle of maximum distance for projectile motion declare initial expr
1. 2405307372; locally 6199255:
$$\sin(2 x) = 2 \sin(x) \cos(x)$$
$$\sin{\left(2 pdg_{1464} \right)} = 2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)}$$
no validation is available for declarations 2405307372:
2405307372:
angle of maximum distance for projectile motion declare initial expr
1. 5438722682; locally 2022953:
$$x = v_0 t \cos(\theta) + x_0$$
$$pdg_{4037} = pdg_{1467} pdg_{5153} \cos{\left(pdg_{1575} \right)} + pdg_{1572}$$
no validation is available for declarations 5438722682:
5438722682:
angle of maximum distance for projectile motion simplify
1. 3607070319; locally 9834994:
$$d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)$$
$$pdg_{1943} = \frac{pdg_{5153}^{2} \sin{\left(\frac{pdg_{3141}}{2} \right)}}{pdg_{1649}}$$
1. 5353282496; locally 6972103:
$$d = \frac{v_0^2}{g}$$
$$pdg_{1943} = \frac{pdg_{5153}^{2}}{pdg_{1649}}$$
LHS diff is 0 RHS diff is pdg5153**2*(sin(pdg3141/2) - 1)/pdg1649 3607070319:
5353282496:
3607070319:
5353282496:
angle of maximum distance for projectile motion boundary condition
1. 4370074654; locally 1654988:
$$t = t_f$$
$$pdg_{1467} = pdg_{2467}$$
1. 2378095808; locally 5891715:
$$x_f = x_0 + d$$
$$pdg_{3652} = pdg_{1572} + pdg_{1943}$$
no validation is available for assumptions 4370074654:
2378095808:
4370074654:
2378095808:
angle of maximum distance for projectile motion substitute LHS of expr 1 into expr 2
1. 2378095808; locally 5891715:
$$x_f = x_0 + d$$
$$pdg_{3652} = pdg_{1572} + pdg_{1943}$$
2. 3485125659; locally 2293278:
$$x_f = v_0 t_f \cos(\theta) + x_0$$
$$pdg_{3652} = pdg_{1572} + pdg_{2467} pdg_{5153} \cos{\left(pdg_{1575} \right)}$$
1. 4268085801; locally 6742208:
$$x_0 + d = v_0 t_f \cos(\theta) + x_0$$
$$pdg_{1572} + pdg_{1943} = pdg_{1572} + pdg_{2467} pdg_{5153} \cos{\left(pdg_{1575} \right)}$$
valid 2378095808:
3485125659:
4268085801:
2378095808:
3485125659:
4268085801:
angle of maximum distance for projectile motion subtract X from both sides
1. 4268085801; locally 6742208:
$$x_0 + d = v_0 t_f \cos(\theta) + x_0$$
$$pdg_{1572} + pdg_{1943} = pdg_{1572} + pdg_{2467} pdg_{5153} \cos{\left(pdg_{1575} \right)}$$
1. 8072682558:
$$x_0$$
$$pdg_{1572}$$
1. 7233558441; locally 6756414:
$$d = v_0 t_f \cos(\theta)$$
$$pdg_{1943} = pdg_{2467} pdg_{5153} \cos{\left(pdg_{1575} \right)}$$
valid 4268085801:
7233558441:
4268085801:
7233558441:
angle of maximum distance for projectile motion change two variables in expr
1. 5438722682; locally 2022953:
$$x = v_0 t \cos(\theta) + x_0$$
$$pdg_{4037} = pdg_{1467} pdg_{5153} \cos{\left(pdg_{1575} \right)} + pdg_{1572}$$
1. 3273630811:
$$x$$
$$pdg_{4037}$$
2. 5194141542:
$$x_f$$
$$pdg_{3652}$$
3. 6732786762:
$$t$$
$$pdg_{1467}$$
4. 6463266449:
$$t_f$$
$$pdg_{2467}$$
1. 3485125659; locally 2293278:
$$x_f = v_0 t_f \cos(\theta) + x_0$$
$$pdg_{3652} = pdg_{1572} + pdg_{2467} pdg_{5153} \cos{\left(pdg_{1575} \right)}$$
valid 5438722682:
3485125659:
5438722682:
3485125659:
angle of maximum distance for projectile motion change two variables in expr
1. 9862900242; locally 1292901:
$$y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0$$
$$pdg_{5647} = - \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{5153} \sin{\left(pdg_{1575} \right)} + pdg_{1469}$$
1. 8406170337:
$$y$$
$$pdg_{5647}$$
2. 8120663858:
$$y_f$$
$$pdg_{7092}$$
3. 2403773761:
$$t$$
$$pdg_{1467}$$
4. 4162188238:
$$t_f$$
$$pdg_{2467}$$
1. 5379546684; locally 8592617:
$$y_f = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0$$
$$pdg_{7092} = pdg_{1469} - \frac{pdg_{1649} pdg_{2467}^{2}}{2} + pdg_{2467} pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
valid 9862900242:
5379546684:
9862900242:
5379546684:
angle of maximum distance for projectile motion substitute LHS of expr 1 into expr 2
1. 8198310977; locally 7336772:
$$0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta) + y_0$$
$$0 = pdg_{1469} - \frac{pdg_{1649} pdg_{2467}^{2}}{2} + pdg_{2467} pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
2. 1650441634; locally 2601896:
$$y_0 = 0$$
$$pdg_{1469} = 0$$
1. 1087417579; locally 7465542:
$$0 = - \frac{1}{2} g t_f^2 + v_0 t_f \sin(\theta)$$
$$0 = - \frac{pdg_{1649} pdg_{2467}^{2}}{2} + pdg_{2467} pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
LHS diff is pdg1469 RHS diff is pdg1469 8198310977:
1650441634:
1087417579:
8198310977:
1650441634:
1087417579:
angle of maximum distance for projectile motion substitute LHS of expr 1 into expr 2
1. 2519058903; locally 7596368:
$$\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)$$
$$\sin{\left(2 pdg_{1575} \right)} = 2 \sin{\left(pdg_{1575} \right)} \cos{\left(pdg_{1575} \right)}$$
2. 2297105551; locally 4362314:
$$d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)$$
$$pdg_{1943} = \frac{2 pdg_{5153}^{2} \sin{\left(pdg_{1575} \right)} \cos{\left(pdg_{1575} \right)}}{pdg_{1649}}$$
1. 8922441655; locally 5129639:
$$d = \frac{v_0^2}{g} \sin(2 \theta)$$
$$pdg_{1943} = \frac{pdg_{5153}^{2} \sin{\left(2 pdg_{1575} \right)}}{pdg_{1649}}$$
valid 2519058903:
2297105551:
8922441655:
2519058903:
2297105551:
8922441655:
angle of maximum distance for projectile motion boundary condition
1. 5373931751; locally 7946350:
$$t = t_f$$
$$pdg_{1467} = pdg_{2467}$$
1. 9112191201; locally 4911015:
$$y_f = 0$$
$$pdg_{7092} = 0$$
no validation is available for assumptions 5373931751:
9112191201:
5373931751:
9112191201:
y(t_f) = y_f = 0
angle of maximum distance for projectile motion declare final expr
1. 1541916015; locally 2728170:
$$\theta = \frac{\pi}{4}$$
$$pdg_{1575} = \frac{pdg_{3141}}{4}$$
no validation is available for declarations 1541916015:
1541916015:
angle of maximum distance for projectile motion multiply both sides by
1. 1191796961; locally 3904454:
$$\frac{1}{2} g t_f = v_0 \sin(\theta)$$
$$\frac{pdg_{1649} pdg_{2467}}{2} = pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
1. 2510804451:
$$2/g$$
$$\frac{2}{pdg_{1649}}$$
1. 4778077984; locally 8982886:
$$t_f = \frac{2 v_0 \sin(\theta)}{g}$$
$$pdg_{2467} = \frac{2 pdg_{5153} \sin{\left(pdg_{1575} \right)}}{pdg_{1649}}$$
valid 1191796961:
4778077984:
1191796961:
4778077984:
angle of maximum distance for projectile motion substitute LHS of expr 1 into expr 2
1. 4778077984; locally 8982886:
$$t_f = \frac{2 v_0 \sin(\theta)}{g}$$
$$pdg_{2467} = \frac{2 pdg_{5153} \sin{\left(pdg_{1575} \right)}}{pdg_{1649}}$$
2. 7233558441; locally 6756414:
$$d = v_0 t_f \cos(\theta)$$
$$pdg_{1943} = pdg_{2467} pdg_{5153} \cos{\left(pdg_{1575} \right)}$$
1. 2297105551; locally 4362314:
$$d = v_0 \frac{2 v_0 \sin(\theta)}{g} \cos(\theta)$$
$$pdg_{1943} = \frac{2 pdg_{5153}^{2} \sin{\left(pdg_{1575} \right)} \cos{\left(pdg_{1575} \right)}}{pdg_{1649}}$$
valid 4778077984:
7233558441:
2297105551:
4778077984:
7233558441:
2297105551:
angle of maximum distance for projectile motion declare initial expr
1. 9862900242; locally 1292901:
$$y = - \frac{1}{2} g t^2 + v_0 t \sin(\theta) + y_0$$
$$pdg_{5647} = - \frac{pdg_{1467}^{2} pdg_{1649}}{2} + pdg_{1467} pdg_{5153} \sin{\left(pdg_{1575} \right)} + pdg_{1469}$$
no validation is available for declarations 9862900242:
9862900242:
angle of maximum distance for projectile motion add X to both sides
1. 2086924031; locally 5115586:
$$0 = - \frac{1}{2} g t_f + v_0 \sin(\theta)$$
$$0 = - \frac{pdg_{1649} pdg_{2467}}{2} + pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
1. 6974054946:
$$\frac{1}{2} g t_f$$
$$\frac{pdg_{1649} pdg_{2467}}{2}$$
1. 1191796961; locally 3904454:
$$\frac{1}{2} g t_f = v_0 \sin(\theta)$$
$$\frac{pdg_{1649} pdg_{2467}}{2} = pdg_{5153} \sin{\left(pdg_{1575} \right)}$$
valid 2086924031:
1191796961:
2086924031:
1191796961:
angle of maximum distance for projectile motion substitute LHS of expr 1 into expr 2
1. 1541916015; locally 2728170:
$$\theta = \frac{\pi}{4}$$
$$pdg_{1575} = \frac{pdg_{3141}}{4}$$
2. 8922441655; locally 5129639:
$$d = \frac{v_0^2}{g} \sin(2 \theta)$$
$$pdg_{1943} = \frac{pdg_{5153}^{2} \sin{\left(2 pdg_{1575} \right)}}{pdg_{1649}}$$
1. 3607070319; locally 9834994:
$$d = \frac{v_0^2}{g} \sin\left(2 \frac{\pi}{4}\right)$$
$$pdg_{1943} = \frac{pdg_{5153}^{2} \sin{\left(\frac{pdg_{3141}}{2} \right)}}{pdg_{1649}}$$
valid 1541916015:
8922441655:
3607070319:
1541916015:
8922441655:
3607070319:
angle of maximum distance for projectile motion declare assumption
1. 1650441634; locally 2601896:
$$y_0 = 0$$
$$pdg_{1469} = 0$$
no validation is available for declarations 1650441634:
1650441634:
angle of maximum distance for projectile motion change variable X to Y
1. 2405307372; locally 6199255:
$$\sin(2 x) = 2 \sin(x) \cos(x)$$
$$\sin{\left(2 pdg_{1464} \right)} = 2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)}$$
1. 7587034465:
$$\theta$$
$$pdg_{1575}$$
2. 7214442790:
$$x$$
$$pdg_{1464}$$
1. 2519058903; locally 7596368:
$$\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)$$
$$\sin{\left(2 pdg_{1575} \right)} = 2 \sin{\left(pdg_{1575} \right)} \cos{\left(pdg_{1575} \right)}$$
LHS diff is sin(2*pdg1464) - sin(2*pdg1575) RHS diff is sin(2*pdg1464) - sin(2*pdg1575) 2405307372:
2519058903:
2405307372:
2519058903:
angle of maximum distance for projectile motion maximum of expr
1. 8922441655; locally 5129639:
$$d = \frac{v_0^2}{g} \sin(2 \theta)$$
$$pdg_{1943} = \frac{pdg_{5153}^{2} \sin{\left(2 pdg_{1575} \right)}}{pdg_{1649}}$$
1. 5667870149:
$$\theta$$
$$pdg_{1575}$$
1. 1541916015; locally 2728170:
$$\theta = \frac{\pi}{4}$$
$$pdg_{1575} = \frac{pdg_{3141}}{4}$$
no check performed 8922441655:
1541916015:
8922441655:
1541916015:
Newton's Law of Gravitation substitute LHS of two expressions into expr
1. 4264859781; locally 8320848:
$$F \propto m_1$$
$$F pdg_{5022} propto$$
2. 4490788873; locally 5440061:
$$F \propto m_2$$
$$F pdg_{4851} propto$$
3. 1571582377; locally 6174613:
$$F_{gravitational} \propto \frac{1}{r^2}$$
$$pdg_{2867} = \frac{k}{pdg_{2530}^{2}}$$
1. 3650814381; locally 1206000:
$$F_{gravitational} \propto \frac{m_1 m_2}{r^2}$$
$$\frac{pdg_{2867} pdg_{4851} pdg_{5022} propto}{pdg_{2530}^{2}}$$
Nothing to split 4264859781:
4490788873:
1571582377:
3650814381:
4264859781:
4490788873:
1571582377:
3650814381:
Newton's Law of Gravitation substitute LHS of expr 1 into expr 2
1. 6026694087; locally 3755872:
$$F_{centripetal} = m \frac{v^2}{r}$$
$$pdg_{1687} = \frac{pdg_{5156} v^{2}}{pdg_{2530}}$$
2. 4820320578; locally 5891249:
$$F_{gravitational} = F_{centripetal}$$
$$pdg_{2867} = pdg_{1687}$$
1. 4267808354; locally 2239910:
$$F_{gravitational} = m \frac{v^2}{r}$$
$$pdg_{2867} = \frac{pdg_{1357}^{2} pdg_{5156}}{pdg_{2530}}$$
LHS diff is 0 RHS diff is pdg5156*(-pdg1357**2 + v**2)/pdg2530 6026694087:
4820320578:
4267808354:
6026694087:
4820320578:
4267808354:
Newton's Law of Gravitation declare initial expr
1. 6785303857; locally 5154120:
$$C = 2 \pi r$$
$$pdg_{3034} = 2 pdg_{2530} pdg_{3141}$$
no validation is available for declarations 6785303857:
6785303857:
Newton's Law of Gravitation declare initial expr
1. 3411994811; locally 9055493:
$$v_{\rm average} = \frac{d}{t}$$
$$pdg_{6709} = \frac{pdg_{1943}}{pdg_{1467}}$$
no validation is available for declarations 3411994811:
3411994811:
Newton's Law of Gravitation declare assumption
1. 4820320578; locally 5891249:
$$F_{gravitational} = F_{centripetal}$$
$$pdg_{2867} = pdg_{1687}$$
no validation is available for declarations 4820320578:
4820320578:
Newton's Law of Gravitation declare final expr
1. 1292735067; locally 8373934:
$$F_{gravitational} = G \frac{m_1 m_2}{r^2}$$
$$pdg_{2867} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{2530}^{2}}$$
no validation is available for declarations 1292735067:
1292735067:
Newton's Law of Gravitation substitute LHS of expr 1 into expr 2
1. 8361238989; locally 6969192:
$$a_{centripetal} = \frac{v^2}{r}$$
$$a_{c*(e*(n*(t*(r*(i*(p*(e*(t*(a*l)))))))))} = \frac{pdg_{1357}^{2}}{pdg_{2530}}$$
2. 5345738321; locally 2020292:
$$F = m a$$
$$pdg_{4202} = pdg_{5156} pdg_{9140}$$
1. 6026694087; locally 3755872:
$$F_{centripetal} = m \frac{v^2}{r}$$
$$pdg_{1687} = \frac{pdg_{5156} v^{2}}{pdg_{2530}}$$
LHS diff is -pdg1687 + pdg4202 RHS diff is pdg5156*(pdg2530*pdg9140 - v**2)/pdg2530 8361238989:
5345738321:
6026694087:
8361238989:
5345738321:
6026694087:
Newton's Law of Gravitation simplify
1. 3004158505; locally 4470678:
$$\frac{T^2}{r} F_{gravitational} = \left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r}$$
$$\frac{pdg_{2867} pdg_{8762}^{2}}{pdg_{2530}} = 4 pdg_{3141}^{2} pdg_{5156}$$
1. 3650370389; locally 7324555:
$$\frac{T^2}{r} F_{gravitational} = 4 \pi^2 m$$
$$\frac{pdg_{2867} pdg_{8762}^{2}}{pdg_{2530}} = 4 pdg_{3141}^{2} pdg_{5156}$$
valid 3004158505:
3650370389:
3004158505:
3650370389:
Newton's Law of Gravitation substitute LHS of expr 1 into expr 2
1. 6785303857; locally 5154120:
$$C = 2 \pi r$$
$$pdg_{3034} = 2 pdg_{2530} pdg_{3141}$$
2. 3411994811; locally 9055493:
$$v_{\rm average} = \frac{d}{t}$$
$$pdg_{6709} = \frac{pdg_{1943}}{pdg_{1467}}$$
1. 5177311762; locally 7653722:
$$v = \frac{2 \pi r}{T}$$
$$pdg_{1357} = \frac{2 pdg_{2530} pdg_{3141}}{pdg_{8762}}$$
LHS diff is -pdg1357 + pdg6709 RHS diff is -2*pdg2530*pdg3141/pdg8762 + pdg1943/pdg1467 6785303857:
3411994811:
5177311762:
6785303857:
3411994811:
5177311762:
Newton's Law of Gravitation change variable X to Y
1. 1848400430; locally 5546471:
$$F \propto m$$
$$F pdg_{5156} propto$$
1. 3876446703:
$$m$$
$$pdg_{5156}$$
2. 7905984866:
$$m_1$$
$$pdg_{5022}$$
1. 4264859781; locally 8320848:
$$F \propto m_1$$
$$F pdg_{5022} propto$$
Nothing to split 1848400430:
4264859781:
1848400430:
4264859781:
Newton's Law of Gravitation simplify
1. 3650814381; locally 1206000:
$$F_{gravitational} \propto \frac{m_1 m_2}{r^2}$$
$$\frac{pdg_{2867} pdg_{4851} pdg_{5022} propto}{pdg_{2530}^{2}}$$
1. 1292735067; locally 8373934:
$$F_{gravitational} = G \frac{m_1 m_2}{r^2}$$
$$pdg_{2867} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{2530}^{2}}$$
Nothing to split 3650814381:
1292735067:
3650814381:
1292735067:
Newton's Law of Gravitation simplify
1. 6268336290; locally 9170078:
$$F_{gravitational} = \frac{m}{r}\left(\frac{2\pi r}{T}\right)^2$$
$$pdg_{2867} = \frac{4 pdg_{2530} pdg_{3141}^{2} pdg_{4851}}{pdg_{8762}^{2}}$$
1. 7672365885; locally 5175707:
$$F_{gravitational} = \frac{4 \pi^2 m r}{T^2}$$
$$pdg_{2867} = \frac{4 pdg_{2530} pdg_{3141}^{2} pdg_{4851}}{pdg_{8762}^{2}}$$
valid 6268336290:
7672365885:
6268336290:
7672365885:
Newton's Law of Gravitation multiply both sides by
1. 7672365885; locally 5175707:
$$F_{gravitational} = \frac{4 \pi^2 m r}{T^2}$$
$$pdg_{2867} = \frac{4 pdg_{2530} pdg_{3141}^{2} pdg_{4851}}{pdg_{8762}^{2}}$$
1. 3448601530:
$$\frac{T^2}{r}$$
$$\frac{pdg_{9491}^{2}}{pdg_{2530}}$$
1. 3004158505; locally 4470678:
$$\frac{T^2}{r} F_{gravitational} = \left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r}$$
$$\frac{pdg_{2867} pdg_{8762}^{2}}{pdg_{2530}} = 4 pdg_{3141}^{2} pdg_{5156}$$
LHS diff is pdg2867*(-pdg8762**2 + pdg9491**2)/pdg2530 RHS diff is 4*pdg3141**2*(pdg4851*pdg9491**2 - pdg5156*pdg8762**2)/pdg8762**2 7672365885:
3004158505:
7672365885:
3004158505:
Newton's Law of Gravitation declare initial expr
1. 5345738321; locally 2020292:
$$F = m a$$
$$pdg_{4202} = pdg_{5156} pdg_{9140}$$
no validation is available for declarations 5345738321:
5345738321:
Newton's Law of Gravitation declare initial expr
1. 8361238989; locally 6969192:
$$a_{centripetal} = \frac{v^2}{r}$$
$$a_{c*(e*(n*(t*(r*(i*(p*(e*(t*(a*l)))))))))} = \frac{pdg_{1357}^{2}}{pdg_{2530}}$$
no validation is available for declarations 8361238989:
8361238989:
Newton's Law of Gravitation change variable X to Y
1. 1848400430; locally 5546471:
$$F \propto m$$
$$F pdg_{5156} propto$$
1. 2346952973:
$$m$$
$$pdg_{5156}$$
2. 9594072504:
$$m_2$$
$$pdg_{4851}$$
1. 4490788873; locally 5440061:
$$F \propto m_2$$
$$F pdg_{4851} propto$$
Nothing to split 1848400430:
4490788873:
1848400430:
4490788873:
Newton's Law of Gravitation substitute LHS of expr 1 into expr 2
1. 5177311762; locally 7653722:
$$v = \frac{2 \pi r}{T}$$
$$pdg_{1357} = \frac{2 pdg_{2530} pdg_{3141}}{pdg_{8762}}$$
2. 4267808354; locally 2239910:
$$F_{gravitational} = m \frac{v^2}{r}$$
$$pdg_{2867} = \frac{pdg_{1357}^{2} pdg_{5156}}{pdg_{2530}}$$
1. 6268336290; locally 9170078:
$$F_{gravitational} = \frac{m}{r}\left(\frac{2\pi r}{T}\right)^2$$
$$pdg_{2867} = \frac{4 pdg_{2530} pdg_{3141}^{2} pdg_{4851}}{pdg_{8762}^{2}}$$
LHS diff is 0 RHS diff is 4*pdg2530*pdg3141**2*(-pdg4851 + pdg5156)/pdg8762**2 5177311762:
4267808354:
6268336290:
5177311762:
4267808354:
6268336290:
Newton's Law of Gravitation declare guess solution
1. 3650370389; locally 7324555:
$$\frac{T^2}{r} F_{gravitational} = 4 \pi^2 m$$
$$\frac{pdg_{2867} pdg_{8762}^{2}}{pdg_{2530}} = 4 pdg_{3141}^{2} pdg_{5156}$$
1. 1571582377; locally 6174613:
$$F_{gravitational} \propto \frac{1}{r^2}$$
$$pdg_{2867} = \frac{k}{pdg_{2530}^{2}}$$
no validation is available for declarations 3650370389:
1571582377:
3650370389:
1571582377:
this is a big leap of logic that is consistent with Kepler's third law of motion
Newton's Law of Gravitation simplify
1. 5345738321; locally 2020292:
$$F = m a$$
$$pdg_{4202} = pdg_{5156} pdg_{9140}$$
1. 1848400430; locally 5546471:
$$F \propto m$$
$$F pdg_{5156} propto$$
Nothing to split 5345738321:
1848400430:
5345738321:
1848400430:
radius for satellite in geostationary orbit change four variables in expr
1. 9226945488; locally 8242154:
$$F = \frac{m v^2}{r}$$
$$pdg_{4202} = \frac{pdg_{1357}^{2} pdg_{5156}}{pdg_{2530}}$$
1. 5089196493:
$$F$$
$$pdg_{4202}$$
2. 1333474099:
$$F_{\rm centripetal}$$
$$pdg_{1687}$$
3. 3342155559:
$$m$$
$$pdg_{5156}$$
4. 2114570475:
$$m_{\rm satellite}$$
$$pdg_{3569}$$
5. 7912578203:
$$v$$
$$pdg_{1357}$$
6. 9789485295:
$$v_{\rm satellite}$$
$$pdg_{4082}$$
1. 4627284246; locally 6845877:
$$F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r}$$
$$pdg_{1687} = \frac{pdg_{3569} pdg_{4082}^{2}}{pdg_{2530}}$$
failed 9226945488:
4627284246:
9226945488:
4627284246:
radius for satellite in geostationary orbit multiply both sides by
1. 3906710072; locally 2871066:
$$G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}$$
$$\frac{pdg_{5458} pdg_{6277}}{pdg_{2530}} = \frac{4 pdg_{2530}^{2} pdg_{3141}^{2}}{pdg_{8762}^{2}}$$
1. 6238632840:
$$r T_{\rm orbit}^2$$
$$pdg_{2530} pdg_{8762}^{2}$$
1. 7010294143; locally 7188516:
$$T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3$$
$$pdg_{5458} pdg_{6277} pdg_{8762}^{2} = 4 pdg_{2530}^{3} pdg_{3141}^{2}$$
valid 3906710072:
7010294143:
3906710072:
7010294143:
radius for satellite in geostationary orbit raise both sides to power
1. 4858693811; locally 6238570:
$$\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3$$
$$\frac{pdg_{5458} pdg_{6277} pdg_{8762}^{2}}{4 pdg_{3141}^{2}} = pdg_{2530}^{3}$$
1. 4319544433:
$$1/3$$
$$\frac{1}{3}$$
1. 2617541067; locally 7139326:
$$\left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r$$
$$\frac{\sqrt[3]{2} \sqrt[3]{\frac{pdg_{5458} pdg_{6277} pdg_{8762}^{2}}{pdg_{3141}^{2}}}}{2} = pdg_{2530}$$
no check is performed 4858693811:
2617541067:
4858693811:
2617541067:
radius for satellite in geostationary orbit divide both sides by
1. 4072200527; locally 4948724:
$$\frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}$$
$$\frac{pdg_{3569} pdg_{4082}^{2}}{pdg_{2530}} = \frac{pdg_{3569} pdg_{5458} pdg_{6277}}{pdg_{2530}^{2}}$$
1. 5359471792:
$$\frac{m_{\rm satellite}}{r}$$
$$\frac{pdg_{3569}}{pdg_{2530}}$$
1. 1994296484; locally 2009493:
$$v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r}$$
$$pdg_{4082}^{2} = \frac{pdg_{5458} pdg_{6277}}{pdg_{2530}}$$
valid 4072200527:
1994296484:
4072200527:
1994296484:
radius for satellite in geostationary orbit change two variables in expr
1. 2617541067; locally 7139326:
$$\left(\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r$$
