SubstituteTensor substitutes the equation(s) EQ into ee, taking care of free and repeated indices such that:

equations in EQ are interpreted as mappings having the free indices as parameters, so for example substituting A[j] = G[j] into A[k] results in G[k];

repeated indices in EQ do not clash with repeated indices in ee, so for example substituting A[j] = G[j]*F[k,k] into A[i]*G[k]*G[k] or into A[k]*A[k] respectively results in G[i]*F[m,m]*G[k]*G[k], and G[k]*F[j,j] * G[k]*F[m,m].

SubstituteTensor can also substitute sub-expressions of type product or sum, similar to what algsubs does, so for example substituting A[j] * B[k] = F[j, k] into A[i] * B[i] * F[j, k] and into A[i] * B[j] * A[k] * B[m] respectively results in F[i, i] * F[j, k] and F[i, j] * F[k, m].

Note: when the left-hand-side of a substitution equation is a tensor function (of type Library:-PhysicsType:-Tensor and also of type function), say T[j,k](X), then not just the indices j,k but also the functionality X is considered a parameter, so for example substituting T[j, k](X) = F[j, k](X) into T[i, i](Y) results in F[i, i](Y). The functionality is not considered a parameter when the left-hand-side is an algebraic expression.

By default, it is expected that the free indices of the left and right hand sides of the substitution equation(s) are the same, and SubstituteTensor will interrupt with an error message when that doesn't check. When the free indices on both sides of a substitution equation are intentionally different, to avoid that interruption you can use the option disregardfreeindices.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{Define}\left(A\,B\,C\,F\,G\right)\:$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\mathrm{EQ}≔A\left[\mathrm{\mu }\right]=G\left[\mathrm{\nu },\mathrm{\alpha }\right]A\left[\mathrm{\alpha }\right]F\left[\mathrm{\mu },\mathrm{\nu }\right]$

$\mathrm{EQ}≔A_{\mathrm{\mu }}=G_{\mathrm{\nu },\mathrm{\alpha }}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{F}_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}$

$\mathrm{Check}\left(\mathrm{EQ}\,\mathrm{all}\right)$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\left(\left[\varnothing \right]\,\left\{\mathrm{\mu }\right\}\right)=\left(\left[\left\{\mathrm{\alpha }\,\mathrm{\nu }\right\}\right]\,\left\{\mathrm{\mu }\right\}\right)$

$\mathrm{SubstituteTensor}\left(\mathrm{EQ}\,A\left[\mathrm{\mu }\right]\right)$

$G_{\mathrm{\nu },\mathrm{\alpha }}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{F}_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}$

$\mathrm{SubstituteTensor}\left(\mathrm{EQ}\,A\left[\mathrm{\nu }\right]\right)$

$G_{\mathrm{\beta },\mathrm{\alpha }}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{F}_{\mathrm{\nu }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\nu }}\mathrm{\beta }}$

$\mathrm{SubstituteTensor}\left(\mathrm{EQ}\,A\left[\mathrm{~nu}\right]\right)$

$G_{\mathrm{\beta },\mathrm{\alpha }}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{F}_{\phantom{}\phantom{\mathrm{\nu },\mathrm{\beta }}}^{\phantom{}\mathrm{\nu },\mathrm{\beta }}$

$\mathrm{ee}≔A\left[\mathrm{\nu }\right]A\left[\mathrm{\nu }\right]$

$\mathrm{ee}≔A_{\mathrm{\nu }}{A}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}$

$\mathrm{res}≔\mathrm{SubstituteTensor}\left(\mathrm{EQ}\,\mathrm{ee}\right)$

$\mathrm{res}≔G_{\mathrm{\lambda },\mathrm{\kappa }}{A}_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}{F}_{\mathrm{\nu }\phantom{\mathrm{\lambda }}}^{\phantom{\mathrm{\nu }}\mathrm{\lambda }}{G}_{\mathrm{\beta },\mathrm{\alpha }}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{F}_{\phantom{}\phantom{\mathrm{\nu },\mathrm{\beta }}}^{\phantom{}\mathrm{\nu },\mathrm{\beta }}$

