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Review curl curl identity

step inference rule input feed output step validity (as per SymPy)
1
  • 111299: 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
2
  • 111935: replace curl with LeviCevita summation contravariant
  • number of inputs: 1; feeds: 0; outputs: 1
  • Replace curl in Eq.~\\ref{eq:#1} with Levi-Cevita contravariant; yields Eq.~\\ref{eq:#2}.
  1. 7575859295
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E}=\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
  1. 7575859300
    \(\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k=\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
 recognized infrule but not yet supported
3
  • 111935: replace curl with LeviCevita summation contravariant
  • number of inputs: 1; feeds: 0; outputs: 1
  • Replace curl in Eq.~\\ref{eq:#1} with Levi-Cevita contravariant; yields Eq.~\\ref{eq:#2}.
  1. 7575859300
    \(\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k=\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
  1. 7575859302
    \(\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n=\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
 recognized infrule but not yet supported
4
  • 111299: declare identity
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an identity.
  1. 7575859304
    \(\epsilon^{i,j,k} \epsilon_{n,j,k}=\delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}\)
 no validation is available for declarations
5
  • 111634: substitute RHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute RHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 7575859302
    \(\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n=\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
  1. 7575859304
    \(\epsilon^{i,j,k} \epsilon_{n,j,k}=\delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}\)
  1. 7575859306
    \(\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n=\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
 'Symbol' object is not callable
6
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 7575859306
    \(\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n=\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
  1. 7575859308
    \(\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right)=\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
 'Symbol' object is not callable
7
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 7575859308
    \(\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right)=\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
  1. 7575859310
    \(\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n=\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
 '(' was never closed (<string>, line 1)
8
  • 111894: replace summation notation with vector notation
  • number of inputs: 1; feeds: 0; outputs: 1
  • Replace summation notation in Eq.~\\ref{eq:#1} with vector notation; yields Eq.~\\ref{eq:#2}.
  1. 7575859310
    \(\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n=\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
  1. 7575859312
    \(\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})=\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
 recognized infrule but not yet supported
9
  • 111345: claim LHS equals RHS
  • number of inputs: 1; feeds: 0; outputs: 0
  • Thus we see that LHS of Eq.~\\ref{eq:#1} is equal to RHS.
  1. 7575859312
    \(\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})=\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
 invalid syntax (<string>, line 0)


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