$$\frac{\sqrt[3]{2} \sqrt[3]{\frac{pdg_{5458} pdg_{6277} pdg_{8762}^{2}}{pdg_{3141}^{2}}}}{2} = pdg_{2530}$$
1. 3846345263:
$$T_{\rm orbit}$$
$$pdg_{8762}$$
2. 5208737840:
$$T_{\rm geostationary\ orbit}$$
$$pdg_{5595}$$
3. 5770088141:
$$r$$
$$pdg_{2530}$$
4. 7053449926:
$$r_{\rm geostationary\ orbit}$$
$$pdg_{7110}$$
1. 1559688463; locally 4507350:
$$\left(\frac{T_{\rm geostationary\ orbit}^2 G m_{\rm Earth}}{4 \pi^2}\right)^{1/3} = r_{\rm geostationary\ orbit}$$
$$\frac{\sqrt[3]{2} \sqrt[3]{\frac{pdg_{5458} pdg_{5595}^{2} pdg_{6277}}{pdg_{3141}^{2}}}}{2} = pdg_{7110}$$
valid 2617541067:
1559688463:
2617541067:
1559688463:
radius for satellite in geostationary orbit substitute LHS of expr 1 into expr 2
1. 9262596735; locally 5369477:
$$d = 2 \pi r$$
$$pdg_{1943} = 2 pdg_{2530} pdg_{3141}$$
2. 5426308937; locally 5114041:
$$v = \frac{d}{t}$$
$$pdg_{1357} = \frac{pdg_{1943}}{pdg_{1467}}$$
1. 4245712581; locally 8090893:
$$v = \frac{2 \pi r}{t}$$
$$pdg_{1357} = \frac{2 pdg_{2530} pdg_{3141}}{pdg_{1467}}$$
valid 9262596735:
5426308937:
4245712581:
9262596735:
5426308937:
4245712581:
radius for satellite in geostationary orbit change four variables in expr
1. 6935745841; locally 2820438:
$$F = G \frac{m_1 m_2}{x^2}$$
$$pdg_{4202} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{4037}^{2}}$$
1. 3398368564:
$$F$$
$$pdg_{4202}$$
2. 3594626260:
$$F_{\rm gravity}$$
$$pdg_{2867}$$
3. 9794128647:
$$m_1$$
$$pdg_{5458}$$
4. 4153613253:
$$m_{\rm Earth}$$
$$pdg_{5458}$$
5. 3088463019:
$$m_2$$
$$pdg_{4851}$$
6. 3486213448:
$$m_{\rm satellite}$$
$$pdg_{3569}$$
7. 4830480629:
$$x$$
$$pdg_{4037}$$
8. 7819443873:
$$r$$
$$pdg_{2530}$$
1. 5563580265; locally 1917654:
$$F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}$$
$$pdg_{2867} = \frac{pdg_{3569} pdg_{5458} pdg_{6277}}{pdg_{2530}^{2}}$$
LHS diff is 0 RHS diff is pdg3569*pdg6277*(pdg5022 - pdg5458)/pdg2530**2 6935745841:
5563580265:
6935745841:
5563580265:
radius for satellite in geostationary orbit change variable X to Y
1. 4245712581; locally 8090893:
$$v = \frac{2 \pi r}{t}$$
$$pdg_{1357} = \frac{2 pdg_{2530} pdg_{3141}}{pdg_{1467}}$$
1. 3722461713:
$$t$$
$$pdg_{1467}$$
2. 9346215480:
$$T_{\rm orbit}$$
$$pdg_{8762}$$
1. 3614055652; locally 2392562:
$$v = \frac{2 \pi r}{T_{\rm orbit}}$$
$$pdg_{1357} = \frac{2 pdg_{2530} pdg_{3141}}{pdg_{8762}}$$
valid 4245712581:
3614055652:
4245712581:
3614055652:
radius for satellite in geostationary orbit raise both sides to power
1. 3614055652; locally 2392562:
$$v = \frac{2 \pi r}{T_{\rm orbit}}$$
$$pdg_{1357} = \frac{2 pdg_{2530} pdg_{3141}}{pdg_{8762}}$$
1. 2754264786:
$$2$$
$$2$$
1. 8059639673; locally 6390693:
$$v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}$$
$$pdg_{1357}^{2} = \frac{4 pdg_{2530}^{2} pdg_{3141}^{2}}{pdg_{8762}^{2}}$$
no check is performed 3614055652:
8059639673:
3614055652:
8059639673:
radius for satellite in geostationary orbit LHS of expr 1 equals LHS of expr 2
1. 1994296484; locally 2009493:
$$v_{\rm satellite}^2 = G \frac{m_{\rm Earth}}{r}$$
$$pdg_{4082}^{2} = \frac{pdg_{5458} pdg_{6277}}{pdg_{2530}}$$
2. 8059639673; locally 6390693:
$$v^2 = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}$$
$$pdg_{1357}^{2} = \frac{4 pdg_{2530}^{2} pdg_{3141}^{2}}{pdg_{8762}^{2}}$$
1. 3906710072; locally 2871066:
$$G \frac{m_{\rm Earth}}{r} = \frac{4 \pi^2 r^2}{T_{\rm orbit}^2}$$
$$\frac{pdg_{5458} pdg_{6277}}{pdg_{2530}} = \frac{4 pdg_{2530}^{2} pdg_{3141}^{2}}{pdg_{8762}^{2}}$$
input diff is -pdg1357**2 + pdg4082**2 diff is 0 diff is 0 1994296484:
8059639673:
3906710072:
1994296484:
8059639673:
3906710072:
radius for satellite in geostationary orbit declare assumption
1. 3920616792; locally 9978909:
$$T_{\rm geostationary orbit} = 24\ {\rm hours}$$
$$pdg_{5595}$$
no validation is available for declarations 3920616792:
3920616792:
radius for satellite in geostationary orbit substitute LHS of two expressions into expr
1. 5563580265; locally 1917654:
$$F_{\rm gravity} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}$$
$$pdg_{2867} = \frac{pdg_{3569} pdg_{5458} pdg_{6277}}{pdg_{2530}^{2}}$$
2. 4627284246; locally 6845877:
$$F_{\rm centripetal} = \frac{m_{\rm satellite} v_{\rm satellite}^2}{r}$$
$$pdg_{1687} = \frac{pdg_{3569} pdg_{4082}^{2}}{pdg_{2530}}$$
3. 3176662571; locally 2154616:
$$F_{\rm centripetal} = F_{\rm gravity}$$
$$pdg_{2867} = pdg_{1687}$$
1. 4072200527; locally 4948724:
$$\frac{m_{\rm satellite} v_{\rm satellite}^2}{r} = G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}$$
$$\frac{pdg_{3569} pdg_{4082}^{2}}{pdg_{2530}} = \frac{pdg_{3569} pdg_{5458} pdg_{6277}}{pdg_{2530}^{2}}$$
failed 5563580265:
4627284246:
3176662571:
4072200527:
5563580265:
4627284246:
3176662571:
4072200527:
radius for satellite in geostationary orbit divide both sides by
1. 7010294143; locally 7188516:
$$T_{\rm orbit}^2 G m_{\rm Earth} = 4 \pi^2 r^3$$
$$pdg_{5458} pdg_{6277} pdg_{8762}^{2} = 4 pdg_{2530}^{3} pdg_{3141}^{2}$$
1. 7556442438:
$$4 \pi^2$$
$$4 pdg_{3141}^{2}$$
1. 4858693811; locally 6238570:
$$\frac{T_{\rm orbit}^2 G m_{\rm Earth}}{4 \pi^2} = r^3$$
$$\frac{pdg_{5458} pdg_{6277} pdg_{8762}^{2}}{4 pdg_{3141}^{2}} = pdg_{2530}^{3}$$
valid 7010294143:
4858693811:
7010294143:
4858693811:
radius for satellite in geostationary orbit declare initial expr
1. 9226945488; locally 8242154:
$$F = \frac{m v^2}{r}$$
$$pdg_{4202} = \frac{pdg_{1357}^{2} pdg_{5156}}{pdg_{2530}}$$
no validation is available for declarations 9226945488:
9226945488:
radius for satellite in geostationary orbit change variable X to Y
1. 6785303857; locally 1115424:
$$C = 2 \pi r$$
$$pdg_{3034} = 2 pdg_{2530} pdg_{3141}$$
1. 1823570358:
$$C$$
$$pdg_{3034}$$
2. 3236313290:
$$d$$
$$pdg_{1943}$$
1. 9262596735; locally 5369477:
$$d = 2 \pi r$$
$$pdg_{1943} = 2 pdg_{2530} pdg_{3141}$$
valid 6785303857:
9262596735:
6785303857:
9262596735:
radius for satellite in geostationary orbit declare assumption
1. 3176662571; locally 2154616:
$$F_{\rm centripetal} = F_{\rm gravity}$$
$$pdg_{2867} = pdg_{1687}$$
no validation is available for declarations 3176662571:
3176662571:
1. 4162950326; locally 3160921:
$$f asmasf$$
$$a^{2} f^{2} m s^{2}$$
1. 4841405183:
$$m$$
$$m$$
1. 1676875597; locally 5881666:
$$F \propto m_1$$
$$F m_{1} propto$$
Nothing to split 4162950326:
1676875597:
4162950326:
1676875597:
1. 5530941257; locally 2600082:
$$asdfagadf = r_{\rm Eath}$$
$$a^{3} df^{2} g s = rEath$$
1. 6396851146:
$$p_A [S]$$
$$S p_{A}$$
1. 5183724025; locally 1944502:
$$F \propto m_1$$
$$F m_{1} propto$$
Nothing to split 5530941257:
5183724025:
5530941257:
5183724025:
equations of motion in 1D with constant acceleration - SUVAT (algebra) swap LHS with RHS
1. 9759901995; locally 4127918:
$$v - v_0 = a t$$
$$pdg_{1357} - pdg_{5153} = pdg_{1467} pdg_{9140}$$
1. 4748157455; locally 5666935:
$$a t = v - v_0$$
$$pdg_{1467} pdg_{9140} = pdg_{1357} - pdg_{5153}$$
valid 9759901995:
4748157455:
9759901995:
4748157455:
equations of motion in 1D with constant acceleration - SUVAT (algebra) simplify
1. 4580545876; locally 8442394:
$$d = v t - a t^2 + \frac{1}{2} a t^2$$
$$pdg_{1943} = pdg_{1357} pdg_{1467} - \frac{pdg_{1467}^{2} pdg_{9140}}{2}$$
1. 6421241247; locally 3917794:
$$d = v t - \frac{1}{2} a t^2$$
$$pdg_{1943} = pdg_{1357} pdg_{1467} - \frac{pdg_{1467}^{2} pdg_{9140}}{2}$$
valid 4580545876:
6421241247:
4580545876:
6421241247:
equations of motion in 1D with constant acceleration - SUVAT (algebra) declare initial expr
1. 3366703541; locally 7864125:
$$a = \frac{v - v_0}{t}$$
$$pdg_{9140} = \frac{pdg_{1357} - pdg_{5153}}{pdg_{1467}}$$
no validation is available for declarations 3366703541:
3366703541:
equations of motion in 1D with constant acceleration - SUVAT (algebra) add X to both sides
1. 4748157455; locally 5666935:
$$a t = v - v_0$$
$$pdg_{1467} pdg_{9140} = pdg_{1357} - pdg_{5153}$$
1. 6417359412:
$$v_0$$
$$pdg_{5153}$$
1. 4798787814; locally 3386860:
$$a t + v_0 = v$$
$$pdg_{1467} pdg_{9140} + pdg_{5153} = pdg_{1357}$$
valid 4748157455:
4798787814:
4748157455:
4798787814:
equations of motion in 1D with constant acceleration - SUVAT (algebra) declare final expr
1. 3462972452; locally 8873965:
$$v = v_0 + a t$$
$$pdg_{1357} = pdg_{1467} pdg_{9140} + pdg_{5153}$$
no validation is available for declarations 3462972452:
3462972452:
equations of motion in 1D with constant acceleration - SUVAT (algebra) raise both sides to power
1. 3462972452; locally 8873965:
$$v = v_0 + a t$$
$$pdg_{1357} = pdg_{1467} pdg_{9140} + pdg_{5153}$$
1. 5799753649:
$$2$$
$$2$$
1. 7215099603; locally 4385757:
$$v^2 = v_0^2 + 2 a t v_0 + a^2 t^2$$
$$pdg_{1357}^{2} = pdg_{1467}^{2} pdg_{9140}^{2} + 2 pdg_{1467} pdg_{5153} pdg_{9140} + pdg_{5153}^{2}$$
no check is performed 3462972452:
7215099603:
3462972452:
7215099603:
equations of motion in 1D with constant acceleration - SUVAT (algebra) divide both sides by
1. 4748157455; locally 5666935:
$$a t = v - v_0$$
$$pdg_{1467} pdg_{9140} = pdg_{1357} - pdg_{5153}$$
1. 2242144313:
$$a$$
$$pdg_{9140}$$
1. 1967582749; locally 8222540:
$$t = \frac{v - v_0}{a}$$
$$pdg_{1467} = \frac{pdg_{1357} - pdg_{5153}}{pdg_{9140}}$$
valid 4748157455:
1967582749:
4748157455:
1967582749:
equations of motion in 1D with constant acceleration - SUVAT (algebra) simplify
1. 1265150401; locally 6881977:
$$d = \frac{2 v_0 + a t}{2} t$$
$$pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1467} pdg_{9140}}{2} + pdg_{5153}\right)$$
1. 9658195023; locally 5385244:
$$d = v_0 t + \frac{1}{2} a t^2$$
$$pdg_{1943} = \frac{pdg_{1467}^{2} pdg_{9140}}{2} + pdg_{1467} pdg_{5153}$$
valid 1265150401:
9658195023:
1265150401:
9658195023:
equations of motion in 1D with constant acceleration - SUVAT (algebra) declare final expr
1. 6421241247; locally 3917794:
$$d = v t - \frac{1}{2} a t^2$$
$$pdg_{1943} = pdg_{1357} pdg_{1467} - \frac{pdg_{1467}^{2} pdg_{9140}}{2}$$
no validation is available for declarations 6421241247:
6421241247:
equations of motion in 1D with constant acceleration - SUVAT (algebra) substitute RHS of expr 1 into expr 2
1. 1967582749; locally 8222540:
$$t = \frac{v - v_0}{a}$$
$$pdg_{1467} = \frac{pdg_{1357} - pdg_{5153}}{pdg_{9140}}$$
2. 8706092970; locally 1476448:
$$d = \left(\frac{v + v_0}{2}\right)t$$
$$pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}\right)$$
1. 5733721198; locally 9270356:
$$d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)$$
$$pdg_{1943} = \frac{\left(pdg_{1357} - pdg_{5153}\right) \left(pdg_{1357} + pdg_{5153}\right)}{2 pdg_{9140}}$$
LHS diff is 0 RHS diff is (pdg1357 + pdg5153)*(-pdg1357 + pdg1467*pdg9140 + pdg5153)/(2*pdg9140) 1967582749:
8706092970:
5733721198:
1967582749:
8706092970:
5733721198:
equations of motion in 1D with constant acceleration - SUVAT (algebra) declare final expr
1. 9658195023; locally 5385244:
$$d = v_0 t + \frac{1}{2} a t^2$$
$$pdg_{1943} = \frac{pdg_{1467}^{2} pdg_{9140}}{2} + pdg_{1467} pdg_{5153}$$
no validation is available for declarations 9658195023:
9658195023:
equations of motion in 1D with constant acceleration - SUVAT (algebra) add X to both sides
1. 8269198922; locally 6814979:
$$2 a d = v^2 - v_0^2$$
$$2 pdg_{1943} pdg_{9140} = pdg_{1357}^{2} - pdg_{5153}^{2}$$
1. 9070454719:
$$v_0^2$$
$$pdg_{5153}^{2}$$
1. 4948763856; locally 7086842:
$$2 a d + v_0^2 = v^2$$
$$2 pdg_{1943} pdg_{9140} + pdg_{5153}^{2} = pdg_{1357}^{2}$$
valid 8269198922:
4948763856:
8269198922:
4948763856:
equations of motion in 1D with constant acceleration - SUVAT (algebra) substitute RHS of expr 1 into expr 2
1. 5144263777; locally 9796063:
$$v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)$$
$$pdg_{1357}$$
2. 9658195023; locally 5385244:
$$d = v_0 t + \frac{1}{2} a t^2$$
$$pdg_{1943} = \frac{pdg_{1467}^{2} pdg_{9140}}{2} + pdg_{1467} pdg_{5153}$$
1. 7939765107; locally 7702534:
$$v^2 = v_0^2 + 2 a d$$
$$pdg_{1357}^{2} = 2 pdg_{1943} pdg_{9140} + pdg_{5153}^{2}$$
Nothing to split 5144263777:
9658195023:
7939765107:
5144263777:
9658195023:
7939765107:
equations of motion in 1D with constant acceleration - SUVAT (algebra) multiply both sides by
1. 9897284307; locally 4622149:
$$\frac{d}{t} = \frac{v + v_0}{2}$$
$$\frac{pdg_{1943}}{pdg_{1467}} = \frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}$$
1. 8865085668:
$$t$$
$$pdg_{1467}$$
1. 8706092970; locally 1476448:
$$d = \left(\frac{v + v_0}{2}\right)t$$
$$pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}\right)$$
valid 9897284307:
8706092970:
9897284307:
8706092970:
equations of motion in 1D with constant acceleration - SUVAT (algebra) declare final expr
1. 7939765107; locally 7702534:
$$v^2 = v_0^2 + 2 a d$$
$$pdg_{1357}^{2} = 2 pdg_{1943} pdg_{9140} + pdg_{5153}^{2}$$
no validation is available for declarations 7939765107:
7939765107:
equations of motion in 1D with constant acceleration - SUVAT (algebra) substitute RHS of expr 1 into expr 2
1. 6457044853; locally 8007427:
$$v - a t = v_0$$
$$pdg_{1357} - pdg_{1467} pdg_{9140} = pdg_{5153}$$
2. 9658195023; locally 5385244:
$$d = v_0 t + \frac{1}{2} a t^2$$
$$pdg_{1943} = \frac{pdg_{1467}^{2} pdg_{9140}}{2} + pdg_{1467} pdg_{5153}$$
1. 1259826355; locally 5577530:
$$d = (v - a t) t + \frac{1}{2} a t^2$$
$$pdg_{1943} = \frac{pdg_{1467}^{2} pdg_{9140}}{2} + pdg_{1467} \left(pdg_{1357} - pdg_{1467} pdg_{9140}\right)$$
valid 6457044853:
9658195023:
1259826355:
6457044853:
9658195023:
1259826355:
equations of motion in 1D with constant acceleration - SUVAT (algebra) simplify
1. 1259826355; locally 5577530:
$$d = (v - a t) t + \frac{1}{2} a t^2$$
$$pdg_{1943} = \frac{pdg_{1467}^{2} pdg_{9140}}{2} + pdg_{1467} \left(pdg_{1357} - pdg_{1467} pdg_{9140}\right)$$
1. 4580545876; locally 8442394:
$$d = v t - a t^2 + \frac{1}{2} a t^2$$
$$pdg_{1943} = pdg_{1357} pdg_{1467} - \frac{pdg_{1467}^{2} pdg_{9140}}{2}$$
valid 1259826355:
4580545876:
1259826355:
4580545876:
equations of motion in 1D with constant acceleration - SUVAT (algebra) LHS of expr 1 equals LHS of expr 2
1. 3411994811; locally 8658331:
$$v_{\rm average} = \frac{d}{t}$$
$$pdg_{6709} = \frac{pdg_{1943}}{pdg_{1467}}$$
2. 6175547907; locally 5013638:
$$v_{\rm average} = \frac{v + v_0}{2}$$
$$pdg_{6709} = \frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}$$
1. 9897284307; locally 4622149:
$$\frac{d}{t} = \frac{v + v_0}{2}$$
$$\frac{pdg_{1943}}{pdg_{1467}} = \frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}$$
valid 3411994811:
6175547907:
9897284307:
3411994811:
6175547907:
9897284307:
equations of motion in 1D with constant acceleration - SUVAT (algebra) substitute RHS of expr 1 into expr 2
1. 3462972452; locally 8873965:
$$v = v_0 + a t$$
$$pdg_{1357} = pdg_{1467} pdg_{9140} + pdg_{5153}$$
2. 8706092970; locally 1476448:
$$d = \left(\frac{v + v_0}{2}\right)t$$
$$pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}\right)$$
1. 7011114072; locally 3069767:
$$d = \frac{(v_0 + a t) + v_0}{2} t$$
$$pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1467} pdg_{9140}}{2} + pdg_{5153}\right)$$
LHS diff is 0 RHS diff is pdg1467*(pdg1357 - pdg1467*pdg9140 - pdg5153)/2 3462972452:
8706092970:
7011114072:
3462972452:
8706092970:
7011114072:
equations of motion in 1D with constant acceleration - SUVAT (algebra) multiply both sides by
1. 5611024898; locally 7103968:
$$d = \frac{1}{2 a} (v^2 - v_0^2)$$
$$pdg_{1943} = \frac{pdg_{1357}^{2} - pdg_{5153}^{2}}{2 pdg_{9140}}$$
1. 5542390646:
$$2 a$$
$$2 pdg_{9140}$$
1. 8269198922; locally 6814979:
$$2 a d = v^2 - v_0^2$$
$$2 pdg_{1943} pdg_{9140} = pdg_{1357}^{2} - pdg_{5153}^{2}$$
valid 5611024898:
8269198922:
5611024898:
8269198922:
equations of motion in 1D with constant acceleration - SUVAT (algebra) multiply both sides by
1. 3366703541; locally 7864125:
$$a = \frac{v - v_0}{t}$$
$$pdg_{9140} = \frac{pdg_{1357} - pdg_{5153}}{pdg_{1467}}$$
1. 7083390553:
$$t$$
$$pdg_{1467}$$
1. 4748157455; locally 5666935:
$$a t = v - v_0$$
$$pdg_{1467} pdg_{9140} = pdg_{1357} - pdg_{5153}$$
valid 3366703541:
4748157455:
3366703541:
4748157455:
equations of motion in 1D with constant acceleration - SUVAT (algebra) subtract X from both sides
1. 3462972452; locally 8873965:
$$v = v_0 + a t$$
$$pdg_{1357} = pdg_{1467} pdg_{9140} + pdg_{5153}$$
1. 9645178657:
$$a t$$
$$pdg_{1467} pdg_{9140}$$
1. 6457044853; locally 8007427:
$$v - a t = v_0$$
$$pdg_{1357} - pdg_{1467} pdg_{9140} = pdg_{5153}$$
valid 3462972452:
6457044853:
3462972452:
6457044853:
equations of motion in 1D with constant acceleration - SUVAT (algebra) simplify
1. 5733721198; locally 9270356:
$$d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right)$$
$$pdg_{1943} = \frac{\left(pdg_{1357} - pdg_{5153}\right) \left(pdg_{1357} + pdg_{5153}\right)}{2 pdg_{9140}}$$
1. 5611024898; locally 7103968:
$$d = \frac{1}{2 a} (v^2 - v_0^2)$$
$$pdg_{1943} = \frac{pdg_{1357}^{2} - pdg_{5153}^{2}}{2 pdg_{9140}}$$
valid 5733721198:
5611024898:
5733721198:
5611024898:
difference of squares
equations of motion in 1D with constant acceleration - SUVAT (algebra) subtract X from both sides
1. 3462972452; locally 8873965:
$$v = v_0 + a t$$
$$pdg_{1357} = pdg_{1467} pdg_{9140} + pdg_{5153}$$
1. 6729698807:
$$v_0$$
$$pdg_{5153}$$
1. 9759901995; locally 4127918:
$$v - v_0 = a t$$
$$pdg_{1357} - pdg_{5153} = pdg_{1467} pdg_{9140}$$
valid 3462972452:
9759901995:
3462972452:
9759901995:
equations of motion in 1D with constant acceleration - SUVAT (algebra) swap LHS with RHS
1. 4798787814; locally 3386860:
$$a t + v_0 = v$$
$$pdg_{1467} pdg_{9140} + pdg_{5153} = pdg_{1357}$$
1. 3462972452; locally 8873965:
$$v = v_0 + a t$$
$$pdg_{1357} = pdg_{1467} pdg_{9140} + pdg_{5153}$$
valid 4798787814:
3462972452:
4798787814:
3462972452:
equations of motion in 1D with constant acceleration - SUVAT (algebra) swap LHS with RHS
1. 4948763856; locally 7086842:
$$2 a d + v_0^2 = v^2$$
$$2 pdg_{1943} pdg_{9140} + pdg_{5153}^{2} = pdg_{1357}^{2}$$
1. 7939765107; locally 7702534:
$$v^2 = v_0^2 + 2 a d$$
$$pdg_{1357}^{2} = 2 pdg_{1943} pdg_{9140} + pdg_{5153}^{2}$$
valid 4948763856:
7939765107:
4948763856:
7939765107:
equations of motion in 1D with constant acceleration - SUVAT (algebra) simplify
1. 7215099603; locally 4385757:
$$v^2 = v_0^2 + 2 a t v_0 + a^2 t^2$$
$$pdg_{1357}^{2} = pdg_{1467}^{2} pdg_{9140}^{2} + 2 pdg_{1467} pdg_{5153} pdg_{9140} + pdg_{5153}^{2}$$
1. 5144263777; locally 9796063:
$$v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right)$$
$$pdg_{1357}$$
Nothing to split 7215099603:
5144263777:
7215099603:
5144263777:
factored 2a out of two terms
equations of motion in 1D with constant acceleration - SUVAT (algebra) declare final expr
1. 8706092970; locally 1476448:
$$d = \left(\frac{v + v_0}{2}\right)t$$
$$pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}\right)$$
no validation is available for declarations 8706092970:
8706092970:
equations of motion in 1D with constant acceleration - SUVAT (algebra) simplify
1. 7011114072; locally 3069767:
$$d = \frac{(v_0 + a t) + v_0}{2} t$$
$$pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1467} pdg_{9140}}{2} + pdg_{5153}\right)$$
1. 1265150401; locally 6881977:
$$d = \frac{2 v_0 + a t}{2} t$$
$$pdg_{1943} = pdg_{1467} \left(\frac{pdg_{1467} pdg_{9140}}{2} + pdg_{5153}\right)$$
valid 7011114072:
1265150401:
7011114072:
1265150401:
equations of motion in 1D with constant acceleration - SUVAT (algebra) declare initial expr
1. 6175547907; locally 5013638:
$$v_{\rm average} = \frac{v + v_0}{2}$$
$$pdg_{6709} = \frac{pdg_{1357}}{2} + \frac{pdg_{5153}}{2}$$
no validation is available for declarations 6175547907:
6175547907:
equations of motion in 1D with constant acceleration - SUVAT (algebra) declare initial expr
1. 3411994811; locally 8658331:
$$v_{\rm average} = \frac{d}{t}$$
$$pdg_{6709} = \frac{pdg_{1943}}{pdg_{1467}}$$
no validation is available for declarations 3411994811:
3411994811:
velocity at distance r of object dropped from infinity substitute LHS of expr 1 into expr 2
1. 5902985919; locally 3470082:
$$\vec{F} = G \frac{m_1 m_2}{x^2} \hat{x}$$

