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ElectromagneticTensor.wl
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(* ::Package:: *)
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_]] :=
ElectromagneticTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
Join[{"\[FormalCapitalPhi]"}, (Superscript["\[FormalCapitalA]", ToString[#1]] & ) /@ Range[Length[matrixRepresentation] - 1]],
Subscript["\[FormalMu]", "0"], True, True] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
index1_, index2_] := ElectromagneticTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates,
metricIndex1, metricIndex2], Join[{"\[FormalCapitalPhi]"}, (Superscript["\[FormalCapitalA]", ToString[#1]] & ) /@
Range[Length[matrixRepresentation] - 1]], Subscript["\[FormalMu]", "0"], index1, index2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List] := ElectromagneticTensor[ResourceFunction["MetricTensor"][matrixRepresentation,
coordinates, metricIndex1, metricIndex2], electromagneticPotential, Subscript["\[FormalMu]", "0"], True, True] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, index1_, index2_] :=
ElectromagneticTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
electromagneticPotential, Subscript["\[FormalMu]", "0"], index1, index2] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_] :=
ElectromagneticTensor[ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
electromagneticPotential, vacuumPermeability, True, True] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] == Length[matrixRepresentation]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["MatrixRepresentation"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newElectromagneticPotential = electromagneticPotential /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; If[index1 === True && index2 === True, electromagneticTensor,
If[index1 === False && index2 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]], If[index1 === True && index2 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,Last[index]]]*
electromagneticTensor[[First[index],#1]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]], If[index1 === False && index2 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
electromagneticTensor[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]], Indeterminate]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ReducedMatrixRepresentation"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newElectromagneticPotential = electromagneticPotential /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; If[index1 === True && index2 === True, FullSimplify[electromagneticTensor],
If[index1 === False && index2 === False, FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]], If[index1 === True && index2 === False,
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,
Last[index]]]*electromagneticTensor[[First[index],#1]] & ) /@ Range[Length[
electromagneticPotential]]]] & ) /@ Tuples[Range[Length[electromagneticPotential]], 2]]]],
If[index1 === False && index2 === True, FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
electromagneticTensor[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]], Indeterminate]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["SymbolicMatrixRepresentation"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newElectromagneticPotential = electromagneticPotential /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> Inactive[D][covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
Inactive[D][covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; If[index1 === True && index2 === True, electromagneticTensor,
If[index1 === False && index2 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]], If[index1 === True && index2 === False,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[#1,Last[index]]]*
electromagneticTensor[[First[index],#1]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]], If[index1 === False && index2 === True,
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
electromagneticTensor[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]], Indeterminate]]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["MetricTensor"] :=
ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["Coordinates"] :=
coordinates /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["CoordinateOneForms"] :=
(If[Head[#1] === Subscript, Subscript[StringJoin["\[FormalD]", ToString[First[#1]]], ToString[Last[#1]]],
If[Head[#1] === Superscript, Superscript[StringJoin["\[FormalD]", ToString[First[#1]]], ToString[Last[#1]]],
StringJoin["\[FormalD]", ToString[#1]]]] & ) /@ coordinates /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["Indices"] :=
{index1, index2} /; SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["CovariantQ"] :=
If[index1 === True && index2 === True, True, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ContravariantQ"] :=
If[index1 === False && index2 === False, True, False] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["MixedQ"] :=
If[(index1 === True && index2 === False) || (index1 === False && index2 === True), True, False] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["Symbol"] :=
If[index1 === True && index2 === True, Subscript["\[FormalCapitalF]", "\[FormalMu]\[FormalNu]"], If[index1 === False && index2 === False,
Superscript["\[FormalCapitalF]", "\[FormalMu]\[FormalNu]"], If[index1 === True && index2 === False, Subsuperscript["\[FormalCapitalF]", "\[FormalMu]", "\[FormalNu]"],
If[index1 === False && index2 === True, Subsuperscript["\[FormalCapitalF]", "\[FormalNu]", "\[FormalMu]"], Indeterminate]]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ElectricField"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, covariantElectricField},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; covariantElectricField =
Normal[SparseArray[(#1 -> electromagneticTensor[[1,#1 + 1]] & ) /@ Range[Length[electromagneticPotential] - 1]]];
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index + 1,#1 + 1]]*
covariantElectricField[[#1]] & ) /@ Range[Length[electromagneticPotential] - 1]]] & ) /@
Range[Length[electromagneticPotential] - 1]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ReducedElectricField"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, covariantElectricField},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; covariantElectricField =
Normal[SparseArray[(#1 -> electromagneticTensor[[1,#1 + 1]] & ) /@ Range[Length[electromagneticPotential] - 1]]];
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index + 1,
#1 + 1]]*covariantElectricField[[#1]] & ) /@ Range[Length[electromagneticPotential] - 1]]] & ) /@
Range[Length[electromagneticPotential] - 1]]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["SymbolicElectricField"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, covariantElectricField},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> Inactive[D][covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
Inactive[D][covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; covariantElectricField =
Normal[SparseArray[(#1 -> electromagneticTensor[[1,#1 + 1]] & ) /@ Range[Length[electromagneticPotential] - 1]]];
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index + 1,#1 + 1]]*
covariantElectricField[[#1]] & ) /@ Range[Length[electromagneticPotential] - 1]]] & ) /@
Range[Length[electromagneticPotential] - 1]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["MagneticField"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, contravariantElectromagneticTensor, leviCivitaTensor, covariantMagneticField},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; leviCivitaTensor = Normal[LeviCivitaTensor[3]]; covariantMagneticField =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((-(1/2))*(leviCivitaTensor[[index,First[#1],Last[#1]]]*
contravariantElectromagneticTensor[[First[#1] + 1,Last[#1] + 1]]) & ) /@ Tuples[Range[
Length[electromagneticPotential] - 1], 2]]] & ) /@ Range[Length[electromagneticPotential] - 1]]];
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index + 1,#1 + 1]]*
covariantMagneticField[[#1]] & ) /@ Range[Length[electromagneticPotential] - 1]]] & ) /@
Range[Length[electromagneticPotential] - 1]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ReducedMagneticField"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, contravariantElectromagneticTensor, leviCivitaTensor, covariantMagneticField},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; leviCivitaTensor = Normal[LeviCivitaTensor[3]]; covariantMagneticField =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((-(1/2))*(leviCivitaTensor[[index,First[#1],Last[#1]]]*
contravariantElectromagneticTensor[[First[#1] + 1,Last[#1] + 1]]) & ) /@ Tuples[Range[
Length[electromagneticPotential] - 1], 2]]] & ) /@ Range[Length[electromagneticPotential] - 1]]];
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index + 1,
#1 + 1]]*covariantMagneticField[[#1]] & ) /@ Range[Length[electromagneticPotential] - 1]]] & ) /@
Range[Length[electromagneticPotential] - 1]]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["SymbolicMagneticField"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, contravariantElectromagneticTensor, leviCivitaTensor, covariantMagneticField},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> Inactive[D][covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
Inactive[D][covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; leviCivitaTensor = Normal[LeviCivitaTensor[3]]; covariantMagneticField =
Normal[SparseArray[(Module[{index = #1}, index -> Total[((-(1/2))*(leviCivitaTensor[[index,First[#1],Last[#1]]]*
contravariantElectromagneticTensor[[First[#1] + 1,Last[#1] + 1]]) & ) /@ Tuples[Range[
Length[electromagneticPotential] - 1], 2]]] & ) /@ Range[Length[electromagneticPotential] - 1]]];
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index + 1,#1 + 1]]*
covariantMagneticField[[#1]] & ) /@ Range[Length[electromagneticPotential] - 1]]] & ) /@
Range[Length[electromagneticPotential] - 1]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ChargeDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, electromagneticDisplacementTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
