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Review curl curl identity

step inference rule input feed output step validity (as per SymPy)
1
  • 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
2
  • 0000111935: 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
  • 0000111935: 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
  • 0000111299: 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
  • 0000111634: 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
    \(\epsilon^{i,j,k} \epsilon_{n,j,k} \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
  2. 7575859304
    \(\epsilon^{i,j,k} \epsilon_{n,j,k} = \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h}\)
  1. 7575859306
    \(\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} - \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \right) \hat{x}_i \nabla_j \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
 LHS diff is nabla**pdg0007930*nabla_{pdg0001552}*pdg0006238**pdg0001592*pdg0008349*(KroneckerDelta(h, pdg0007930)*KroneckerDelta(pdg0008304, pdg0009690) - KroneckerDelta(pdg0001552, pdg0008304)*KroneckerDelta(pdg0007930, pdg0009690)) + (-pdg0001552 + pdg0009690)**2*(pdg0001552 - pdg0001592)*(pdg0001552 - pdg0007984)*(-pdg0001592 + pdg0009690)*(-pdg0007984 + pdg0009690)/4 RHS diff is nabla**2*pdg0004326*(nabla - 1) - KroneckerDelta(h, pdg0007930)*KroneckerDelta(pdg0008304, pdg0009690) + KroneckerDelta(pdg0001552, pdg0008304)*KroneckerDelta(pdg0007930, pdg0009690)
6
  • 0000111457: 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
    \(\left( \delta^{l}_{\ \ j} \delta^{m}_{\ \ k} \hat{x}_i \nabla_j \nabla^m E^n\right)-\left( \delta^{l}_{\ \ k} \delta^{m}_{\ \ h} \hat{x}_i \nabla_j \nabla^m E^n \right) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
 Not evaluated due to missing term in SymPy
7
  • 0000111457: 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
    \(\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
 Not evaluated due to missing term in SymPy
8
  • 0000111894: 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
  • 0000111345: 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
    \(\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E}) = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
Not evaluated due to missing term in SymPy

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