2. 7882872592; locally 6798426:
$$W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r}$$

1. 3566149658; locally 7300369:
$$W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx$$

failed 5902985919:
7882872592:
3566149658:
5902985919:
7882872592:
3566149658:
velocity at distance r of object dropped from infinity declare initial expr
1. 5902985919; locally 3470082:
$$\vec{F} = G \frac{m_1 m_2}{x^2} \hat{x}$$

no validation is available for declarations 5902985919:
5902985919:
https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation#Modern_form
velocity at distance r of object dropped from infinity substitute LHS of expr 1 into expr 2
1. 8049905441; locally 9781919:
$$\Delta KE = KE_{\rm final} - KE_{\rm initial}$$

2. 1114820451; locally 9835406:
$$W_{\rm by\ system} = \Delta KE$$

1. 5779256336; locally 8118190:
$$W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial}$$

valid 8049905441:
1114820451:
5779256336:
8049905441:
1114820451:
5779256336:
velocity at distance r of object dropped from infinity declare initial expr
1. 2924222857; locally 1712972:
$$v_{\rm initial} = v(r=\infty)$$

no validation is available for declarations 2924222857:
2924222857:
velocity at distance r of object dropped from infinity simplify
1. 5596822289; locally 5818573:
$$W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right)$$

1. 2061086175; locally 2429271:
$$W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)$$

LHS diff is 0 RHS diff is pdg5022*pdg6277*(-pdg4851 + pdg4851(-1/pdg2530)) 5596822289:
2061086175:
5596822289:
2061086175:
velocity at distance r of object dropped from infinity declare initial expr
1. 8357234146; locally 5104592:
$$KE = \frac{1}{2} m v^2$$
$$pdg_{4929} = \frac{pdg_{1357}^{2} pdg_{5156}}{2}$$
no validation is available for declarations 8357234146:
8357234146:
velocity at distance r of object dropped from infinity declare final expr
1. 2005061870; locally 3435796:
$$v(r) = \sqrt{\frac{2 G m_2}{r}}$$

no validation is available for declarations 2005061870:
2005061870:
velocity at distance r of object dropped from infinity evaluate definite integral
1. 8405272745; locally 9707318:
$$W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx$$

1. 5596822289; locally 5818573:
$$W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right)$$

LHS diff is 0 RHS diff is pdg4851*pdg5022*pdg6277*(1 + 1/pdg2530) 8405272745:
5596822289:
8405272745:
5596822289:
velocity at distance r of object dropped from infinity simplify
1. 2061086175; locally 2429271:
$$W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)$$

1. 4393670960; locally 4947999:
$$W_{\rm to\ system} = \frac{G m_1 m_2}{r}$$

LHS diff is 0 RHS diff is -pdg5022*pdg6277*(pdg2530*pdg4851(-1/pdg2530) + pdg4851)/pdg2530 2061086175:
4393670960:
2061086175:
4393670960:
velocity at distance r of object dropped from infinity change variable X to Y
1. 5846639423; locally 7112224:
$$v_{\rm final} = \sqrt{\frac{2 G m_2}{r}}$$

1. 6599829782:
$$v_{\rm final}$$

2. 3531380618:
$$v(r)$$

1. 2005061870; locally 3435796:
$$v(r) = \sqrt{\frac{2 G m_2}{r}}$$

valid 5846639423:
2005061870:
5846639423:
2005061870:
velocity at distance r of object dropped from infinity simplify
1. 3566149658; locally 7300369:
$$W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx$$

1. 8405272745; locally 9707318:
$$W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx$$

valid 3566149658:
8405272745:
3566149658:
8405272745:
velocity at distance r of object dropped from infinity substitute LHS of expr 1 into expr 2
1. 3214170322; locally 8462685:
$$v(r=\infty) = 0$$