Total[(D[electromagneticDisplacementTensor[[1,#1]], newCoordinates[[#1]]] & ) /@
Range[Length[newElectromagneticPotential]]] /. (ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ReducedChargeDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, electromagneticDisplacementTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
FullSimplify[Total[(D[electromagneticDisplacementTensor[[1,#1]], newCoordinates[[#1]]] & ) /@
Range[Length[newElectromagneticPotential]]] /. (ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["SymbolicChargeDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, electromagneticDisplacementTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> Inactive[D][covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
Inactive[D][covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
Total[(Inactive[D][electromagneticDisplacementTensor[[1,#1]], newCoordinates[[#1]]] & ) /@
Range[Length[newElectromagneticPotential]]] /. (ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["CurrentDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, electromagneticDisplacementTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
Normal[SparseArray[(Module[{index = #1}, index -> Total[(D[electromagneticDisplacementTensor[[index + 1,#1]],
newCoordinates[[#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential] - 1]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ReducedCurrentDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, electromagneticDisplacementTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(D[electromagneticDisplacementTensor[[index + 1,
#1]], newCoordinates[[#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential] - 1]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["SymbolicCurrentDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, electromagneticDisplacementTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> Inactive[D][covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
Inactive[D][covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inactive[D][electromagneticDisplacementTensor[[index + 1,
#1]], newCoordinates[[#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential] - 1]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["SpacetimeCurrentDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, electromagneticDisplacementTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
Normal[SparseArray[(Module[{index = #1}, index -> Total[(D[electromagneticDisplacementTensor[[index,#1]],
newCoordinates[[#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]] /. (ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ReducedSpacetimeCurrentDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, electromagneticDisplacementTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(D[electromagneticDisplacementTensor[[index,
#1]], newCoordinates[[#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["SymbolicSpacetimeCurrentDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, electromagneticDisplacementTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> Inactive[D][covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
Inactive[D][covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inactive[D][electromagneticDisplacementTensor[[index,#1]],
newCoordinates[[#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]] /. (ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ElectromagneticStressEnergyTensor"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
electromagneticTensor[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; ResourceFunction["StressEnergyTensor"][
ResourceFunction["MetricTensor"][matrixRepresentation, coordinates, metricIndex1, metricIndex2],
stressEnergyTensor, False, False]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["PoyntingVector"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor,
poyntingCovector}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newElectromagneticPotential = electromagneticPotential /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
electromagneticTensor[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; poyntingCovector =
((1/2)*(stressEnergyTensor[[1,#1 + 1]] + stressEnergyTensor[[#1 + 1,1]]) & ) /@
Range[Length[stressEnergyTensor] - 1];
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index + 1,#1 + 1]]*
poyntingCovector[[#1]] & ) /@ Range[Length[stressEnergyTensor] - 1]]] & ) /@
Range[Length[stressEnergyTensor] - 1]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ReducedPoyntingVector"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor,
poyntingCovector}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newElectromagneticPotential = electromagneticPotential /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
electromagneticTensor[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; poyntingCovector =
((1/2)*(stressEnergyTensor[[1,#1 + 1]] + stressEnergyTensor[[#1 + 1,1]]) & ) /@
Range[Length[stressEnergyTensor] - 1]; FullSimplify[
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index + 1,#1 + 1]]*
poyntingCovector[[#1]] & ) /@ Range[Length[stressEnergyTensor] - 1]]] & ) /@
