## review derivation: electric field wave equation: from time dependent to time independent

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
5 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:
6 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:
2 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:
3 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:
4 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:
7 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:
10 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:
11 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:
9 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: failed
9485384858:
1858578388:
4585828572: N/A
9485384858:
1 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:
8 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: failed
4585828572: N/A
Physics Derivation Graph: Steps for electric field wave equation: from time dependent to time independent

## Symbols for this derivation

symbol ID category latex scope dimension name value Used in derivations references
4621 variable i
$$i$$
['imaginary'] dimensionless imaginary unit
74
2718 constant \exp
$$\exp$$
['real'] dimensionless e 2.71828   unitless
8
4326 variable \vec{E}
$$\vec{E}$$
complex dimensionless electric field
9
7940 constant \epsilon_0
$$\epsilon_0$$
real
• electric charge: 2
• length: -3
• mass: -1
• time: 2
vacuum permittivity, permittivity of free space or electric constant or the distributed capacitance of the vacuum 8.8541878128E-{12}   F/m
14
1467 variable t
$$t$$
['real']
• time: 1
time
121
2321 variable \omega
$$\omega$$
['real']
• time: -1
angular frequency
26
4567 constant c
$$c$$
['real']
• length: 1
• time: -1
speed of light in vacuum 299792458   meters/second
32
6197 constant \mu_0
$$\mu_0$$
real
• electric charge: -2
• length: 1
• mass: 1
vacuum permeability, permeability of free space, permeability of vacuum, or magnetic constant 1.25663706212E^{-6}   N/A^2
8
6238 variable E
$$E$$
real dimensionless electric field
20
9472 variable \vec{r}
$$\vec{r}$$
real
• length: 1
• str_note
24
MESSAGES:
• local variable 'all_df' referenced before assignment
• in step 1002928: name 'electric_charge' is not defined
• in step 1002928: name 'electric_charge' is not defined
• in step 2319391: name 'electric_charge' is not defined
• in step 2319391: name 'electric_charge' is not defined
• in step 4585829: name 'electric_charge' is not defined
• in step 4858282: name 'electric_charge' is not defined
• in step 4858282: name 'electric_charge' is not defined
• in step 4955966: name 'electric_charge' is not defined
• in step 4955966: name 'electric_charge' is not defined
• in step 5839535: name 'electric_charge' is not defined
• in step 5839535: name 'electric_charge' is not defined
• in step 7419980: name 'electric_charge' is not defined
• in step 8485758: name 'electric_charge' is not defined