2. 2924222857; locally 1712972:
$$v_{\rm initial} = v(r=\infty)$$

1. 2998709778; locally 6923850:
$$v_{\rm initial} = 0$$

Nothing to split 3214170322:
2924222857:
2998709778:
3214170322:
2924222857:
2998709778:
velocity at distance r of object dropped from infinity declare initial expr
1. 1114820451; locally 9835406:
$$W_{\rm by\ system} = \Delta KE$$

no validation is available for declarations 1114820451:
1114820451:
velocity at distance r of object dropped from infinity substitute LHS of expr 1 into expr 2
1. 9510328252; locally 7110498:
$$KE_{\rm initial} = 0$$

2. 5779256336; locally 8118190:
$$W_{\rm by\ system} = KE_{\rm final} - KE_{\rm initial}$$

1. 5850144586; locally 2751634:
$$W_{\rm by\ system} = KE_{\rm final}$$

valid 9510328252:
5779256336:
5850144586:
9510328252:
5779256336:
5850144586:
velocity at distance r of object dropped from infinity declare initial expr
1. 8049905441; locally 9781919:
$$\Delta KE = KE_{\rm final} - KE_{\rm initial}$$

no validation is available for declarations 8049905441:
8049905441:
velocity at distance r of object dropped from infinity change three variables in expr
1. 8357234146; locally 5104592:
$$KE = \frac{1}{2} m v^2$$
$$pdg_{4929} = \frac{pdg_{1357}^{2} pdg_{5156}}{2}$$
1. 3731774096:
$$KE$$

2. 3350802342:
$$KE_{\rm initial}$$

3. 5904227750:
$$m$$

4. 6281834543:
$$m_1$$

5. 8066819515:
$$v$$

6. 3274176452:
$$v_{\rm initial}$$

1. 6091977310; locally 9031887:
$$KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2$$

valid 8357234146:
6091977310:
8357234146:
6091977310:
velocity at distance r of object dropped from infinity substitute LHS of expr 1 into expr 2
1. 9081138616; locally 6536576:
$$W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2$$

2. 2907404069; locally 2619766:
$$W_{\rm by\ system} = W_{\rm to\ system}$$

1. 4947831649; locally 8655239:
$$\frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system}$$

valid 9081138616:
2907404069:
4947831649:
9081138616:
2907404069:
4947831649:
velocity at distance r of object dropped from infinity declare initial expr
1. 2907404069; locally 2619766:
$$W_{\rm by\ system} = W_{\rm to\ system}$$

no validation is available for declarations 2907404069:
2907404069:
velocity at distance r of object dropped from infinity substitute LHS of expr 1 into expr 2
1. 4393670960; locally 4947999:
$$W_{\rm to\ system} = \frac{G m_1 m_2}{r}$$

2. 4947831649; locally 8655239:
$$\frac{1}{2} m_1 v_{\rm final}^2 = W_{\rm to\ system}$$

1. 6892595652; locally 2942416:
$$\frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r}$$

valid 4393670960:
4947831649:
6892595652:
4393670960:
4947831649:
6892595652:
velocity at distance r of object dropped from infinity substitute LHS of expr 1 into expr 2
1. 2998709778; locally 6923850:
$$v_{\rm initial} = 0$$

2. 6091977310; locally 9031887:
$$KE_{\rm initial} = \frac{1}{2} m_1 v_{\rm initial}^2$$

1. 9510328252; locally 7110498:
$$KE_{\rm initial} = 0$$

valid 2998709778:
6091977310:
9510328252:
2998709778:
6091977310:
9510328252:
velocity at distance r of object dropped from infinity change three variables in expr
1. 8357234146; locally 5104592:
$$KE = \frac{1}{2} m v^2$$
$$pdg_{4929} = \frac{pdg_{1357}^{2} pdg_{5156}}{2}$$
1. 4587046017:
$$KE$$

2. 3939572542:
$$KE_{\rm final}$$

3. 9350720370:
$$m$$

4. 3166466250:
$$m_1$$

5. 6038673136:
$$v$$

6. 1616666229:
$$v_{\rm final}$$

1. 8552710882; locally 1397156:
$$KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2$$

failed 8357234146:
8552710882:
8357234146:
8552710882:
velocity at distance r of object dropped from infinity declare initial expr
1. 7882872592; locally 6798426:
$$W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r}$$

no validation is available for declarations 7882872592:
7882872592:
velocity at distance r of object dropped from infinity square root both sides
1. 7112646057; locally 4594601:
$$v_{\rm final}^2 = \frac{2 G m_2}{r}$$

1. 5846639423; locally 7112224:
$$v_{\rm final} = \sqrt{\frac{2 G m_2}{r}}$$

2. 5693047217; locally 1366396:
$$v_{\rm final} = -\sqrt{\frac{2 G m_2}{r}}$$

no check performed 7112646057:
5846639423:
5693047217:
7112646057:
5846639423:
5693047217:
velocity at distance r of object dropped from infinity substitute LHS of expr 1 into expr 2
1. 8552710882; locally 1397156:
$$KE_{\rm final} = \frac{1}{2} m_1 v_{\rm final}^2$$

2. 5850144586; locally 2751634:
$$W_{\rm by\ system} = KE_{\rm final}$$

1. 9081138616; locally 6536576:
$$W_{\rm by\ system} = \frac{1}{2} m_1 v_{\rm final}^2$$

valid 8552710882:
5850144586:
9081138616:
8552710882:
5850144586:
9081138616:
velocity at distance r of object dropped from infinity multiply both sides by
1. 6892595652; locally 2942416:
$$\frac{1}{2} m_1 v_{\rm final}^2 = \frac{G m_1 m_2}{r}$$

1. 7410526982:
$$2/m_1$$

1. 7112646057; locally 4594601:
$$v_{\rm final}^2 = \frac{2 G m_2}{r}$$

valid 6892595652:
7112646057:
6892595652:
7112646057:
velocity at distance r of object dropped from infinity declare initial expr
1. 3214170322; locally 8462685:
$$v(r=\infty) = 0$$