Range[Length[stressEnergyTensor] - 1]]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["SymbolicPoyntingVector"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor,
poyntingCovector}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newElectromagneticPotential = electromagneticPotential /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> Inactive[D][covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
Inactive[D][covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
electromagneticTensor[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; poyntingCovector =
((1/2)*(stressEnergyTensor[[1,#1 + 1]] + stressEnergyTensor[[#1 + 1,1]]) & ) /@
Range[Length[stressEnergyTensor] - 1];
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[index + 1,#1 + 1]]*
poyntingCovector[[#1]] & ) /@ Range[Length[stressEnergyTensor] - 1]]] & ) /@
Range[Length[stressEnergyTensor] - 1]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["MaxwellStressTensor"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
electromagneticTensor[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]];
-Normal[SparseArray[(#1 -> stressEnergyTensor[[First[#1] + 1,Last[#1] + 1]] & ) /@
Tuples[Range[Length[electromagneticPotential] - 1], 2]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ReducedMaxwellStressTensor"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
electromagneticTensor[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]];
FullSimplify[-Normal[SparseArray[(#1 -> stressEnergyTensor[[First[#1] + 1,Last[#1] + 1]] & ) /@
Tuples[Range[Length[electromagneticPotential] - 1], 2]]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["SymbolicMaxwellStressTensor"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, contravariantElectromagneticTensor, mixedElectromagneticTensor, stressEnergyTensor},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> Inactive[D][covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
Inactive[D][covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],First[#1]]]*
Inverse[matrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@ Tuples[Range[Length[electromagneticPotential]],
2]]]; mixedElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[matrixRepresentation][[First[index],#1]]*
electromagneticTensor[[#1,Last[index]]] & ) /@ Range[Length[electromagneticPotential]]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]]; stressEnergyTensor =
(1/vacuumPermeability)*Normal[SparseArray[
(Module[{index = #1}, index -> Total[(contravariantElectromagneticTensor[[First[index],#1]]*
mixedElectromagneticTensor[[Last[index],#1]] & ) /@ Range[Length[electromagneticPotential]]] -
(1/4)*Total[(Inverse[matrixRepresentation][[First[index],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]]*contravariantElectromagneticTensor[[First[#1],Last[#1]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[electromagneticPotential]], 2]]];
-Normal[SparseArray[(#1 -> stressEnergyTensor[[First[#1] + 1,Last[#1] + 1]] & ) /@
Tuples[Range[Length[electromagneticPotential] - 1], 2]]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["LagrangianDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, contravariantElectromagneticTensor, electromagneticDisplacementTensor, currentDensity},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],First[#1]]]*
Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
currentDensity = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(D[electromagneticDisplacementTensor[[index,#1]], newCoordinates[[
#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]];
(Sqrt[-Det[newMatrixRepresentation]]/(4*vacuumPermeability))*
Total[(electromagneticTensor[[First[#1],Last[#1]]]*contravariantElectromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]] +
Total[(covariantElectromagneticPotential[[#1]]*currentDensity[[#1]] & ) /@
Range[Length[newElectromagneticPotential]]] /. (ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ReducedLagrangianDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, contravariantElectromagneticTensor, electromagneticDisplacementTensor, currentDensity},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],First[#1]]]*
Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
currentDensity = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(D[electromagneticDisplacementTensor[[index,#1]], newCoordinates[[
#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]];
FullSimplify[(Sqrt[-Det[newMatrixRepresentation]]/(4*vacuumPermeability))*
Total[(electromagneticTensor[[First[#1],Last[#1]]]*contravariantElectromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]] +
Total[(covariantElectromagneticPotential[[#1]]*currentDensity[[#1]] & ) /@
Range[Length[newElectromagneticPotential]]] /. (ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["SymbolicLagrangianDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, contravariantElectromagneticTensor, electromagneticDisplacementTensor, currentDensity},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> Inactive[D][covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
