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Review Maxwell equations to electric field wave equation

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.
  1. 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.
  1. 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.
  1. 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}.
  1. 1314864131
    \(\vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}\)
  1. 0005626421
    \(t\)
  1. 1314464131
    \(\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)
 Not evaluated due to missing term in SymPy
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}.
  1. 9991999979
    \(\vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}\)
  1. 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}.
  1. 1314464131
    \(\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)
  2. 9291999979
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}\)
  1. 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.
  1. 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}.
  1. 9919999981
    \(\rho = 0\)
  2. 9999999981
    \(\vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0\)
  1. 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.
  1. 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}.
  1. 7466829492
    \(\vec{ \nabla} \cdot \vec{E} = 0\)
  2. 7575859295
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
  1. 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**2*pdg0004326 + nabla(-nabla**2*pdg0006238)
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}.
  1. 3947269979
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)
  2. 1636453295
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E}\)
  1. 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.
  1. 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

Steps and expressions for Maxwell equations to electric field wave equation

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