no validation is available for declarations 3214170322:
3214170322:
starting velocity at infinity is zero
coefficient of isothermal compressibility using the equation of state for an ideal gas declare final expr
1. 9718685793; locally 2206759:
$$\kappa_T = \frac{1}{P}$$
$$pdg_{4645} = \frac{1}{pdg_{8134}}$$
no validation is available for declarations 9718685793:
9718685793:
coefficient of isothermal compressibility using the equation of state for an ideal gas simplify
1. 1190768176; locally 3915956:
$$\kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T$$
$$pdg_{4645} = - \frac{pdg_{2834} pdg_{7343} pdg_{8179} \frac{d}{d pdg_{8134}} \frac{1}{pdg_{8134}}}{pdg_{7586}}$$
1. 3605073197; locally 6275836:
$$\kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right)$$
$$pdg_{4645} = \frac{pdg_{2834} pdg_{7343} pdg_{8179}}{pdg_{7586} pdg_{8134}^{2}}$$
valid 1190768176:
3605073197:
1190768176:
3605073197:
coefficient of isothermal compressibility using the equation of state for an ideal gas declare initial expr
1. 9781951738; locally 4239912:
$$\kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T$$
$$pdg_{4645} = - \frac{\frac{d}{d pdg_{8134}} pdg_{7586}}{pdg_{7586}}$$
no validation is available for declarations 9781951738:
9781951738:
coefficient of isothermal compressibility using the equation of state for an ideal gas divide both sides by
1. 8435841627; locally 4454896:
$$P V = n R T$$
$$pdg_{7586} pdg_{8134} = pdg_{2834} pdg_{7343} pdg_{8179}$$
1. 6296166842:
$$P$$
$$pdg_{8134}$$
1. 3497828859; locally 5840241:
$$V = \frac{n R T}{P}$$
$$pdg_{7586} = \frac{pdg_{2834} pdg_{7343} pdg_{8179}}{pdg_{8134}}$$
valid 8435841627:
3497828859:
8435841627:
3497828859:
coefficient of isothermal compressibility using the equation of state for an ideal gas substitute LHS of expr 1 into expr 2
1. 8435841627; locally 4454896:
$$P V = n R T$$
$$pdg_{7586} pdg_{8134} = pdg_{2834} pdg_{7343} pdg_{8179}$$
2. 3605073197; locally 6275836:
$$\kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right)$$
$$pdg_{4645} = \frac{pdg_{2834} pdg_{7343} pdg_{8179}}{pdg_{7586} pdg_{8134}^{2}}$$
1. 9847143017; locally 1003658:
$$\kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right)$$
$$pdg_{4645} = \frac{1}{pdg_{8134}}$$
valid 8435841627:
3605073197:
9847143017:
8435841627:
3605073197:
9847143017:
coefficient of isothermal compressibility using the equation of state for an ideal gas simplify
1. 8368984890; locally 5196207:
$$\kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T$$
$$pdg_{4645} = - \frac{\frac{\partial}{\partial pdg_{8134}} \frac{pdg_{2834} pdg_{7343} pdg_{8179}}{pdg_{8134}}}{pdg_{7586}}$$
1. 1190768176; locally 3915956:
$$\kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T$$
$$pdg_{4645} = - \frac{pdg_{2834} pdg_{7343} pdg_{8179} \frac{d}{d pdg_{8134}} \frac{1}{pdg_{8134}}}{pdg_{7586}}$$
valid 8368984890:
1190768176:
8368984890:
1190768176:
coefficient of isothermal compressibility using the equation of state for an ideal gas simplify
1. 9847143017; locally 1003658:
$$\kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right)$$
$$pdg_{4645} = \frac{1}{pdg_{8134}}$$
1. 9718685793; locally 2206759:
$$\kappa_T = \frac{1}{P}$$
$$pdg_{4645} = \frac{1}{pdg_{8134}}$$
valid 9847143017:
9718685793:
9847143017:
9718685793:
coefficient of isothermal compressibility using the equation of state for an ideal gas substitute LHS of expr 1 into expr 2
1. 3497828859; locally 5840241:
$$V = \frac{n R T}{P}$$
$$pdg_{7586} = \frac{pdg_{2834} pdg_{7343} pdg_{8179}}{pdg_{8134}}$$
2. 9781951738; locally 4239912:
$$\kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T$$
$$pdg_{4645} = - \frac{\frac{d}{d pdg_{8134}} pdg_{7586}}{pdg_{7586}}$$
1. 8368984890; locally 5196207:
$$\kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T$$
$$pdg_{4645} = - \frac{\frac{\partial}{\partial pdg_{8134}} \frac{pdg_{2834} pdg_{7343} pdg_{8179}}{pdg_{8134}}}{pdg_{7586}}$$
LHS diff is 0 RHS diff is -(pdg2834*pdg7343*pdg8179 - pdg7586*pdg8134)/(pdg7586*pdg8134**2) 3497828859:
9781951738:
8368984890:
3497828859:
9781951738:
8368984890:
coefficient of isothermal compressibility using the equation of state for an ideal gas declare initial expr
1. 8435841627; locally 4454896:
$$P V = n R T$$
$$pdg_{7586} pdg_{8134} = pdg_{2834} pdg_{7343} pdg_{8179}$$
no validation is available for declarations 8435841627:
8435841627:
speed of Earth around Sun declare final expr
1. 4180845508; locally 1001745:
$$v_{\rm Earth\ orbit} = 29.8 \frac{{\rm km}}{{\rm sec}}$$
$$pdg_{7427} = 29.8$$
no validation is available for declarations 4180845508:
4180845508:
speed of Earth around Sun substitute LHS of expr 1 into expr 2
1. 6348260313; locally 5753220:
$$C_{\rm Earth\ orbit} = 2 \pi r_{\rm Earth\ orbit}$$
$$pdg_{1534} = 2 pdg_{3141} pdg_{6081}$$
2. 3046191961; locally 5320197:
$$v_{\rm Earth\ orbit} = \frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}$$
$$pdg_{7427} = \frac{pdg_{1534}}{pdg_{5344}}$$
1. 3080027960; locally 9129246:
$$v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}$$
$$pdg_{7427} = \frac{2 pdg_{3141} pdg_{6081}}{pdg_{5344}}$$
valid 6348260313:
3046191961:
3080027960:
6348260313:
3046191961:
3080027960:
speed of Earth around Sun simplify
1. 6998364753; locally 8698819:
$$v_{\rm Earth\ orbit} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}$$
$$pdg_{7427} = 0.632911392405063 pdg_{3141}$$
1. 4180845508; locally 1001745:
$$v_{\rm Earth\ orbit} = 29.8 \frac{{\rm km}}{{\rm sec}}$$
$$pdg_{7427} = 29.8$$
LHS diff is 0 RHS diff is 0.632911392405063*pdg3141 - 29.8 6998364753:
4180845508:
6998364753:
4180845508:
speed of Earth around Sun substitute LHS of expr 1 into expr 2
1. 8721295221; locally 9417128:
$$t_{\rm Earth\ orbit} = 3.16 10^7 {\rm seconds}$$
$$pdg_{5344} = 3$$
2. 3080027960; locally 9129246:
$$v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}$$
$$pdg_{7427} = \frac{2 pdg_{3141} pdg_{6081}}{pdg_{5344}}$$
1. 4593428198; locally 1441436:
$$v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}}$$
$$pdg_{7427} = 0.632911392405063 pdg_{3141} pdg_{6081}$$
LHS diff is 0 RHS diff is 0.0337552742616034*pdg3141*pdg6081 8721295221:
3080027960:
4593428198:
8721295221:
3080027960:
4593428198:
speed of Earth around Sun declare initial expr
1. 6785303857; locally 7959026:
$$C = 2 \pi r$$
$$pdg_{3034} = 2 pdg_{2530} pdg_{3141}$$
no validation is available for declarations 6785303857:
6785303857:
circumference of a circle
speed of Earth around Sun declare assumption
1. 3472836147; locally 4133484:
$$r_{\rm Earth\ orbit} = 1.496\ 10^8 {\rm km}$$
$$pdg_{6081} = 1.496$$
no validation is available for declarations 3472836147:
3472836147:
speed of Earth around Sun change variable X to Y
1. 5426308937; locally 1131405:
$$v = \frac{d}{t}$$
$$pdg_{1357} = \frac{pdg_{1943}}{pdg_{1467}}$$
1. 1277713901:
$$d$$
$$pdg_{1943}$$
2. 7476820482:
$$C$$
$$pdg_{3034}$$
1. 6946088325; locally 7360652:
$$v = \frac{C}{t}$$
$$pdg_{1357} = \frac{pdg_{3034}}{pdg_{1467}}$$
valid 5426308937:
6946088325:
5426308937:
6946088325:
speed of Earth around Sun declare assumption
1. 7175416299; locally 9494155:
$$t_{\rm Earth\ orbit} = 1 {\rm year}$$
$$pdg_{5344} = 1$$
no validation is available for declarations 7175416299:
7175416299:
speed of Earth around Sun change three variables in expr
1. 6946088325; locally 7360652:
$$v = \frac{C}{t}$$
$$pdg_{1357} = \frac{pdg_{3034}}{pdg_{1467}}$$
1. 4057686137:
$$C$$
$$pdg_{3034}$$
2. 7708501762:
$$C_{\rm Earth\ orbit}$$
$$pdg_{1534}$$
3. 9753878784:
$$v$$
$$pdg_{1357}$$
4. 9601500174:
$$v_{\rm Earth\ orbit}$$
$$pdg_{7427}$$
5. 8135396036:
$$t$$
$$pdg_{1467}$$
6. 4470433702:
$$t_{\rm Earth\ orbit}$$
$$pdg_{5344}$$
1. 3046191961; locally 5320197:
$$v_{\rm Earth\ orbit} = \frac{C_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}$$
$$pdg_{7427} = \frac{pdg_{1534}}{pdg_{5344}}$$
valid 6946088325:
3046191961:
6946088325:
3046191961:
speed of Earth around Sun change two variables in expr
1. 6785303857; locally 7959026:
$$C = 2 \pi r$$
$$pdg_{3034} = 2 pdg_{2530} pdg_{3141}$$
1. 4057686137:
$$C$$
$$pdg_{3034}$$
2. 6239815585:
$$C_{\rm Earth\ orbit}$$
$$pdg_{1534}$$
3. 2346150725:
$$r$$
$$pdg_{2530}$$
4. 4202292449:
$$r_{\rm Earth\ orbit}$$
$$pdg_{6081}$$
1. 6348260313; locally 5753220:
$$C_{\rm Earth\ orbit} = 2 \pi r_{\rm Earth\ orbit}$$
$$pdg_{1534} = 2 pdg_{3141} pdg_{6081}$$
valid 6785303857:
6348260313:
6785303857:
6348260313:
speed of Earth around Sun multiply RHS by unity
1. 7175416299; locally 9494155:
$$t_{\rm Earth\ orbit} = 1 {\rm year}$$
$$pdg_{5344} = 1$$
1. 3219318145:
$$\frac{365 {\rm days}}{1 {\rm year}} \frac{24 {\rm hours}}{1 {\rm day}} \frac{60 {\rm minutes}}{1 {\rm hour}} \frac{60 {\rm seconds}}{1 {\rm minute}}$$
$$365$$
1. 8721295221; locally 9417128:
$$t_{\rm Earth\ orbit} = 3.16 10^7 {\rm seconds}$$
$$pdg_{5344} = 3$$
feed diff is 364 LHS diff is 0 RHS diff is 362 7175416299:
8721295221:
7175416299:
8721295221:
speed of Earth around Sun declare initial expr
1. 5426308937; locally 1131405:
$$v = \frac{d}{t}$$
$$pdg_{1357} = \frac{pdg_{1943}}{pdg_{1467}}$$
no validation is available for declarations 5426308937:
5426308937:
speed of Earth around Sun substitute LHS of expr 1 into expr 2
1. 3472836147; locally 4133484:
$$r_{\rm Earth\ orbit} = 1.496\ 10^8 {\rm km}$$
$$pdg_{6081} = 1.496$$
2. 4593428198; locally 1441436:
$$v_{\rm Earth\ orbit} = \frac{2 \pi r_{\rm Earth\ orbit}}{3.16\ 10^7 {\rm seconds}}$$
$$pdg_{7427} = 0.632911392405063 pdg_{3141} pdg_{6081}$$
1. 6998364753; locally 8698819:
$$v_{\rm Earth\ orbit} = \frac{2 \pi \left( 1.496\ 10^8 {\rm km} \right)}{3.16\ 10^7 {\rm seconds}}$$
$$pdg_{7427} = 0.632911392405063 pdg_{3141}$$
LHS diff is 0 RHS diff is 0.313924050632911*pdg3141 3472836147:
4593428198:
6998364753:
3472836147:
4593428198:
6998364753:
first law of thermodynamics multiply both sides by
1. 3464107376; locally 2714175:
$$\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p$$
$$pdg_{4686} = \frac{\frac{d}{d pdg_{7343}} pdg_{7586}}{pdg_{7586}}$$
1. 5074423401:
$$V$$
$$pdg_{7586}$$
1. 6397683463; locally 7939101:
$$V \alpha = \left( \frac{\partial V}{\partial T} \right)_p$$
$$pdg_{4686} pdg_{7586} = \frac{d}{d pdg_{7343}} pdg_{7586}$$
valid 3464107376:
6397683463:
3464107376:
6397683463:
first law of thermodynamics substitute LHS of two expressions into expr
1. 1085150613; locally 4576755:
$$C_V = \left(\frac{\partial U}{\partial T}\right)_V$$
$$pdg_{6682} = \frac{d}{d pdg_{7343}} pdg_{5786}$$
2. 5634116660; locally 7384950:
$$\pi_T = \left(\frac{\partial U}{\partial V}\right)_T$$
$$pdg_{5480} = \frac{d}{d pdg_{7586}} pdg_{5786}$$
3. 9941599459; locally 2445123:
$$dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV$$
$$dU = \frac{d}{d pdg_{7343}} pdg_{5786}$$
1. 5002539602; locally 5358683:
$$dU = C_V dT + \pi_T dV$$
$$dU = dT pdg_{6682} + dV pdg_{5480}$$
failed 1085150613:
5634116660:
9941599459:
5002539602:
1085150613:
5634116660:
9941599459:
5002539602:
first law of thermodynamics declare initial expr
1. 3464107376; locally 2714175:
$$\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p$$
$$pdg_{4686} = \frac{\frac{d}{d pdg_{7343}} pdg_{7586}}{pdg_{7586}}$$
no validation is available for declarations 3464107376:
3464107376:
first law of thermodynamics declare initial expr
1. 1815398659; locally 7368252:
$$U = Q + W$$
$$pdg_{5786} = pdg_{1088} + pdg_{9432}$$
no validation is available for declarations 1815398659:
1815398659:
first law of thermodynamics divide both sides by
1. 5002539602; locally 5358683:
$$dU = C_V dT + \pi_T dV$$
$$dU = dT pdg_{6682} + dV pdg_{5480}$$
1. 8854422847:
$$dT$$
$$pdg_{7343}$$
1. 6055078815; locally 3830663:
$$\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p$$
$$\frac{d}{d pdg_{7343}} pdg_{5786}$$
Nothing to split 5002539602:
6055078815:
5002539602:
6055078815:
first law of thermodynamics simplify
1. 2257410739; locally 1136968:
$$\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha$$
$$\frac{d}{d pdg_{7343}} pdg_{5786}$$
1. 7826132469; locally 1189259:
$$\left(\frac{\partial U}{\partial T}\right)_p = C_V + \pi_T V \alpha$$
$$\frac{d}{d pdg_{7343}} pdg_{5786}$$
Nothing to split 2257410739:
7826132469:
2257410739:
7826132469:
first law of thermodynamics declare initial expr
1. 5634116660; locally 7384950:
$$\pi_T = \left(\frac{\partial U}{\partial V}\right)_T$$
$$pdg_{5480} = \frac{d}{d pdg_{7586}} pdg_{5786}$$
no validation is available for declarations 5634116660:
5634116660:
first law of thermodynamics substitute LHS of expr 1 into expr 2
1. 6397683463; locally 7939101:
$$V \alpha = \left( \frac{\partial V}{\partial T} \right)_p$$
$$pdg_{4686} pdg_{7586} = \frac{d}{d pdg_{7343}} pdg_{7586}$$
2. 6055078815; locally 3830663:
$$\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p$$
$$\frac{d}{d pdg_{7343}} pdg_{5786}$$
1. 2257410739; locally 1136968:
$$\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha$$
$$\frac{d}{d pdg_{7343}} pdg_{5786}$$
Nothing to split 6397683463:
6055078815:
2257410739:
6397683463:
6055078815:
2257410739:
first law of thermodynamics declare initial expr
1. 9941599459; locally 2445123:
$$dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV$$
$$dU = \frac{d}{d pdg_{7343}} pdg_{5786}$$
no validation is available for declarations 9941599459:
9941599459:
hold volume constant in first term; hold temperature constant in second term
first law of thermodynamics declare initial expr
1. 3547519267; locally 8155541:
$$S = k_{\rm Boltzmann} \ln \Omega$$
$$pdg_{1394} = pdg_{1157} \log{\left(pdg_{3434} \right)}$$
no validation is available for declarations 3547519267:
3547519267:
first law of thermodynamics declare initial expr
1. 1085150613; locally 4576755:
$$C_V = \left(\frac{\partial U}{\partial T}\right)_V$$
$$pdg_{6682} = \frac{d}{d pdg_{7343}} pdg_{5786}$$
no validation is available for declarations 1085150613:
1085150613:
first law of thermodynamics declare initial expr
1. 9781951738; locally 9670239:
$$\kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T$$
$$pdg_{4645} = - \frac{\frac{d}{d pdg_{8134}} pdg_{7586}}{pdg_{7586}}$$
no validation is available for declarations 9781951738:
9781951738:
optics: Law of refraction to Brewster's angle declare identity
1. 8588429722; locally 3940135:
$$\sin( 90^{\circ} - x ) = \cos( x )$$
$$- \sin{\left(pdg_{1464} - 90 \right)} = \cos{\left(pdg_{1464} \right)}$$
no validation is available for declarations 8588429722:
8588429722:
optics: Law of refraction to Brewster's angle substitute LHS of expr 1 into expr 2
1. 6831637424; locally 7426234:
$$\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )$$
$$pdg_{4928}$$
2. 7696214507; locally 4962698:
$$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )$$
$$pdg_{2941} \sin{\left(pdg_{4928} \right)} = - pdg_{1958} \sin{\left(pdg_{4928} - 90 \right)}$$
1. 3061811650; locally 9701820:
$$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )$$
$$pdg_{2941} \sin{\left(pdg_{4928} \right)} = pdg_{1958} \cos{\left(pdg_{4928} \right)}$$
Nothing to split 6831637424:
7696214507:
3061811650:
6831637424:
7696214507:
3061811650:
optics: Law of refraction to Brewster's angle declare final expr
1. 8495187962; locally 8186016:
$$\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }$$
$$pdg_{4928} = \operatorname{atan}{\left(\frac{pdg_{2941}}{pdg_{1958}} \right)}$$
no validation is available for declarations 8495187962:
8495187962:
optics: Law of refraction to Brewster's angle declare initial expr
1. 6450985774; locally 9932375:
$$n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )$$
$$pdg_{2941} \sin{\left(pdg_{3509} \right)} = pdg_{1958} \sin{\left(pdg_{7545} \right)}$$
no validation is available for declarations 6450985774:
6450985774:
optics: Law of refraction to Brewster's angle declare identity
1. 4968680693; locally 2621708:
$$\tan( x ) = \frac{ \sin( x )}{\cos( x )}$$
$$\tan{\left(pdg_{1464} \right)} = \frac{\sin{\left(pdg_{1464} \right)}}{\cos{\left(pdg_{1464} \right)}}$$
no validation is available for declarations 4968680693:
4968680693:
optics: Law of refraction to Brewster's angle divide both sides by
1. 3061811650; locally 9701820:
$$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )$$
$$pdg_{2941} \sin{\left(pdg_{4928} \right)} = pdg_{1958} \cos{\left(pdg_{4928} \right)}$$
1. 7857757625:
$$n_1$$
$$pdg_{2941}$$
1. 9756089533; locally 9314305:
$$\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )$$
$$\sin{\left(pdg_{4928} \right)} = \frac{pdg_{1958} \cos{\left(pdg_{4928} \right)}}{pdg_{2941}}$$
valid 3061811650:
9756089533:
3061811650:
9756089533:
optics: Law of refraction to Brewster's angle substitute LHS of expr 1 into expr 2
1. 1310571337; locally 3893026:
$$\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}$$
$$pdg_{4928}$$
2. 2575937347; locally 4176694:
$$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )$$
$$pdg_{2941} \sin{\left(pdg_{4928} \right)} = pdg_{1958} \sin{\left(pdg_{2243} \right)}$$
1. 7696214507; locally 4962698:
$$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )$$
$$pdg_{2941} \sin{\left(pdg_{4928} \right)} = - pdg_{1958} \sin{\left(pdg_{4928} - 90 \right)}$$
Nothing to split 1310571337:
2575937347:
7696214507:
1310571337:
2575937347:
7696214507:
optics: Law of refraction to Brewster's angle change two variables in expr
1. 6450985774; locally 9932375:
$$n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )$$
$$pdg_{2941} \sin{\left(pdg_{3509} \right)} = pdg_{1958} \sin{\left(pdg_{7545} \right)}$$
1. 7154592211:
$$\theta_2$$
$$pdg_{7545}$$
2. 6353701615:
$$\theta_{\rm refracted}$$
$$pdg_{2243}$$
3. 2773628333:
$$\theta_1$$
$$pdg_{3509}$$
4. 9029795851:
$$\theta_{\rm Brewster}$$
$$pdg_{4928}$$
1. 2575937347; locally 4176694:
$$n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )$$
$$pdg_{2941} \sin{\left(pdg_{4928} \right)} = pdg_{1958} \sin{\left(pdg_{2243} \right)}$$
valid 6450985774:
2575937347:
6450985774:
2575937347:
optics: Law of refraction to Brewster's angle change variable X to Y
1. 4968680693; locally 2621708:
$$\tan( x ) = \frac{ \sin( x )}{\cos( x )}$$
$$\tan{\left(pdg_{1464} \right)} = \frac{\sin{\left(pdg_{1464} \right)}}{\cos{\left(pdg_{1464} \right)}}$$
1. 7321695558:
$$\theta_{\rm Brewster}$$
$$pdg_{4928}$$
2. 9906920183:
$$x$$
$$pdg_{1464}$$
1. 4501377629; locally 1898054:
$$\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}$$
$$\tan{\left(pdg_{4928} \right)} = \frac{\sin{\left(pdg_{4928} \right)}}{\cos{\left(pdg_{4928} \right)}}$$
LHS diff is tan(pdg1464) - tan(pdg4928) RHS diff is tan(pdg1464) - tan(pdg4928) 4968680693:
4501377629:
4968680693:
4501377629:
optics: Law of refraction to Brewster's angle substitute LHS of expr 1 into expr 2
1. 4501377629; locally 1898054:
$$\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}$$
$$\tan{\left(pdg_{4928} \right)} = \frac{\sin{\left(pdg_{4928} \right)}}{\cos{\left(pdg_{4928} \right)}}$$
2. 2768857871; locally 8585856:
$$\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}$$
$$\frac{\sin{\left(pdg_{4928} \right)}}{\cos{\left(pdg_{4928} \right)}} = \frac{pdg_{1958}}{pdg_{2941}}$$
1. 3417126140; locally 5179630:
$$\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }$$
$$\tan{\left(pdg_{4928} \right)} = \frac{pdg_{1958}}{pdg_{2941}}$$
valid 4501377629:
2768857871:
3417126140:
4501377629:
2768857871:
3417126140:
optics: Law of refraction to Brewster's angle declare initial expr
1. 8945218208; locally 5563180:
$$\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}$$
$$pdg_{4928}$$
no validation is available for declarations 8945218208:
8945218208:
optics: Law of refraction to Brewster's angle apply function to both sides of expression
1. 3417126140; locally 5179630:
$$\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }$$
$$\tan{\left(pdg_{4928} \right)} = \frac{pdg_{1958}}{pdg_{2941}}$$
1. 5453995431:
$$\arctan{ x }$$
$$\operatorname{atan}{\left(pdg_{1464} \right)}$$
2. 6023986360:
$$x$$
$$pdg_{1464}$$
1. 8495187962; locally 8186016:
$$\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }$$
$$pdg_{4928} = \operatorname{atan}{\left(\frac{pdg_{2941}}{pdg_{1958}} \right)}$$
no check performed 3417126140:
8495187962:
3417126140:
8495187962:
optics: Law of refraction to Brewster's angle divide both sides by
1. 9756089533; locally 9314305:
$$\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )$$
$$\sin{\left(pdg_{4928} \right)} = \frac{pdg_{1958} \cos{\left(pdg_{4928} \right)}}{pdg_{2941}}$$
1. 5632428182:
$$\cos( \theta_{\rm Brewster} )$$
$$\cos{\left(pdg_{4928} \right)}$$
1. 2768857871; locally 8585856:
$$\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}$$
$$\frac{\sin{\left(pdg_{4928} \right)}}{\cos{\left(pdg_{4928} \right)}} = \frac{pdg_{1958}}{pdg_{2941}}$$
valid 9756089533:
2768857871:
9756089533:
2768857871:
optics: Law of refraction to Brewster's angle change variable X to Y
1. 8588429722; locally 3940135:
$$\sin( 90^{\circ} - x ) = \cos( x )$$
$$- \sin{\left(pdg_{1464} - 90 \right)} = \cos{\left(pdg_{1464} \right)}$$
1. 7375348852:
$$\theta_{\rm Brewster}$$
$$pdg_{4928}$$
2. 1512581563:
$$x$$
$$pdg_{1464}$$
1. 6831637424; locally 7426234:
$$\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )$$
$$pdg_{4928}$$
Nothing to split 8588429722:
6831637424:
8588429722:
6831637424:
optics: Law of refraction to Brewster's angle subtract X from both sides
1. 8945218208; locally 5563180:
$$\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}$$
$$pdg_{4928}$$
1. 9025853427:
$$\theta_{\rm Brewster}$$
$$pdg_{4928}$$
1. 1310571337; locally 3893026:
$$\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}$$
$$pdg_{4928}$$
Nothing to split 8945218208:
1310571337:
8945218208:
1310571337:
mass of the Earth replace constant with value
1. 9440616166; locally 9133599:
$$m_{\rm Earth} = \frac{g_{\rm Earth} r_{\rm Earth}^2}{G}$$
$$pdg_{5458} = \frac{pdg_{3236}^{2} pdg_{7557}}{pdg_{6277}}$$
1. 2091584724:
$$g_{\rm Earth}$$
$$pdg_{7557}$$
2. 9590696981:
$$9.80665$$
$$9.80665$$
3. 7816982139:
$$m/s^2$$
$$\frac{\text{m}^{2}}{\text{s}^{2}}$$
1. 7846240076; locally 2593741:
$$m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{G}$$
$$pdg_{5458} = \frac{9 pdg_{3236}^{2}}{pdg_{6277}}$$
no check performed 9440616166:
7846240076:
9440616166:
7846240076:
mass of the Earth divide both sides by
1. 9407192813; locally 7388891:
$$G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} = m g_{\rm Earth}$$
$$\frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}^{2}} = pdg_{5156} pdg_{7557}$$
1. 3246378279:
$$m$$
$$pdg_{5156}$$
1. 2308660627; locally 9159337:
$$G \frac{m_{\rm Earth}}{r_{\rm Earth}^2} = g_{\rm Earth}$$
$$\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}^{2}} = pdg_{7557}$$
valid 9407192813:
2308660627:
9407192813:
2308660627:
mass of the Earth replace constant with value
1. 7112613117; locally 1218257:
$$m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}$$
$$pdg_{5458}$$
1. 7935917166:
$$r_{\rm Earth}$$
$$pdg_{3236}$$
2. 3723096423:
$$6.3781*10^6$$
$$6378100.0$$
3. 7560908617:
$$m$$
$$pdg_{5156}$$
1. 1132941271; locally 9815516:
$$m_{\rm Earth} = \frac{(9.80665 m/s^2) (6.3781*10^6 m)^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}$$
$$pdg_{5458}$$
Nothing to split 7112613117:
1132941271:
7112613117:
1132941271:
mass of the Earth change variable X to Y
1. 5345738321; locally 3843242:
$$F = m a$$
$$pdg_{4202} = pdg_{5156} pdg_{9140}$$
1. 9881106100:
$$a$$
$$pdg_{9140}$$
2. 5781435087:
$$g$$
$$pdg_{1649}$$
1. 2484824786; locally 6779814:
$$F = m g$$
$$pdg_{4202} = pdg_{1649} pdg_{5156}$$
valid 5345738321:
2484824786:
5345738321:
2484824786:
mass of the Earth change three variables in expr
1. 6935745841; locally 2737346:
$$F = G \frac{m_1 m_2}{x^2}$$
$$pdg_{4202} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{4037}^{2}}$$
1. 3921072591:
$$m_1$$
$$pdg_{5458}$$
2. 1193980495:
$$m_{\rm Earth}$$
$$pdg_{5458}$$
3. 4651061153:
$$m_2$$
$$pdg_{4851}$$
4. 9903988330:
$$m$$
$$pdg_{5156}$$
5. 3353418803:
$$x$$
$$pdg_{4037}$$
6. 6535639720:
$$r_{\rm Earth}$$
$$pdg_{3236}$$
1. 8661803554; locally 6771172:
$$F = G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2}$$
$$pdg_{4202} = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}^{2}}$$
LHS diff is 0 RHS diff is pdg5156*pdg6277*(pdg5022 - pdg5458)/pdg3236**2 6935745841:
8661803554:
6935745841:
8661803554:
mass of the Earth simplify
1. 1132941271; locally 9815516:
$$m_{\rm Earth} = \frac{(9.80665 m/s^2) (6.3781*10^6 m)^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}$$
$$pdg_{5458}$$
1. 3364286646; locally 1635641:
$$m_{\rm Earth} = 5.972*10^{24} kg$$
$$pdg_{5458} = 5.972 \cdot 10^{24} kg$$
Nothing to split 1132941271:
3364286646:
1132941271:
3364286646:
mass of the Earth replace constant with value
1. 7846240076; locally 2593741:
$$m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{G}$$
$$pdg_{5458} = \frac{9 pdg_{3236}^{2}}{pdg_{6277}}$$
1. 7326066466:
$$G$$
$$pdg_{6277}$$
2. 9956609318:
$$6.67430*10^{-11}$$
$$6.6743 \cdot 10^{-11}$$
3. 2957211007:
$$m^3 kg^{-1} s^{-2}$$
$$\frac{\text{m}^{3}}{\text{s}^{2}}$$
1. 7112613117; locally 1218257:
$$m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}$$
$$pdg_{5458}$$
Nothing to split 7846240076:
7112613117:
7846240076:
7112613117:
mass of the Earth declare initial expr
1. 5345738321; locally 3843242:
$$F = m a$$
$$pdg_{4202} = pdg_{5156} pdg_{9140}$$
no validation is available for declarations 5345738321:
5345738321:
mass of the Earth LHS of expr 1 equals LHS of expr 2
1. 8661803554; locally 6771172:
$$F = G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2}$$
$$pdg_{4202} = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}^{2}}$$
2. 4800170179; locally 6086107:
$$F = m g_{\rm Earth}$$
$$pdg_{4202} = pdg_{5156} pdg_{7557}$$
1. 9407192813; locally 7388891:
$$G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} = m g_{\rm Earth}$$
$$\frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}^{2}} = pdg_{5156} pdg_{7557}$$
valid 8661803554:
4800170179:
9407192813:
8661803554:
4800170179:
9407192813:
mass of the Earth declare initial expr
1. 6935745841; locally 2737346:
$$F = G \frac{m_1 m_2}{x^2}$$
$$pdg_{4202} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{4037}^{2}}$$
no validation is available for declarations 6935745841:
6935745841:
mass of the Earth multiply both sides by
1. 2308660627; locally 9159337:
$$G \frac{m_{\rm Earth}}{r_{\rm Earth}^2} = g_{\rm Earth}$$
$$\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}^{2}} = pdg_{7557}$$
1. 2685587762:
$$\frac{r_{\rm Earth}^2}{G}$$
$$\frac{pdg_{3236}^{2}}{pdg_{6277}}$$
1. 9440616166; locally 9133599:
$$m_{\rm Earth} = \frac{g_{\rm Earth} r_{\rm Earth}^2}{G}$$
$$pdg_{5458} = \frac{pdg_{3236}^{2} pdg_{7557}}{pdg_{6277}}$$
valid 2308660627:
9440616166:
2308660627:
9440616166:
mass of the Earth change variable X to Y
1. 2484824786; locally 6779814:
$$F = m g$$
$$pdg_{4202} = pdg_{1649} pdg_{5156}$$
1. 9355039511:
$$g$$
$$pdg_{1649}$$
2. 2232825726:
$$g_{\rm Earth}$$
$$pdg_{7557}$$
1. 4800170179; locally 6086107:
$$F = m g_{\rm Earth}$$
$$pdg_{4202} = pdg_{5156} pdg_{7557}$$
valid 2484824786:
4800170179:
2484824786:
4800170179:
double intensity when phase is coherent (optics) substitute LHS of expr 1 into expr 2
1. 6774684564; locally 7781977:
$$\theta = \phi$$
$$pdg_{1575} = pdg_{8586}$$
2. 8497631728; locally 5493675:
$$I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )$$
$$pdg_{7882} = 2 \cos{\left(pdg_{1575} - pdg_{8586} \right)} \left|{pdg_{4453}}\right| \left|{pdg_{4698}}\right| + \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
1. 8283354808; locally 2413866:
$$I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )$$
$$pdg_{8251} = \left|{pdg_{4453}}\right|^{2} + 2 \left|{pdg_{4453}}\right| \left|{pdg_{4698}}\right| + \left|{pdg_{4698}}\right|^{2}$$
LHS diff is pdg7882 - pdg8251 RHS diff is 0 6774684564:
8497631728:
8283354808:
6774684564:
8497631728:
8283354808:
double intensity when phase is coherent (optics) substitute LHS of expr 1 into expr 2
1. 7107090465; locally 2303305:
$$B B^* = |B|^2$$
$$pdg_{4698} \overline{pdg_{4698}} = \left|{pdg_{4698}}\right|^{2}$$
2. 5125940051; locally 4729665:
$$I = |A|^2 + B B^* + A B^* + B A^*$$
$$pdg_{7882} = pdg_{4453} \overline{pdg_{4698}} + pdg_{4698} \overline{pdg_{4453}} + pdg_{4698} \overline{pdg_{4698}} + \left|{pdg_{4453}}\right|^{2}$$
1. 1525861537; locally 8296872:
$$I = |A|^2 + |B|^2 + A B^* + B A^*$$
$$pdg_{7882} = pdg_{4453} \overline{pdg_{4698}} + pdg_{4698} \overline{pdg_{4453}} + \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
valid 7107090465:
5125940051:
1525861537:
7107090465:
5125940051:
1525861537:
double intensity when phase is coherent (optics) substitute LHS of expr 1 into expr 2
1. 8602221482; locally 4842351:
$$\langle \cos(\theta - \phi) \rangle = 0$$
$$\cos{\left(pdg_{1575} - pdg_{8586} \right)} = 0$$
2. 8497631728; locally 5493675:
$$I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )$$
$$pdg_{7882} = 2 \cos{\left(pdg_{1575} - pdg_{8586} \right)} \left|{pdg_{4453}}\right| \left|{pdg_{4698}}\right| + \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
1. 6240206408; locally 8093224:
$$I_{\rm incoherent} = |A|^2 + |B|^2$$
$$pdg_{2435} = \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
LHS diff is -pdg2435 + pdg7882 RHS diff is 0 8602221482:
8497631728:
6240206408:
8602221482:
8497631728:
6240206408:
double intensity when phase is coherent (optics) substitute LHS of four expressions into expr
1. 4192519596; locally 7875296:
$$B = |B| \exp(i \phi)$$
$$pdg_{4698} = e^{pdg_{4621} pdg_{8586}} \left|{pdg_{4698}}\right|$$
2. 4504256452; locally 1174231:
$$B^* = |B| \exp(-i \phi)$$
$$\overline{pdg_{4698}} = e^{- pdg_{4621} pdg_{8586}} \left|{pdg_{4698}}\right|$$
3. 1357848476; locally 2018605:
$$A = |A| \exp(i \theta)$$
$$pdg_{4453} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{4453}}\right|$$
1. 7621705408; locally 1405078:
$$I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)$$
$$pdg_{7882} = e^{pdg_{1575} pdg_{4621}} e^{- pdg_{4621} pdg_{8586}} \left|{pdg_{4453} pdg_{4698} \left|{\left|{pdg_{4453}}\right| + e^{- pdg_{1575} pdg_{4621}} e^{pdg_{4621} pdg_{8586}} \left|{pdg_{4698}}\right|}\right|}\right| + \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
no check performed 4192519596:
4504256452:
1357848476:
7621705408:
4192519596:
4504256452:
1357848476:
7621705408:
double intensity when phase is coherent (optics) change variable X to Y
1. 4182362050; locally 4809503:
$$Z = |Z| \exp( i \theta )$$
$$pdg_{3192} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{3192}}\right|$$
1. 2064205392:
$$A$$
$$pdg_{4453}$$
2. 1894894315:
$$Z$$
$$pdg_{3192}$$
1. 1357848476; locally 2018605:
$$A = |A| \exp(i \theta)$$
$$pdg_{4453} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{4453}}\right|$$
LHS diff is pdg3192 - pdg4453 RHS diff is (Abs(pdg3192) - Abs(pdg4453))*exp(pdg1575*pdg4621) 4182362050:
1357848476:
4182362050:
1357848476:
double intensity when phase is coherent (optics) declare initial expr
1. 2719691582; locally 9739736:
$$|A| = |B|$$
$$\left|{pdg_{4453}}\right| = \left|{pdg_{4698}}\right|$$
no validation is available for declarations 2719691582:
2719691582:
double intensity when phase is coherent (optics) simplify
1. 6529793063; locally 5409843:
$$I_{\rm incoherent} = |A|^2 + |A|^2$$
$$pdg_{2435} = 2 \left|{pdg_{4453}}\right|^{2}$$
1. 3060393541; locally 3246829:
$$I_{\rm incoherent} = 2|A|^2$$
$$pdg_{2435} = 2 \left|{pdg_{4453}}\right|^{2}$$
valid 6529793063:
3060393541:
6529793063:
3060393541:
double intensity when phase is coherent (optics) substitute LHS of expr 1 into expr 2
1. 2700934933; locally 8635275:
$$2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)$$
$$2 \cos{\left(pdg_{1464} \right)} = e^{pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)} + e^{- pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)}$$
2. 3085575328; locally 5595798:
$$I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))$$
$$pdg_{7882} = \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2} + e^{- pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)} \left|{pdg_{4453} pdg_{4698} \left|{e^{pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)} \left|{pdg_{4698}}\right| + \left|{pdg_{4453}}\right|}\right|}\right|$$
1. 8497631728; locally 5493675:
$$I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi )$$
$$pdg_{7882} = 2 \cos{\left(pdg_{1575} - pdg_{8586} \right)} \left|{pdg_{4453}}\right| \left|{pdg_{4698}}\right| + \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
LHS diff is 0 RHS diff is (-2*exp(pdg4621*(pdg1575 - pdg8586))*cos(pdg1575 - pdg8586)*Abs(pdg4453*pdg4698) + Abs(pdg4453*pdg4698*Abs(exp(pdg4621*(pdg1575 - pdg8586))*Abs(pdg4698) + Abs(pdg4453))))*exp(-pdg4621*(pdg1575 - pdg8586)) 2700934933:
3085575328:
8497631728:
2700934933:
3085575328:
8497631728:
double intensity when phase is coherent (optics) conjugate both sides
1. 4192519596; locally 7875296:
$$B = |B| \exp(i \phi)$$
$$pdg_{4698} = e^{pdg_{4621} pdg_{8586}} \left|{pdg_{4698}}\right|$$
1. 4504256452; locally 1174231:
$$B^* = |B| \exp(-i \phi)$$
$$\overline{pdg_{4698}} = e^{- pdg_{4621} pdg_{8586}} \left|{pdg_{4698}}\right|$$
no check performed 4192519596:
4504256452:
4192519596:
4504256452:
double intensity when phase is coherent (optics) declare final expr
1. 6556875579; locally 6088608:
$$\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2$$
$$\frac{pdg_{8251}}{pdg_{2435}} = 2$$
no validation is available for declarations 6556875579:
6556875579:
double intensity when phase is coherent (optics) change variable X to Y
1. 3350830826; locally 4362190:
$$Z Z^* = |Z|^2$$
$$pdg_{3192}$$
1. 9761485403:
$$Z$$
$$pdg_{3192}$$
2. 8710504862:
$$A$$
$$pdg_{4453}$$
1. 4075539836; locally 3404497:
$$A A^* = |A|^2$$
$$pdg_{4453} \overline{pdg_{4453}} = \left|{pdg_{4453}}\right|^{2}$$
Nothing to split 3350830826:
4075539836:
3350830826:
4075539836:
double intensity when phase is coherent (optics) declare initial expr
1. 8396997949; locally 6461198:
$$I = | A + B |^2$$
$$pdg_{7882} = \left|{pdg_{4453} + pdg_{4698}}\right|^{2}$$
no validation is available for declarations 8396997949:
8396997949:
double intensity when phase is coherent (optics) simplify
1. 7621705408; locally 1405078:
$$I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi)$$
$$pdg_{7882} = e^{pdg_{1575} pdg_{4621}} e^{- pdg_{4621} pdg_{8586}} \left|{pdg_{4453} pdg_{4698} \left|{\left|{pdg_{4453}}\right| + e^{- pdg_{1575} pdg_{4621}} e^{pdg_{4621} pdg_{8586}} \left|{pdg_{4698}}\right|}\right|}\right| + \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
1. 3085575328; locally 5595798:
$$I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi))$$
$$pdg_{7882} = \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2} + e^{- pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)} \left|{pdg_{4453} pdg_{4698} \left|{e^{pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)} \left|{pdg_{4698}}\right| + \left|{pdg_{4453}}\right|}\right|}\right|$$
LHS diff is 0 RHS diff is (-exp(pdg4621*pdg8586)*Abs(pdg4453*pdg4698*Abs(exp(pdg4621*(pdg1575 - pdg8586))*Abs(pdg4698) + Abs(pdg4453))) + exp(pdg1575*pdg4621 + pdg4621*(pdg1575 - pdg8586) - (re(pdg1575) - re(pdg8586))*re(pdg4621) + (im(pdg1575) - im(pdg8586))*im(pdg4621))*Abs(pdg4453*pdg4698*Abs(exp(pdg4621*(pdg1575 - pdg8586))*Abs(pdg4453) + Abs(pdg4698))))*exp(-pdg1575*pdg4621) 7621705408:
3085575328:
7621705408:
3085575328:
double intensity when phase is coherent (optics) change variable X to Y
1. 4585932229; locally 7002927:
$$\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$$
$$\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}$$
1. 4935235303:
$$x$$
$$pdg_{4037}$$
2. 2293352649:
$$\theta - \phi$$
$$pdg_{1575} - pdg_{8586}$$
1. 3660957533; locally 9190817:
$$\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)$$
$$\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)}}{2} + \frac{e^{- pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)}}{2}$$
