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Review Maxwell equations to electric field wave equation

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.			9991999979 $\vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}$	no validation is available for declarations
1.3	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			1314864131 $\vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}$	no validation is available for declarations
1.6	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			9999999981 $\vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0$	no validation is available for declarations
3	0000111680: partially differentiate with respect to number of inputs: 1; feeds: 1; outputs: 1 Partially differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.	1314864131 $\vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}$	0005626421 $t$	1314464131 $\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$	recognized infrule but not yet supported
4	0000111776: take curl of both sides number of inputs: 1; feeds: 0; outputs: 1 Apply curl to both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	9991999979 $\vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}$		9291999979 $\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}$	recognized infrule but not yet supported
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}.	1314464131 $\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$ 9291999979 $\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}$		3947269979 $\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$	Not evaluated due to missing term in SymPy
7	0000111104: declare assumption number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an assumption.			9919999981 $\rho = 0$	no validation is available for declarations
8	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}.	9919999981 $\rho = 0$ 9999999981 $\vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0$		7466829492 $\vec{ \nabla} \cdot \vec{E} = 0$	LHS diff is nabla*pdg0004326 - dot(pdg0006238, nabla) RHS diff is 0
9	0000111299: declare identity number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an identity.			7575859295 $\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$	no validation is available for declarations
10	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}.	7466829492 $\vec{ \nabla} \cdot \vec{E} = 0$ 7575859295 $\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})$		1636453295 $\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E}$	LHS diff is -cross(nabla, cross(nabla, pdg0004326)) + cross(pdg0006238, cross(nabla, nabla)) RHS diff is nabla*2pdg0004326 + nabla(-nabla*2pdg0006238)
11	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}.	3947269979 $\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$ 1636453295 $\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E}$		8494839423 $\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$	Not evaluated due to missing term in SymPy
12	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	8494839423 $\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}$			no validation is available for declarations

Symbols used in Maxwell equations to electric field wave equation

0000007940: $ \epsilon_0 $
0000006197: $ \mu_0 $
0000003935: $ \rho $
0000004326: $ \vec{E} $
0000002069: $ \vec{H} $
0000006238: $ E $
0000001467: $ t $

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timing of Neo4j queries:

0.006 seconds for pdg_app/to_review_derivation 86eeed65-d390-4a1b-8ffe-80655374b048
0.009 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivation8fa7f23a-47d1-49e2-9f20-8d23f2fcc38a
0.003 seconds for compute/input_feed_output_infrule_for_step: get_inference_rule_connected_to_step_ID8fa7f23a-47d1-49e2-9f20-8d23f2fcc38a
0.003 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type HAS_INPUT8fa7f23a-47d1-49e2-9f20-8d23f2fcc38a
0.002 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_FEED8fa7f23a-47d1-49e2-9f20-8d23f2fcc38a
0.002 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUT8fa7f23a-47d1-49e2-9f20-8d23f2fcc38a
0.002 seconds for compute/get_dict_of_steps_in_derivation: get_sequence_index_for_step8fa7f23a-47d1-49e2-9f20-8d23f2fcc38a
0.003 seconds for pdg_app/ 86eeed65-d390-4a1b-8ffe-80655374b048

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.			9991999979 \(\vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}\)	no validation is available for declarations
1.3	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			1314864131 \(\vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}\)	no validation is available for declarations
1.6	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			9999999981 \(\vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0\)	no validation is available for declarations
3	0000111680: partially differentiate with respect to number of inputs: 1; feeds: 1; outputs: 1 Partially differentiate Eq.~\ref{eq:#2} with respect to $#1$; yields Eq.~\ref{eq:#3}.	1314864131 \(\vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}\)	0005626421 \(t\)	1314464131 \(\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)	recognized infrule but not yet supported
4	0000111776: take curl of both sides number of inputs: 1; feeds: 0; outputs: 1 Apply curl to both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	9991999979 \(\vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}\)		9291999979 \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}\)	recognized infrule but not yet supported
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}.	1314464131 \(\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\) 9291999979 \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}\)		3947269979 \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)	Not evaluated due to missing term in SymPy
7	0000111104: declare assumption number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an assumption.			9919999981 \(\rho = 0\)	no validation is available for declarations
8	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}.	9919999981 \(\rho = 0\) 9999999981 \(\vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0\)		7466829492 \(\vec{ \nabla} \cdot \vec{E} = 0\)	LHS diff is nabla*pdg0004326 - dot(pdg0006238, nabla) RHS diff is 0
9	0000111299: declare identity number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an identity.			7575859295 \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)	no validation is available for declarations
10	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}.	7466829492 \(\vec{ \nabla} \cdot \vec{E} = 0\) 7575859295 \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)		1636453295 \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E}\)	LHS diff is -cross(nabla, cross(nabla, pdg0004326)) + cross(pdg0006238, cross(nabla, nabla)) RHS diff is nabla*2pdg0004326 + nabla(-nabla*2pdg0006238)
11	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}.	3947269979 \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\) 1636453295 \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E}\)		8494839423 \(\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)	Not evaluated due to missing term in SymPy
12	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	8494839423 \(\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)			no validation is available for declarations