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Review electric field wave equation: from time dependent to time independent

abstract:
reference:

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			8494839423 $\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$	no validation is available for declarations
2	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			8572852424 $\vec{E} = E( \vec{r},t)$	no validation is available for declarations
3	0000111237: declare guess solution number of inputs: 1; feeds: 0; outputs: 1 Judicious choice as a guessed solution to Eq.~\ref{eq:#1} is Eq.~\ref{eq:#2},	8494839423 $\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$		9499428242 $E( \vec{r},t) = E( \vec{r})\exp(i \omega t)$	no validation is available for declarations
4	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	8494839423 $\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$ 8572852424 $\vec{E} = E( \vec{r},t)$		9394939493 $\nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t)$	LHS diff is -nabla*2pdg0006238(pdg0009472, pdg0001467) + pdg0004326 RHS diff is (-partialpdg0006197pdg0007940 + pdg0001467*2)pdg0006238(pdg0009472, pdg0001467)/pdg0001467**2
5	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	9394939493 $\nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t)$ 9499428242 $E( \vec{r},t) = E( \vec{r})\exp(i \omega t)$		2029293929 $\nabla^2 E( \vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t)$	LHS diff is -nabla*2pdg0006238(pdg0009472)exp(pdg0001467pdg0002321pdg0004621) + pdg0006238(pdg0009472, pdg0001467) RHS diff is (-partialpdg0006197pdg0007940exp(pdg0001467pdg0002321pdg0004621) + pdg0001467*2pdg0002718(pdg0001467pdg0002321pdg0004621))pdg0006238(pdg0009472)/pdg0001467*2
6	0000111649: differentiate with respect to number of inputs: 1; feeds: 1; outputs: 1 Differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.	2029293929 $\nabla^2 E( \vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t)$	4476266504 $t$	4985825552 $\nabla^2 E( \vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t)$	LHS diff is -nabla*2pdg0006238(pdg0009472)exp(pdg0001467pdg0002321pdg0004621) RHS diff is -pdg00023212pdg0004621*2pdg0006197pdg0007940pdg0006238(pdg0009472)exp(pdg0001467pdg0002321*pdg0004621)
7	0000111649: differentiate with respect to number of inputs: 1; feeds: 1; outputs: 1 Differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.	4985825552 $\nabla^2 E( \vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t)$	0003232242 $t$	1858578388 $\nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t)$	LHS diff is nabla*2(pdg0002321pdg0004621 - 1)pdg0006238(pdg0009472)exp(pdg0001467pdg0002321pdg0004621) RHS diff is pdg00023212pdg0006197pdg0007940(pdg0002321pdg00046213 + 1)pdg0006238(pdg0009472)exp(pdg0001467pdg0002321*pdg0004621)
8	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			4585828572 $\epsilon_0 \mu_0 = \frac{1}{c^2}$	no validation is available for declarations
9	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	4585828572 $\epsilon_0 \mu_0 = \frac{1}{c^2}$ 1858578388 $\nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t)$		9485384858 $\nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t)$	LHS diff is nabla*2(-pdg0002718(pdg0001467pdg0002321pdg0004621) + exp(pdg0001467pdg0002321pdg0004621))pdg0006238(pdg0009472) RHS diff is pdg00023212(pdg0002718(pdg0001467pdg0002321pdg0004621) - exp(pdg0001467pdg0002321pdg0004621))pdg0006238(pdg0009472)/pdg0004567*2
10	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	9485384858 $\nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t)$		3485475729 $\nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r})$	LHS diff is nabla*2(pdg0002718(pdg0001467pdg0002321pdg0004621) - 1)pdg0006238(pdg0009472) RHS diff is pdg00023212(1 - pdg0002718(pdg0001467pdg0002321pdg0004621))pdg0006238(pdg0009472)/pdg0004567*2
11	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	3485475729 $\nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r})$			no validation is available for declarations

Symbols used in electric field wave equation: from time dependent to time independent

0000007940: $ \epsilon_0 $
0000002718: $ \exp $
0000006197: $ \mu_0 $
0000002321: $ \omega $
0000004326: $ \vec{E} $
0000009472: $ \vec{r} $
0000004567: $ c $
0000006238: $ E $
0000004621: $ i $
0000001467: $ t $