LHS diff is 0 RHS diff is exp(pdg1464*pdg4621)/2 - exp(-pdg1575*pdg4621 + pdg4621*pdg8586)/2 - exp(pdg1575*pdg4621 - pdg4621*pdg8586)/2 + exp(-pdg1464*pdg4621)/2 4585932229:
3660957533:
4585932229:
3660957533:
double intensity when phase is coherent (optics) change variable X to Y
1. 3350830826; locally 4362190:
$$Z Z^* = |Z|^2$$
$$pdg_{3192}$$
1. 4437214608:
$$Z$$
$$pdg_{3192}$$
2. 5623794884:
$$A + B$$
$$pdg_{4453} + pdg_{4698}$$
1. 2236639474; locally 4137499:
$$(A + B)(A + B)^* = |A + B|^2$$
$$\left(pdg_{4453} + pdg_{4698}\right)^{2} = \left|{pdg_{4453} + pdg_{4698}}\right|^{2}$$
Nothing to split 3350830826:
2236639474:
3350830826:
2236639474:
double intensity when phase is coherent (optics) divide expr 1 by expr 2
1. 1172039918; locally 7442815:
$$I_{\rm coherent} = 4 |A|^2$$
$$pdg_{8251} = 4 \left|{pdg_{4453}}\right|^{2}$$
2. 3060393541; locally 3246829:
$$I_{\rm incoherent} = 2|A|^2$$
$$pdg_{2435} = 2 \left|{pdg_{4453}}\right|^{2}$$
1. 6556875579; locally 6088608:
$$\frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2$$
$$\frac{pdg_{8251}}{pdg_{2435}} = 2$$
no check performed 1172039918:
3060393541:
6556875579:
1172039918:
3060393541:
6556875579:
double intensity when phase is coherent (optics) change variable X to Y
1. 3350830826; locally 4362190:
$$Z Z^* = |Z|^2$$
$$pdg_{3192}$$
1. 6529120965:
$$B$$
$$pdg_{4698}$$
2. 1511199318:
$$Z$$
$$pdg_{3192}$$
1. 7107090465; locally 2303305:
$$B B^* = |B|^2$$
$$pdg_{4698} \overline{pdg_{4698}} = \left|{pdg_{4698}}\right|^{2}$$
Nothing to split 3350830826:
7107090465:
3350830826:
7107090465:
double intensity when phase is coherent (optics) simplify
1. 8046208134; locally 2139818:
$$I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2$$
$$pdg_{8251} = 4 \left|{pdg_{4453}}\right|^{2}$$
1. 1172039918; locally 7442815:
$$I_{\rm coherent} = 4 |A|^2$$
$$pdg_{8251} = 4 \left|{pdg_{4453}}\right|^{2}$$
valid 8046208134:
1172039918:
8046208134:
1172039918:
double intensity when phase is coherent (optics) multiply expr 1 by expr 2
1. 4182362050; locally 4809503:
$$Z = |Z| \exp( i \theta )$$
$$pdg_{3192} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{3192}}\right|$$
2. 1928085940; locally 5663009:
$$Z^* = |Z| \exp( -i \theta )$$
$$pdg_{3192}$$
1. 9191880568; locally 4577339:
$$Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )$$
$$pdg_{3192}$$
Nothing to split 4182362050:
1928085940:
9191880568:
4182362050:
1928085940:
9191880568:
double intensity when phase is coherent (optics) distribute conjugate to factors
1. 1020854560; locally 9192406:
$$I = (A + B)(A + B)^*$$
$$pdg_{7882} = \left(pdg_{4453} + pdg_{4698}\right) \left(\overline{pdg_{4453}} + \overline{pdg_{4698}}\right)$$
1. 6306552185; locally 2300056:
$$I = (A + B)(A^* + B^*)$$
$$pdg_{7882} = \left(pdg_{4453} + pdg_{4698}\right) \left(\overline{pdg_{4453}} + \overline{pdg_{4698}}\right)$$
no check performed 1020854560:
6306552185:
1020854560:
6306552185:
double intensity when phase is coherent (optics) declare initial expr
1. 6774684564; locally 7781977:
$$\theta = \phi$$
$$pdg_{1575} = pdg_{8586}$$
no validation is available for declarations 6774684564:
6774684564:
double intensity when phase is coherent (optics) declare initial expr
1. 4182362050; locally 4809503:
$$Z = |Z| \exp( i \theta )$$
$$pdg_{3192} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{3192}}\right|$$
no validation is available for declarations 4182362050:
4182362050:
double intensity when phase is coherent (optics) change two variables in expr
1. 7607271250; locally 5513927:
$$\theta$$
$$pdg_{1575}$$
1. 4182362050:
$$Z = |Z| \exp( i \theta )$$
$$pdg_{3192} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{3192}}\right|$$
2. 1742775076:
$$Z$$
$$pdg_{3192}$$
3. 4583868070:
$$B$$
$$pdg_{4698}$$
1. 4192519596; locally 7875296:
$$B = |B| \exp(i \phi)$$
$$pdg_{4698} = e^{pdg_{4621} pdg_{8586}} \left|{pdg_{4698}}\right|$$
Nothing to split 7607271250:
4192519596:
7607271250:
4192519596:
double intensity when phase is coherent (optics) substitute LHS of expr 1 into expr 2
1. 4075539836; locally 3404497:
$$A A^* = |A|^2$$
$$pdg_{4453} \overline{pdg_{4453}} = \left|{pdg_{4453}}\right|^{2}$$
2. 8065128065; locally 9934418:
$$I = A A^* + B B^* + A B^* + B A^*$$
$$pdg_{7882} = pdg_{4453} \overline{pdg_{4453}} + pdg_{4453} \overline{pdg_{4698}} + pdg_{4698} \overline{pdg_{4453}} + pdg_{4698} \overline{pdg_{4698}}$$
1. 5125940051; locally 4729665:
$$I = |A|^2 + B B^* + A B^* + B A^*$$
$$pdg_{7882} = pdg_{4453} \overline{pdg_{4698}} + pdg_{4698} \overline{pdg_{4453}} + pdg_{4698} \overline{pdg_{4698}} + \left|{pdg_{4453}}\right|^{2}$$
valid 4075539836:
8065128065:
5125940051:
4075539836:
8065128065:
5125940051:
double intensity when phase is coherent (optics) declare initial expr
1. 8602221482; locally 4842351:
$$\langle \cos(\theta - \phi) \rangle = 0$$
$$\cos{\left(pdg_{1575} - pdg_{8586} \right)} = 0$$
no validation is available for declarations 8602221482:
8602221482:
double intensity when phase is coherent (optics) simplify
1. 9191880568; locally 4577339:
$$Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta )$$
$$pdg_{3192}$$
1. 3350830826; locally 4362190:
$$Z Z^* = |Z|^2$$
$$pdg_{3192}$$
Nothing to split 9191880568:
3350830826:
9191880568:
3350830826:
double intensity when phase is coherent (optics) substitute LHS of expr 1 into expr 2
1. 2236639474; locally 4137499:
$$(A + B)(A + B)^* = |A + B|^2$$
$$\left(pdg_{4453} + pdg_{4698}\right)^{2} = \left|{pdg_{4453} + pdg_{4698}}\right|^{2}$$
2. 8396997949; locally 6461198:
$$I = | A + B |^2$$
$$pdg_{7882} = \left|{pdg_{4453} + pdg_{4698}}\right|^{2}$$
1. 1020854560; locally 9192406:
$$I = (A + B)(A + B)^*$$
$$pdg_{7882} = \left(pdg_{4453} + pdg_{4698}\right) \left(\overline{pdg_{4453}} + \overline{pdg_{4698}}\right)$$
LHS diff is 0 RHS diff is -(pdg4453 + pdg4698)*(conjugate(pdg4453) + conjugate(pdg4698)) + Abs(pdg4453 + pdg4698)**2 2236639474:
8396997949:
1020854560:
2236639474:
8396997949:
1020854560:
double intensity when phase is coherent (optics) substitute LHS of expr 1 into expr 2
1. 2719691582; locally 9739736:
$$|A| = |B|$$
$$\left|{pdg_{4453}}\right| = \left|{pdg_{4698}}\right|$$
2. 6240206408; locally 8093224:
$$I_{\rm incoherent} = |A|^2 + |B|^2$$
$$pdg_{2435} = \left|{pdg_{4453}}\right|^{2} + \left|{pdg_{4698}}\right|^{2}$$
1. 6529793063; locally 5409843:
$$I_{\rm incoherent} = |A|^2 + |A|^2$$
$$pdg_{2435} = 2 \left|{pdg_{4453}}\right|^{2}$$
LHS diff is 0 RHS diff is -2*Abs(pdg4453)**2 + 2*Abs(pdg4698)**2 2719691582:
6240206408:
6529793063:
2719691582:
6240206408:
6529793063:
double intensity when phase is coherent (optics) substitute LHS of expr 1 into expr 2
1. 2719691582; locally 9739736:
$$|A| = |B|$$
$$\left|{pdg_{4453}}\right| = \left|{pdg_{4698}}\right|$$
2. 8283354808; locally 2413866:
$$I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 )$$
$$pdg_{8251} = \left|{pdg_{4453}}\right|^{2} + 2 \left|{pdg_{4453}}\right| \left|{pdg_{4698}}\right| + \left|{pdg_{4698}}\right|^{2}$$
1. 8046208134; locally 2139818:
$$I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2$$
$$pdg_{8251} = 4 \left|{pdg_{4453}}\right|^{2}$$
LHS diff is 0 RHS diff is -4*Abs(pdg4453)**2 + 4*Abs(pdg4698)**2 2719691582:
8283354808:
8046208134:
2719691582:
8283354808:
8046208134:
double intensity when phase is coherent (optics) simplify
1. 6306552185; locally 2300056:
$$I = (A + B)(A^* + B^*)$$
$$pdg_{7882} = \left(pdg_{4453} + pdg_{4698}\right) \left(\overline{pdg_{4453}} + \overline{pdg_{4698}}\right)$$
1. 8065128065; locally 9934418:
$$I = A A^* + B B^* + A B^* + B A^*$$
$$pdg_{7882} = pdg_{4453} \overline{pdg_{4453}} + pdg_{4453} \overline{pdg_{4698}} + pdg_{4698} \overline{pdg_{4453}} + pdg_{4698} \overline{pdg_{4698}}$$
valid 6306552185:
8065128065:
6306552185:
8065128065:
double intensity when phase is coherent (optics) conjugate both sides
1. 4182362050; locally 4809503:
$$Z = |Z| \exp( i \theta )$$
$$pdg_{3192} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{3192}}\right|$$
1. 1928085940; locally 5663009:
$$Z^* = |Z| \exp( -i \theta )$$
$$pdg_{3192}$$
Nothing to split 4182362050:
1928085940:
4182362050:
1928085940:
double intensity when phase is coherent (optics) conjugate both sides
1. 1357848476; locally 2018605:
$$A = |A| \exp(i \theta)$$
$$pdg_{4453} = e^{pdg_{1575} pdg_{4621}} \left|{pdg_{4453}}\right|$$
1. 6555185548; locally 1584527:
$$A^* = |A| \exp(-i \theta)$$
$$\overline{pdg_{4453}} = e^{- pdg_{1575} pdg_{4621}} \left|{pdg_{4453}}\right|$$
no check performed 1357848476:
6555185548:
1357848476:
6555185548:
double intensity when phase is coherent (optics) multiply both sides by
1. 3660957533; locally 9190817:
$$\cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)$$
$$\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)}}{2} + \frac{e^{- pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)}}{2}$$
1. 3967985562:
$$2$$
$$2$$
1. 2700934933; locally 8635275:
$$2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)$$
$$2 \cos{\left(pdg_{1464} \right)} = e^{pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)} + e^{- pdg_{4621} \left(pdg_{1575} - pdg_{8586}\right)}$$
valid 3660957533:
2700934933:
3660957533:
2700934933:
Lorentz transformation expr 1 is equivalent to expr 2 under the condition
1. 4287102261; locally 6319661:
$$x^2 + y^2 + z^2 = c^2 t^2$$
$$pdg_{4037}^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1467}^{2} pdg_{4567}^{2}$$
2. 1586866563; locally 4202425:
$$\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)$$
$$- 2 pdg_{1357} pdg_{1467} pdg_{1790}^{2} pdg_{4037} + pdg_{4037}^{2} \left(pdg_{1790}^{2} - \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}}\right) + pdg_{5647}^{2} + pdg_{6728}^{2} - \frac{2 pdg_{1467} pdg_{4037} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357}} = pdg_{1467}^{2} \left(- pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{1790}^{2} pdg_{4567}^{2}\right)$$
1. 1916173354; locally 3640931:
$$-\gamma^2 v^2 + c^2 \gamma^2 = c^2$$
$$- pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{1790}^{2} pdg_{4567}^{2} = pdg_{4567}^{2}$$
no check performed 4287102261:
1586866563:
1916173354:
4287102261:
1586866563:
1916173354:
based on the comparison of the t^2 terms
Lorentz transformation declare initial expr
1. 4662369843; locally 5427510:
$$x' = \gamma (x - v t)$$
$$pdg_{5456} = pdg_{1790} \left(- pdg_{1357} pdg_{1467} + pdg_{1464}\right)$$
no validation is available for declarations 4662369843:
4662369843:
equation 1-13 on page 21 in \cite{1999_Tipler_Llewellyn}
Lorentz transformation declare assumption
1. 8515803375; locally 7666907:
$$z' = z$$
$$pdg_{4306} = pdg_{6728}$$
no validation is available for declarations 8515803375:
8515803375:
Lorentz transformation swap LHS with RHS
1. 8730201316; locally 7546640:
$$\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'$$
$$pdg_{1790}$$
1. 5148266645; locally 1693888:
$$t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t$$
$$pdg_{1790}$$
Nothing to split 8730201316:
5148266645:
8730201316:
5148266645:
Lorentz transformation declare initial expr
1. 4287102261; locally 6319661:
$$x^2 + y^2 + z^2 = c^2 t^2$$
$$pdg_{4037}^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1467}^{2} pdg_{4567}^{2}$$
no validation is available for declarations 4287102261:
4287102261:
Lorentz transformation add X to both sides
1. 5763749235; locally 8195408:
$$-c^2 + c^2 \gamma^2 = v^2 \gamma^2$$
$$pdg_{1790}^{2} pdg_{4567}^{2} - pdg_{4567}^{2} = pdg_{1357}^{2} pdg_{1790}^{2}$$
1. 6408214498:
$$c^2$$
$$pdg_{4567}^{2}$$
1. 2999795755; locally 6913493:
$$c^2 \gamma^2 = v^2 \gamma^2 + c^2$$
$$pdg_{1790}^{2} pdg_{4567}^{2} = pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{4567}^{2}$$
valid 5763749235:
2999795755:
5763749235:
2999795755:
Lorentz transformation subtract X from both sides
1. 7741202861; locally 6463955:
$$x = \gamma^2 x - \gamma^2 v t + \gamma v t'$$
$$pdg_{4037} = - pdg_{1357} pdg_{1467} pdg_{1790}^{2} + pdg_{1357} pdg_{1790} pdg_{4989} + pdg_{1790}^{2} pdg_{4037}$$
1. 7337056406:
$$\gamma^2 x$$
$$pdg_{1790}^{2} pdg_{4037}$$
1. 4139999399; locally 8494407:
$$x - \gamma^2 x = - \gamma^2 v t + \gamma v t'$$
$$- pdg_{1790}^{2} pdg_{4037} + pdg_{4037} = - pdg_{1357} pdg_{1467} pdg_{1790}^{2} + pdg_{1357} pdg_{1790} pdg_{4989}$$
valid 7741202861:
4139999399:
7741202861:
4139999399:
Lorentz transformation divide both sides by
1. 2417941373; locally 7403799:
$$- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2$$
$$- \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}} = 1 - pdg_{1790}^{2}$$
1. 5787469164:
$$1 - \gamma^2$$
$$1 - pdg_{1790}^{2}$$
1. 1639827492; locally 4052253:
$$- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1$$
$$- \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357}^{2} pdg_{1790}^{2}} = 1$$
valid 2417941373:
1639827492:
2417941373:
1639827492:
Lorentz transformation multiply both sides by
1. 1639827492; locally 4052253:
$$- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1$$
$$- \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357}^{2} pdg_{1790}^{2}} = 1$$
1. 5669500954:
$$v^2 \gamma^2$$
$$pdg_{1357}^{2} pdg_{1790}^{2}$$
1. 5763749235; locally 8195408:
$$-c^2 + c^2 \gamma^2 = v^2 \gamma^2$$
$$pdg_{1790}^{2} pdg_{4567}^{2} - pdg_{4567}^{2} = pdg_{1357}^{2} pdg_{1790}^{2}$$
valid 1639827492:
5763749235:
1639827492:
5763749235:
Lorentz transformation simplify
1. 1974334644; locally 5995189:
$$\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'$$
$$pdg_{1467} pdg_{1790} + \frac{\operatorname{pdg_{4037}}{\left(1 - pdg_{1790}^{2} \right)}}{pdg_{1357} pdg_{1790}} = pdg_{4989}$$
1. 8730201316; locally 7546640:
$$\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t'$$
$$pdg_{1790}$$
Nothing to split 1974334644:
8730201316:
1974334644:
8730201316:
Lorentz transformation simplify
1. 3426941928; locally 4471422:
$$x = \gamma ( \gamma (x - v t) + v t' )$$
$$pdg_{4037} = pdg_{1790} \left(pdg_{1357} pdg_{4989} + pdg_{1790} \left(- pdg_{1357} pdg_{1467} + pdg_{4037}\right)\right)$$
1. 2096918413; locally 7169020:
$$x = \gamma ( \gamma x - \gamma v t + v t' )$$
$$pdg_{4037} = \operatorname{pdg_{1790}}{\left(- pdg_{1357} pdg_{1467} pdg_{1790} + pdg_{1357} pdg_{4989} + pdg_{1790} pdg_{4037} \right)}$$
LHS diff is 0 RHS diff is pdg1790*(pdg1357*pdg4989 - pdg1790*(pdg1357*pdg1467 - pdg4037)) - pdg1790(-pdg1357*pdg1467*pdg1790 + pdg1357*pdg4989 + pdg1790*pdg4037) 3426941928:
2096918413:
3426941928:
2096918413:
Lorentz transformation expr 1 is equivalent to expr 2 under the condition
1. 4287102261; locally 6319661:
$$x^2 + y^2 + z^2 = c^2 t^2$$
$$pdg_{4037}^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1467}^{2} pdg_{4567}^{2}$$
2. 1586866563; locally 4202425:
$$\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)$$
$$- 2 pdg_{1357} pdg_{1467} pdg_{1790}^{2} pdg_{4037} + pdg_{4037}^{2} \left(pdg_{1790}^{2} - \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}}\right) + pdg_{5647}^{2} + pdg_{6728}^{2} - \frac{2 pdg_{1467} pdg_{4037} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357}} = pdg_{1467}^{2} \left(- pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{1790}^{2} pdg_{4567}^{2}\right)$$
1. 3182633789; locally 2562123:
$$\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1$$
$$pdg_{1790}^{2} - \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}} = 1$$
no check performed 4287102261:
1586866563:
3182633789:
4287102261:
1586866563:
3182633789:
based on the comparison of the x^2 terms
Lorentz transformation subtract X from both sides
1. 3182633789; locally 2562123:
$$\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1$$
$$pdg_{1790}^{2} - \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}} = 1$$
1. 5284610349:
$$\gamma^2$$
$$pdg_{1790}^{2}$$
1. 2417941373; locally 7403799:
$$- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2$$
$$- \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}} = 1 - pdg_{1790}^{2}$$
valid 3182633789:
2417941373:
3182633789:
2417941373:
solve for \gamma
Lorentz transformation square root both sides
1. 7906112355; locally 7595841:
$$\gamma^2 = \frac{c^2}{c^2 - \gamma^2}$$
$$pdg_{1790}^{2} = \frac{pdg_{4567}^{2}}{- pdg_{1790}^{2} + pdg_{4567}^{2}}$$
1. 1528310784; locally 3040283:
$$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
$$pdg_{1790} = \frac{1}{\sqrt{- \frac{pdg_{1357}^{2}}{pdg_{4567}^{2}} + 1}}$$
2. 8360117126; locally 6010461:
$$\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}}$$
$$pdg_{1790} = - \frac{1}{\sqrt{- \frac{pdg_{1357}^{2}}{pdg_{4567}^{2}} + 1}}$$
no check performed 7906112355:
1528310784:
8360117126:
7906112355:
1528310784:
8360117126:
Lorentz transformation simplify
1. 9805063945; locally 4326342:
$$\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2$$
$$pdg_{1790}^{2} \left(- pdg_{1357} pdg_{1467} + pdg_{4037}\right)^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1790}^{2} pdg_{4567}^{2} \left(pdg_{1467} + \frac{pdg_{4037} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357} pdg_{1790}^{2}}\right)^{2}$$
1. 1935543849; locally 6066191:
$$\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2$$
$$pdg_{1357}^{2} pdg_{1467}^{2} pdg_{1790}^{2} - 2 pdg_{1357} pdg_{1467} pdg_{1790}^{2} pdg_{4037} + pdg_{1790}^{2} pdg_{4037}^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1467}^{2} pdg_{1790}^{2} pdg_{4567}^{2} + \frac{2 pdg_{1467} pdg_{4037} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1790}} + \frac{pdg_{4037}^{2} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1790}^{2}}$$
LHS diff is 0 RHS diff is pdg4567**2*(-pdg1357**2*pdg1467**2*pdg1790**4 + 2*pdg1357**2*pdg1467*pdg1790*pdg4037*(pdg1790**2 - 1) + pdg1357**2*pdg4037**2*(pdg1790**2 - 1) + (pdg1357*pdg1467*pdg1790**2 - pdg4037*(pdg1790**2 - 1))**2)/(pdg1357**2*pdg1790**2) 9805063945:
1935543849:
9805063945:
1935543849:
expanded the squared terms
Lorentz transformation declare final expr
1. 1528310784; locally 3040283:
$$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$
$$pdg_{1790} = \frac{1}{\sqrt{- \frac{pdg_{1357}^{2}}{pdg_{4567}^{2}} + 1}}$$