Inactive[D][covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; contravariantElectromagneticTensor =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],First[#1]]]*
Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
currentDensity = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inactive[D][electromagneticDisplacementTensor[[index,#1]],
newCoordinates[[#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]];
(Sqrt[-Det[newMatrixRepresentation]]/(4*vacuumPermeability))*
Total[(electromagneticTensor[[First[#1],Last[#1]]]*contravariantElectromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]] +
Total[(covariantElectromagneticPotential[[#1]]*currentDensity[[#1]] & ) /@
Range[Length[newElectromagneticPotential]]] /. (ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["LorentzForceDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, electromagneticDisplacementTensor, currentDensity},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
currentDensity = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(D[electromagneticDisplacementTensor[[index,#1]], newCoordinates[[
#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]];
Normal[SparseArray[(Module[{index = #1}, index -> Total[(electromagneticTensor[[index,#1]]*currentDensity[[
#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]] /. (ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ReducedLorentzForceDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, electromagneticDisplacementTensor, currentDensity},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> D[covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
D[covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
currentDensity = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(D[electromagneticDisplacementTensor[[index,#1]], newCoordinates[[
#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]];
FullSimplify[Normal[SparseArray[(Module[{index = #1}, index -> Total[(electromagneticTensor[[index,#1]]*
currentDensity[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]] /. (ToExpression[#1] -> #1 & ) /@
Select[coordinates, StringQ]]] /; SymbolName[metricTensor] === "MetricTensor" &&
Length[Dimensions[matrixRepresentation]] == 2 && Length[coordinates] == Length[matrixRepresentation] &&
BooleanQ[metricIndex1] && BooleanQ[metricIndex2] && Length[electromagneticPotential] ==
Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["SymbolicLorentzForceDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor, electromagneticDisplacementTensor, currentDensity},
newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newElectromagneticPotential = electromagneticPotential /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*
newElectromagneticPotential[[#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]]; electromagneticTensor =
Normal[SparseArray[(#1 -> Inactive[D][covariantElectromagneticPotential[[Last[#1]]], newCoordinates[[First[#1]]]] -
Inactive[D][covariantElectromagneticPotential[[First[#1]]], newCoordinates[[Last[#1]]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]]; electromagneticDisplacementTensor =
(Sqrt[-Det[newMatrixRepresentation]]/vacuumPermeability)*
Normal[SparseArray[(Module[{index = #1}, index -> Total[(Inverse[newMatrixRepresentation][[First[index],
First[#1]]]*Inverse[newMatrixRepresentation][[Last[#1],Last[index]]]*electromagneticTensor[[First[#1],
Last[#1]]] & ) /@ Tuples[Range[Length[newElectromagneticPotential]], 2]]] & ) /@
Tuples[Range[Length[newElectromagneticPotential]], 2]]];
currentDensity = Normal[SparseArray[
(Module[{index = #1}, index -> Total[(Inactive[D][electromagneticDisplacementTensor[[index,#1]],
newCoordinates[[#1]]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]];
Normal[SparseArray[(Module[{index = #1}, index -> Total[(electromagneticTensor[[index,#1]]*currentDensity[[
#1]] & ) /@ Range[Length[newElectromagneticPotential]]]] & ) /@
Range[Length[newElectromagneticPotential]]]] /. (ToExpression[#1] -> #1 & ) /@ Select[coordinates, StringQ]] /;
SymbolName[metricTensor] === "MetricTensor" && Length[Dimensions[matrixRepresentation]] == 2 &&
Length[coordinates] == Length[matrixRepresentation] && BooleanQ[metricIndex1] && BooleanQ[metricIndex2] &&
Length[electromagneticPotential] == Length[matrixRepresentation] && BooleanQ[index1] && BooleanQ[index2]
ElectromagneticTensor[(metricTensor_)[matrixRepresentation_List, coordinates_List, metricIndex1_, metricIndex2_],
electromagneticPotential_List, vacuumPermeability_, index1_, index2_]["ElectromagneticDisplacementDensity"] :=
Module[{newMatrixRepresentation, newElectromagneticPotential, newCoordinates, covariantElectromagneticPotential,
electromagneticTensor}, newMatrixRepresentation = matrixRepresentation /. (#1 -> ToExpression[#1] & ) /@
Select[coordinates, StringQ]; newElectromagneticPotential = electromagneticPotential /.
(#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
newCoordinates = coordinates /. (#1 -> ToExpression[#1] & ) /@ Select[coordinates, StringQ];
covariantElectromagneticPotential =
Normal[SparseArray[(Module[{index = #1}, index -> Total[(newMatrixRepresentation[[index,#1]]*