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0.005 seconds for pdg_app/to_review_derivation f1f1615b-9443-4237-9a6b-6c96a1735bc2
0.006 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivationb2fc425d-ec8d-40e0-9620-782d15080e99
0.002 seconds for compute/input_feed_output_infrule_for_step: get_inference_rule_connected_to_step_IDb2fc425d-ec8d-40e0-9620-782d15080e99
0.002 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type HAS_INPUTb2fc425d-ec8d-40e0-9620-782d15080e99
0.002 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_FEEDb2fc425d-ec8d-40e0-9620-782d15080e99
0.002 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUTb2fc425d-ec8d-40e0-9620-782d15080e99
0.002 seconds for compute/get_dict_of_steps_in_derivation: get_sequence_index_for_stepb2fc425d-ec8d-40e0-9620-782d15080e99
0.004 seconds for pdg_app/ f1f1615b-9443-4237-9a6b-6c96a1735bc2

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			8494839423 \(\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)	no validation is available for declarations
2	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			8572852424 \(\vec{E} = E( \vec{r},t)\)	no validation is available for declarations
3	0000111237: declare guess solution number of inputs: 1; feeds: 0; outputs: 1 Judicious choice as a guessed solution to Eq.~\ref{eq:#1} is Eq.~\ref{eq:#2},	8494839423 \(\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)		9499428242 \(E( \vec{r},t) = E( \vec{r})\exp(i \omega t)\)	no validation is available for declarations
4	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	8494839423 \(\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\) 8572852424 \(\vec{E} = E( \vec{r},t)\)		9394939493 \(\nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t)\)	LHS diff is -nabla*2pdg0006238(pdg0009472, pdg0001467) + pdg0004326 RHS diff is (-partialpdg0006197pdg0007940 + pdg0001467*2)pdg0006238(pdg0009472, pdg0001467)/pdg0001467**2
5	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	9394939493 \(\nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t)\) 9499428242 \(E( \vec{r},t) = E( \vec{r})\exp(i \omega t)\)		2029293929 \(\nabla^2 E( \vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t)\)	LHS diff is -nabla*2pdg0006238(pdg0009472)exp(pdg0001467pdg0002321pdg0004621) + pdg0006238(pdg0009472, pdg0001467) RHS diff is (-partialpdg0006197pdg0007940exp(pdg0001467pdg0002321pdg0004621) + pdg0001467*2pdg0002718(pdg0001467pdg0002321pdg0004621))pdg0006238(pdg0009472)/pdg0001467*2
6	0000111649: differentiate with respect to number of inputs: 1; feeds: 1; outputs: 1 Differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.	2029293929 \(\nabla^2 E( \vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t)\)	4476266504 \(t\)	4985825552 \(\nabla^2 E( \vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t)\)	LHS diff is -nabla*2pdg0006238(pdg0009472)exp(pdg0001467pdg0002321pdg0004621) RHS diff is -pdg00023212pdg0004621*2pdg0006197pdg0007940pdg0006238(pdg0009472)exp(pdg0001467pdg0002321*pdg0004621)
7	0000111649: differentiate with respect to number of inputs: 1; feeds: 1; outputs: 1 Differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.	4985825552 \(\nabla^2 E( \vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t)\)	0003232242 \(t\)	1858578388 \(\nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t)\)	LHS diff is nabla*2(pdg0002321pdg0004621 - 1)pdg0006238(pdg0009472)exp(pdg0001467pdg0002321pdg0004621) RHS diff is pdg00023212pdg0006197pdg0007940(pdg0002321pdg00046213 + 1)pdg0006238(pdg0009472)exp(pdg0001467pdg0002321*pdg0004621)
8	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			4585828572 \(\epsilon_0 \mu_0 = \frac{1}{c^2}\)	no validation is available for declarations
9	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	4585828572 \(\epsilon_0 \mu_0 = \frac{1}{c^2}\) 1858578388 \(\nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t)\)		9485384858 \(\nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t)\)	LHS diff is nabla*2(-pdg0002718(pdg0001467pdg0002321pdg0004621) + exp(pdg0001467pdg0002321pdg0004621))pdg0006238(pdg0009472) RHS diff is pdg00023212(pdg0002718(pdg0001467pdg0002321pdg0004621) - exp(pdg0001467pdg0002321pdg0004621))pdg0006238(pdg0009472)/pdg0004567*2
10	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	9485384858 \(\nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t)\)		3485475729 \(\nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r})\)	LHS diff is nabla*2(pdg0002718(pdg0001467pdg0002321pdg0004621) - 1)pdg0006238(pdg0009472) RHS diff is pdg00023212(1 - pdg0002718(pdg0001467pdg0002321pdg0004621))pdg0006238(pdg0009472)/pdg0004567*2
11	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	3485475729 \(\nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r})\)			no validation is available for declarations