no validation is available for declarations 1528310784:
1528310784:
Lorentz factor definition
Lorentz transformation declare initial expr
1. 1201689765; locally 5649086:
$$x'^2 + y'^2 + z'^2 = c^2 t'^2$$
$$pdg_{1888}^{2} + pdg_{4306}^{2} + pdg_{5456}^{2} = pdg_{4567}^{2} pdg_{4989}^{2}$$
no validation is available for declarations 1201689765:
1201689765:
Lorentz transformation simplify
1. 2096918413; locally 7169020:
$$x = \gamma ( \gamma x - \gamma v t + v t' )$$
$$pdg_{4037} = \operatorname{pdg_{1790}}{\left(- pdg_{1357} pdg_{1467} pdg_{1790} + pdg_{1357} pdg_{4989} + pdg_{1790} pdg_{4037} \right)}$$
1. 7741202861; locally 6463955:
$$x = \gamma^2 x - \gamma^2 v t + \gamma v t'$$
$$pdg_{4037} = - pdg_{1357} pdg_{1467} pdg_{1790}^{2} + pdg_{1357} pdg_{1790} pdg_{4989} + pdg_{1790}^{2} pdg_{4037}$$
LHS diff is 0 RHS diff is pdg1357*pdg1467*pdg1790**2 - pdg1357*pdg1790*pdg4989 - pdg1790**2*pdg4037 + pdg1790(-pdg1357*pdg1467*pdg1790 + pdg1357*pdg4989 + pdg1790*pdg4037) 2096918413:
7741202861:
2096918413:
7741202861:
Lorentz transformation declare assumption
1. 7057864873; locally 6316097:
$$y' = y$$
$$pdg_{1888} = pdg_{5647}$$
no validation is available for declarations 7057864873:
7057864873:
Lorentz transformation substitute LHS of expr 1 into expr 2
1. 4662369843; locally 5427510:
$$x' = \gamma (x - v t)$$
$$pdg_{5456} = pdg_{1790} \left(- pdg_{1357} pdg_{1467} + pdg_{1464}\right)$$
2. 2983053062; locally 2283140:
$$x = \gamma (x' + v t')$$
$$pdg_{4037} = pdg_{1790} \left(pdg_{1357} pdg_{4989} + pdg_{5456}\right)$$
1. 3426941928; locally 4471422:
$$x = \gamma ( \gamma (x - v t) + v t' )$$
$$pdg_{4037} = pdg_{1790} \left(pdg_{1357} pdg_{4989} + pdg_{1790} \left(- pdg_{1357} pdg_{1467} + pdg_{4037}\right)\right)$$
LHS diff is 0 RHS diff is pdg1790**2*(pdg1464 - pdg4037) 4662369843:
2983053062:
3426941928:
4662369843:
2983053062:
3426941928:
solve output expr for t'
Lorentz transformation divide both sides by
1. 9409776983; locally 6047713:
$$x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'$$
$$pdg_{1790}$$
1. 2226340358:
$$\gamma v$$
$$pdg_{1357} pdg_{1790}$$
1. 1974334644; locally 5995189:
$$\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t'$$
$$pdg_{1467} pdg_{1790} + \frac{\operatorname{pdg_{4037}}{\left(1 - pdg_{1790}^{2} \right)}}{pdg_{1357} pdg_{1790}} = pdg_{4989}$$
Nothing to split 9409776983:
1974334644:
9409776983:
1974334644:
Lorentz transformation simplify
1. 1935543849; locally 6066191:
$$\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2$$
$$pdg_{1357}^{2} pdg_{1467}^{2} pdg_{1790}^{2} - 2 pdg_{1357} pdg_{1467} pdg_{1790}^{2} pdg_{4037} + pdg_{1790}^{2} pdg_{4037}^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1467}^{2} pdg_{1790}^{2} pdg_{4567}^{2} + \frac{2 pdg_{1467} pdg_{4037} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1790}} + \frac{pdg_{4037}^{2} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1790}^{2}}$$
1. 1586866563; locally 4202425:
$$\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)$$
$$- 2 pdg_{1357} pdg_{1467} pdg_{1790}^{2} pdg_{4037} + pdg_{4037}^{2} \left(pdg_{1790}^{2} - \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}}\right) + pdg_{5647}^{2} + pdg_{6728}^{2} - \frac{2 pdg_{1467} pdg_{4037} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357}} = pdg_{1467}^{2} \left(- pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{1790}^{2} pdg_{4567}^{2}\right)$$
LHS diff is pdg1357**2*pdg1467**2*pdg1790**2 - 2*pdg1467*pdg1790**2*pdg4037*pdg4567**2/pdg1357 + 2*pdg1467*pdg4037*pdg4567**2/pdg1357 + pdg4037**2*pdg4567**2*(pdg1790**2 - 1)**2/(pdg1357**2*pdg1790**2) RHS diff is (pdg1357**2*pdg1467**2*pdg1790**4 - 2*pdg1467*pdg1790*pdg4037*pdg4567**2*(pdg1790**2 - 1) - pdg4037**2*pdg4567**2*(pdg1790**2 - 1))/pdg1790**2 1935543849:
1586866563:
1935543849:
1586866563:
grouped by terms for x^2, xt, and t^2
Lorentz transformation factor out X from LHS
1. 2542420160; locally 5207615:
$$c^2 \gamma^2 - v^2 \gamma^2 = c^2$$
$$- pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{1790}^{2} pdg_{4567}^{2} = pdg_{4567}^{2}$$
1. 7743841045:
$$\gamma^2$$
$$pdg_{1790}^{2}$$
1. 7513513483; locally 8842089:
$$\gamma^2 (c^2 - v^2) = c^2$$
$$pdg_{1790}^{2} \left(- pdg_{1357}^{2} + pdg_{4567}^{2}\right) = pdg_{4567}^{2}$$
valid 2542420160:
7513513483:
2542420160:
7513513483:
Lorentz transformation expr 1 is equivalent to expr 2 under the condition
1. 4287102261; locally 6319661:
$$x^2 + y^2 + z^2 = c^2 t^2$$
$$pdg_{4037}^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1467}^{2} pdg_{4567}^{2}$$
2. 1586866563; locally 4202425:
$$\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right)$$
$$- 2 pdg_{1357} pdg_{1467} pdg_{1790}^{2} pdg_{4037} + pdg_{4037}^{2} \left(pdg_{1790}^{2} - \frac{pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)^{2}}{pdg_{1357}^{2} pdg_{1790}^{2}}\right) + pdg_{5647}^{2} + pdg_{6728}^{2} - \frac{2 pdg_{1467} pdg_{4037} pdg_{4567}^{2} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357}} = pdg_{1467}^{2} \left(- pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{1790}^{2} pdg_{4567}^{2}\right)$$
1. 2076171250; locally 6685577:
$$-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0$$
$$pdg_{1790}$$
Nothing to split 4287102261:
1586866563:
2076171250:
4287102261:
1586866563:
2076171250:
based on the comparison of the (x t) terms
Lorentz transformation substitute LHS of four expressions into expr
1. 8515803375; locally 7666907:
$$z' = z$$
$$pdg_{4306} = pdg_{6728}$$
2. 7057864873; locally 6316097:
$$y' = y$$
$$pdg_{1888} = pdg_{5647}$$
3. 5148266645; locally 1693888:
$$t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t$$
$$pdg_{1790}$$
4. 4662369843; locally 5427510:
$$x' = \gamma (x - v t)$$
$$pdg_{5456} = pdg_{1790} \left(- pdg_{1357} pdg_{1467} + pdg_{1464}\right)$$
5. 1201689765; locally 5649086:
$$x'^2 + y'^2 + z'^2 = c^2 t'^2$$
$$pdg_{1888}^{2} + pdg_{4306}^{2} + pdg_{5456}^{2} = pdg_{4567}^{2} pdg_{4989}^{2}$$
1. 9805063945; locally 4326342:
$$\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2$$
$$pdg_{1790}^{2} \left(- pdg_{1357} pdg_{1467} + pdg_{4037}\right)^{2} + pdg_{5647}^{2} + pdg_{6728}^{2} = pdg_{1790}^{2} pdg_{4567}^{2} \left(pdg_{1467} + \frac{pdg_{4037} \left(1 - pdg_{1790}^{2}\right)}{pdg_{1357} pdg_{1790}^{2}}\right)^{2}$$
Nothing to split 8515803375:
7057864873:
5148266645:
4662369843:
1201689765:
9805063945:
8515803375:
7057864873:
5148266645:
4662369843:
1201689765:
9805063945:
Lorentz transformation add X to both sides
1. 9031609275; locally 3992172:
$$x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'$$
$$pdg_{1790}$$
1. 8014566709:
$$\gamma^2 v t$$
$$pdg_{1357} pdg_{1467} pdg_{1790}^{2}$$
1. 9409776983; locally 6047713:
$$x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t'$$
$$pdg_{1790}$$
Nothing to split 9031609275:
9409776983:
9031609275:
9409776983:
Lorentz transformation subtract X from both sides
1. 2999795755; locally 6913493:
$$c^2 \gamma^2 = v^2 \gamma^2 + c^2$$
$$pdg_{1790}^{2} pdg_{4567}^{2} = pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{4567}^{2}$$
1. 3412946408:
$$v^2 \gamma^2$$
$$pdg_{1357}^{2} pdg_{1790}^{2}$$
1. 2542420160; locally 5207615:
$$c^2 \gamma^2 - v^2 \gamma^2 = c^2$$
$$- pdg_{1357}^{2} pdg_{1790}^{2} + pdg_{1790}^{2} pdg_{4567}^{2} = pdg_{4567}^{2}$$
valid 2999795755:
2542420160:
2999795755:
2542420160:
Lorentz transformation divide both sides by
1. 7513513483; locally 8842089:
$$\gamma^2 (c^2 - v^2) = c^2$$
$$pdg_{1790}^{2} \left(- pdg_{1357}^{2} + pdg_{4567}^{2}\right) = pdg_{4567}^{2}$$
1. 8571466509:
$$c^2 - \gamma^2$$
$$- pdg_{1790}^{2} + pdg_{4567}^{2}$$
1. 7906112355; locally 7595841:
$$\gamma^2 = \frac{c^2}{c^2 - \gamma^2}$$
$$pdg_{1790}^{2} = \frac{pdg_{4567}^{2}}{- pdg_{1790}^{2} + pdg_{4567}^{2}}$$
LHS diff is pdg1790**2*(pdg1357**2 - pdg1790**2)/(pdg1790**2 - pdg4567**2) RHS diff is 0 7513513483:
7906112355:
7513513483:
7906112355:
Lorentz transformation factor out X from LHS
1. 4139999399; locally 8494407:
$$x - \gamma^2 x = - \gamma^2 v t + \gamma v t'$$
$$- pdg_{1790}^{2} pdg_{4037} + pdg_{4037} = - pdg_{1357} pdg_{1467} pdg_{1790}^{2} + pdg_{1357} pdg_{1790} pdg_{4989}$$
1. 3495403335:
$$x$$
$$pdg_{1464}$$
1. 9031609275; locally 3992172:
$$x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t'$$
$$pdg_{1790}$$
Nothing to split 4139999399:
9031609275:
4139999399:
9031609275:
Lorentz transformation declare initial expr
1. 2983053062; locally 2283140:
$$x = \gamma (x' + v t')$$
$$pdg_{4037} = pdg_{1790} \left(pdg_{1357} pdg_{4989} + pdg_{5456}\right)$$
no validation is available for declarations 2983053062:
2983053062:
equation 1-14 on page 21 in \cite{1999_Tipler_Llewellyn}
upper limit on velocity in condensed matter declare initial expr
1. 8106885760; locally 9431422:
$$\alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}$$
$$pdg_{1370} = \frac{pdg_{1999}^{2}}{4 pdg_{1054} pdg_{3141} pdg_{4567} pdg_{7940}}$$
no validation is available for declarations 8106885760:
8106885760:
upper limit on velocity in condensed matter multiply both sides by
1. 8106885760; locally 9431422:
$$\alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}$$
$$pdg_{1370} = \frac{pdg_{1999}^{2}}{4 pdg_{1054} pdg_{3141} pdg_{4567} pdg_{7940}}$$
1. 8857931498:
$$c$$
$$pdg_{4567}$$
1. 5838268428; locally 6181437:
$$\alpha c = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}$$
$$pdg_{1370} pdg_{4567} = \frac{pdg_{1999}^{2}}{4 pdg_{1054} pdg_{3141} pdg_{7940}}$$
valid 8106885760:
5838268428:
8106885760:
5838268428:
upper limit on velocity in condensed matter declare initial expr
1. 1556389363; locally 5961293:
$$E_{\rm Rydberg} = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}$$
$$pdg_{9838} = \frac{pdg_{1999}^{4} pdg_{2515}}{32 pdg_{1054}^{2} pdg_{3141}^{2} pdg_{7940}^{2}}$$
no validation is available for declarations 1556389363:
1556389363:
upper limit on velocity in condensed matter substitute RHS of expr 1 into expr 2
1. 8688588981; locally 7834577:
$$a^3 \rho = m$$
$$pdg_{3935} pdg_{5854}^{3} = pdg_{9863}$$
2. 8090924099; locally 5077893:
$$v = \sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }$$
$$pdg_{2077} = \sqrt{\frac{pdg_{2241} pdg_{6235}}{pdg_{3935} pdg_{5854}^{3}}}$$
1. 7837519722; locally 5020923:
$$v = \sqrt{f} \sqrt{\frac{E}{m}}$$
$$pdg_{2077} = \sqrt{pdg_{6235}} \sqrt{\frac{pdg_{2241}}{pdg_{9863}}}$$
LHS diff is 0 RHS diff is -sqrt(pdg6235)*sqrt(pdg2241/pdg9863) + sqrt(pdg2241*pdg6235/(pdg3935*pdg5854**3)) 8688588981:
8090924099:
7837519722:
8688588981:
8090924099:
7837519722:
upper limit on velocity in condensed matter maximum of expr
1. 2897612567; locally 8323044:
$$v = \alpha c \sqrt{ \frac{m_e}{A m_p} }$$
$$pdg_{2077} = pdg_{1370} pdg_{4567} \sqrt{\frac{pdg_{2515}}{pdg_{3285} pdg_{5916}}}$$
1. 6259833695:
$$A$$
$$pdg_{3285}$$
1. 7701249282; locally 9568206:
$$v_u = \alpha c \sqrt{ \frac{m_e}{m_p} }$$
$$pdg_{4635} = pdg_{1370} pdg_{4567} \sqrt{\frac{pdg_{2515}}{pdg_{5916}}}$$
no check performed 2897612567:
7701249282:
2897612567:
7701249282:
upper limit on velocity in condensed matter substitute LHS of expr 1 into expr 2
1. 5646314683; locally 6979804:
$$m = A m_p$$
$$pdg_{9863} = pdg_{3285} pdg_{5916}$$
2. 5789289057; locally 5883117:
$$v = \alpha c \sqrt{ \frac{m_e}{2 m} }$$
$$pdg_{2077} = \frac{\sqrt{2} pdg_{1370} pdg_{4567} \sqrt{\frac{pdg_{2515}}{pdg_{9863}}}}{2}$$
1. 2897612567; locally 8323044:
$$v = \alpha c \sqrt{ \frac{m_e}{A m_p} }$$
$$pdg_{2077} = pdg_{1370} pdg_{4567} \sqrt{\frac{pdg_{2515}}{pdg_{3285} pdg_{5916}}}$$
LHS diff is 0 RHS diff is pdg1370*pdg4567*sqrt(pdg2515/(pdg3285*pdg5916))*(-2 + sqrt(2))/2 5646314683:
5789289057:
2897612567:
5646314683:
5789289057:
2897612567:
upper limit on velocity in condensed matter declare final expr
1. 7701249282; locally 9568206:
$$v_u = \alpha c \sqrt{ \frac{m_e}{m_p} }$$
$$pdg_{4635} = pdg_{1370} pdg_{4567} \sqrt{\frac{pdg_{2515}}{pdg_{5916}}}$$
no validation is available for declarations 7701249282:
7701249282:
upper limit on velocity in condensed matter substitute LHS of expr 1 into expr 2
1. 4107032818; locally 6901924:
$$E_{\rm Rydberg} = E$$
$$pdg_{9838} = pdg_{2241}$$
2. 1556389363; locally 5961293:
$$E_{\rm Rydberg} = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}$$
$$pdg_{9838} = \frac{pdg_{1999}^{4} pdg_{2515}}{32 pdg_{1054}^{2} pdg_{3141}^{2} pdg_{7940}^{2}}$$
1. 3291685884; locally 3642765:
$$E = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}$$
$$pdg_{2241} = \frac{pdg_{1999}^{4} pdg_{2515}}{32 pdg_{1054}^{2} pdg_{3141}^{2} pdg_{7940}^{2}}$$
valid 4107032818:
1556389363:
3291685884:
4107032818:
1556389363:
3291685884:
upper limit on velocity in condensed matter declare assumption
1. 4107032818; locally 6901924:
$$E_{\rm Rydberg} = E$$
$$pdg_{9838} = pdg_{2241}$$
no validation is available for declarations 4107032818:
4107032818:
upper limit on velocity in condensed matter simplify
1. 3935058307; locally 2063484:
$$v = \sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }$$
$$pdg_{2077} = \frac{\sqrt{2} \sqrt{\frac{pdg_{1999}^{4} pdg_{2515}}{pdg_{1054}^{2} pdg_{3141}^{2} pdg_{7940}^{2} pdg_{9863}}}}{8}$$
1. 9640720571; locally 4586348:
$$v = \frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}$$
$$pdg_{2077} = \frac{\sqrt{2} pdg_{1999}^{2} \sqrt{\frac{pdg_{2515}}{pdg_{9863}}}}{8 pdg_{1054} pdg_{3141} pdg_{7940}}$$
LHS diff is 0 RHS diff is sqrt(2)*(pdg1054*pdg3141*pdg7940*sqrt(pdg1999**4*pdg2515/(pdg1054**2*pdg3141**2*pdg7940**2*pdg9863)) - pdg1999**2*sqrt(pdg2515/pdg9863))/(8*pdg1054*pdg3141*pdg7940) 3935058307:
9640720571:
3935058307:
9640720571:
upper limit on velocity in condensed matter multiply both sides by
1. 8908736791; locally 2438445:
$$\rho = \frac{m}{a^3}$$
$$pdg_{3935} = \frac{pdg_{9863}}{pdg_{5854}^{3}}$$
1. 2397692197:
$$a^3$$
$$pdg_{5854}^{3}$$
1. 8688588981; locally 7834577:
$$a^3 \rho = m$$
$$pdg_{3935} pdg_{5854}^{3} = pdg_{9863}$$
valid 8908736791:
8688588981:
8908736791:
8688588981:
upper limit on velocity in condensed matter declare initial expr
1. 9376481176; locally 2178289:
$$K = f \frac{E}{a^3}$$
$$K = \frac{pdg_{2241} pdg_{6235}}{pdg_{5854}^{3}}$$
no validation is available for declarations 9376481176:
9376481176:
upper limit on velocity in condensed matter declare initial expr
1. 8908736791; locally 2438445:
$$\rho = \frac{m}{a^3}$$
$$pdg_{3935} = \frac{pdg_{9863}}{pdg_{5854}^{3}}$$
no validation is available for declarations 8908736791:
8908736791:
upper limit on velocity in condensed matter substitute LHS of expr 1 into expr 2
1. 9376481176; locally 2178289:
$$K = f \frac{E}{a^3}$$
$$K = \frac{pdg_{2241} pdg_{6235}}{pdg_{5854}^{3}}$$
2. 6504442697; locally 9155336:
$$v = \sqrt{ \frac{K}{\rho} }$$
$$pdg_{2077} = \sqrt{\frac{K}{pdg_{3935}}}$$
1. 8090924099; locally 5077893:
$$v = \sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }$$
$$pdg_{2077} = \sqrt{\frac{pdg_{2241} pdg_{6235}}{pdg_{3935} pdg_{5854}^{3}}}$$
valid 9376481176:
6504442697:
8090924099:
9376481176:
6504442697:
8090924099:
upper limit on velocity in condensed matter declare initial expr
1. 4560648264; locally 1719451:
$$v = \sqrt{ \frac{K + (4/3) G}{\rho} }$$
$$pdg_{2077} = \sqrt{\frac{pdg_{1466} + \frac{4 pdg_{3033}}{3}}{pdg_{3935}}}$$
no validation is available for declarations 4560648264:
4560648264:
upper limit on velocity in condensed matter substitute LHS of expr 1 into expr 2
1. 3291685884; locally 3642765:
$$E = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}$$
$$pdg_{2241} = \frac{pdg_{1999}^{4} pdg_{2515}}{32 pdg_{1054}^{2} pdg_{3141}^{2} pdg_{7940}^{2}}$$
2. 9854442418; locally 4534919:
$$v = \sqrt{\frac{E}{m}}$$
$$pdg_{2077} = \sqrt{\frac{pdg_{2241}}{pdg_{9863}}}$$
1. 3935058307; locally 2063484:
$$v = \sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }$$
$$pdg_{2077} = \frac{\sqrt{2} \sqrt{\frac{pdg_{1999}^{4} pdg_{2515}}{pdg_{1054}^{2} pdg_{3141}^{2} pdg_{7940}^{2} pdg_{9863}}}}{8}$$
valid 3291685884:
9854442418:
3935058307:
3291685884:
9854442418:
3935058307:
upper limit on velocity in condensed matter declare initial expr
1. 5646314683; locally 6979804:
$$m = A m_p$$
$$pdg_{9863} = pdg_{3285} pdg_{5916}$$
no validation is available for declarations 5646314683:
5646314683:
upper limit on velocity in condensed matter drop non-dominant term
1. 7837519722; locally 5020923:
$$v = \sqrt{f} \sqrt{\frac{E}{m}}$$
$$pdg_{2077} = \sqrt{pdg_{6235}} \sqrt{\frac{pdg_{2241}}{pdg_{9863}}}$$
1. 3685779219:
$$\sqrt{f} \approx 2$$
$$2 approx \sqrt{pdg_{6235}}$$
1. 9854442418; locally 4534919:
$$v = \sqrt{\frac{E}{m}}$$
$$pdg_{2077} = \sqrt{\frac{pdg_{2241}}{pdg_{9863}}}$$
no check performed 7837519722:
9854442418:
7837519722:
9854442418:
upper limit on velocity in condensed matter drop non-dominant term
1. 4560648264; locally 1719451:
$$v = \sqrt{ \frac{K + (4/3) G}{\rho} }$$
$$pdg_{2077} = \sqrt{\frac{pdg_{1466} + \frac{4 pdg_{3033}}{3}}{pdg_{3935}}}$$
1. 9674924517:
$$K >> G$$
$$pdg_{1466} > pdg_{3033}$$
1. 6504442697; locally 9155336:
$$v = \sqrt{ \frac{K}{\rho} }$$
$$pdg_{2077} = \sqrt{\frac{K}{pdg_{3935}}}$$
no check performed 4560648264:
6504442697:
4560648264:
6504442697:
upper limit on velocity in condensed matter substitute LHS of expr 1 into expr 2
1. 5838268428; locally 6181437:
$$\alpha c = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}$$
$$pdg_{1370} pdg_{4567} = \frac{pdg_{1999}^{2}}{4 pdg_{1054} pdg_{3141} pdg_{7940}}$$
2. 9640720571; locally 4586348:
$$v = \frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}$$
$$pdg_{2077} = \frac{\sqrt{2} pdg_{1999}^{2} \sqrt{\frac{pdg_{2515}}{pdg_{9863}}}}{8 pdg_{1054} pdg_{3141} pdg_{7940}}$$
1. 5789289057; locally 5883117:
$$v = \alpha c \sqrt{ \frac{m_e}{2 m} }$$
$$pdg_{2077} = \frac{\sqrt{2} pdg_{1370} pdg_{4567} \sqrt{\frac{pdg_{2515}}{pdg_{9863}}}}{2}$$
LHS diff is 0 RHS diff is sqrt(2)*sqrt(pdg2515/pdg9863)*(-4*pdg1054*pdg1370*pdg3141*pdg4567*pdg7940 + pdg1999**2)/(8*pdg1054*pdg3141*pdg7940) 5838268428:
9640720571:
5789289057:
5838268428:
9640720571:
5789289057:
equation of motion for a spring declare initial expr