$\mathrm{Check}\left(\mathrm{res}\,\mathrm{all}\right)$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\left[\left\{\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\kappa }\,\mathrm{\lambda }\,\mathrm{\nu }\right\}\right],\varnothing$

$\mathrm{SubstituteTensor}\left(A\left[\mathrm{\mu }\right]\left(X\right)=B\left[\mathrm{\mu }\right]\left(X\right)\,A\left[\mathrm{\nu }\right]\left(Y\right)\right)$

$B_{\mathrm{\nu }}\left(Y\right)$

$\mathrm{EQ2}≔A\left[\mathrm{\mu }\right]\left(X\right)B\left[\mathrm{\nu }\right]\left(Y\right)=G\left[\mathrm{\mu },\mathrm{\nu }\right]\left(X\right)$

$\mathrm{EQ2}≔A_{\mathrm{\mu }}\left(X\right)B_{\mathrm{\nu }}\left(Y\right)=G_{\mathrm{\mu },\mathrm{\nu }}\left(X\right)$

$\mathrm{ee2}≔A\left[\mathrm{\alpha }\right]\left(X\right)B\left[\mathrm{\beta }\right]\left(Y\right)A\left[\mathrm{\rho }\right]\left(X\right)B\left[\mathrm{\rho }\right]\left(Y\right)$

$\mathrm{ee2}≔A_{\mathrm{\alpha }}\left(X\right)B_{\mathrm{\beta }}\left(Y\right)A_{\mathrm{\rho }}\left(X\right)B_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(Y\right)$

$\mathrm{indets}\left(\mathrm{ee2}\right)$

$\left\{X\,Y\,A_{\mathrm{\alpha }}\left(X\right)\,A_{\mathrm{\rho }}\left(X\right)\,B_{\mathrm{\beta }}\left(Y\right)\,B_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(Y\right)\right\}$

$\mathrm{SubstituteTensor}\left(\mathrm{EQ2}\,\mathrm{ee2}\right)$

$G_{\mathrm{\alpha },\mathrm{\beta }}\left(X\right)G_{\mathrm{\rho }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\rho }}\mathrm{\rho }}\left(X\right)$

$\mathrm{Setup}\left(\mathrm{spinorindices}=\mathrm{lowercaselatin\_is}\right)$

$\left[\mathrm{spinorindices}=\mathrm{lowercaselatin\_is}\right]$

$\mathrm{Define}\left(T\left[\mathrm{\mu },i,j\right]\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{B\,C\,A_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,F_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}\,G_{\mathrm{\nu },\mathrm{\alpha }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,T_{\mathrm{\mu },i,j}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

$\mathrm{SubstituteTensor}\left(\mathrm{Dgamma}\left[\mathrm{\mu }\right]\left[i,j\right]=T\left[\mathrm{\mu },i,j\right]\,\mathrm{Dgamma}\left[\mathrm{~alpha}\right]\left[\mathrm{~k},m\right]\right)$

$T_{\phantom{}\phantom{\mathrm{\alpha },k}m}^{\phantom{}\mathrm{\alpha },k\phantom{m}}$

$\mathrm{SubstituteTensor}\left(\mathrm{Dgamma}\left[\mathrm{\mu }\right]\left[i,j\right]=1\,\mathrm{Dgamma}\left[\mathrm{~alpha}\right]\left[\mathrm{~k},m\right]\right)$

$\mathrm{Free\; indices\; on\; both\; sides\; of\; the\; equation\; are\; different:\; found}\left\{i\,j\,\mathrm{\mu }\right\}\mathrm{on\; the\; left-hand\; side\; and}\varnothing \mathrm{on\; the\; right-hand\; side}$

Error, (in Physics:-SubstituteTensor) free indices on both sides of the equation are different

$\mathrm{SubstituteTensor}\left(\mathrm{Dgamma}\left[\mathrm{\mu }\right]\left[i,j\right]=1\,\mathrm{Dgamma}\left[\mathrm{~alpha}\right]\left[\mathrm{~k},m\right]\,\mathrm{disregardfreeindices}\right)$

$1$

The Physics[SubstituteTensor] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.

The Physics[SubstituteTensor] command was updated in Maple 2020.