Physics Derivation Graph navigation Sign in

review derivation: Euler equation proof

This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.


Hold the mouse over a node to highlight that node and its neighbors. You can zoom in/out. You can pan the image. You can move nodes by clicking and dragging.

Notes for this derivation:

Options
Alternate views of this derivation:
Edit this content:    

To edit a step, click on the number in the "Index" column in the table below

Clicking on the step index will take you to the page where you can edit that step.

Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
10 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: error for dim with 9482923849
 4923339482:
9482923849: N/A
7 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:
5 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:
3 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:
8 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:
2 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: error for dim with 9492920340
9429829482:
9492920340: N/A
9429829482:
6 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:
1 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: error for dim with 9492920340
 9492920340: N/A
4 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: error for dim with 9492920340
9482984922:
9848294829:
9492920340: N/A
9482984922:
9848294829:
12 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: error for dim with 4938429483
 4938429483: N/A
9 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:
11 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: error for dim with 9482923849
9492920340: error for dim with 9492920340
4938429483: error for dim with 4938429483
 9482923849: N/A
9492920340: N/A
4938429483: N/A
Physics Derivation Graph: Steps for Euler equation proof

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
2718 constant \exp
\(\exp\)		 ['real'] dimensionless e 2.71828   unitless
 8
9199 variable dx
\(dx\)		 ['real']
  • length: 1
 15
1464 variable x
\(x\)		 ['real'] dimensionless 140
1452 variable y
\(y\)		 ['real'] dimensionless 13
5842 variable dy
\(dy\)		 ['real']
  • length: 1
 differential displacement along y axis
  • str_note
2
4621 variable i
\(i\)		 ['imaginary'] dimensionless imaginary unit 74
MESSAGE: