The SpaceTimeVector command is a representation for spacetime indexed vectors. Note that having defined X as a label representing a sequence of spacetime coordinates using Coordinates or Setup, the Physics commands already understand $X^{\mathrm{\mu }}$ as a representation for a spacetime vector.

In a galilean system (Euclidean or Minkowski), both $X^{\mathrm{\mu }}$ and the differential of the coordinates $d{X}^{\mathrm{\mu }}$ are vectors (tensors with 1 index), and so both $\partial \left(A^{\mathrm{\mu }}\right)$ and $\partial {}_{}^{\mathrm{\mu }}$ are vectors, the latter representing

where $X_{\mathrm{\mu }}=g_{\mathrm{\mu },\mathrm{\nu }}{X}_{}^{\mathrm{\nu }}$ is the covariant spacetime vector. However, unlike the galilean case, in a curvilinear system of coordinates, $X^{\mathrm{\mu }}$ is not a vector, only $d{X}^{\mathrm{\mu }}$ is and the formula above for $\partial {}_{}^{\mathrm{\mu }}$ loses its meaning; instead, the convention used in the Physics package (it becomes the one above only in the galilean case) is

Likely, from $X_{\mathrm{\nu }}=g_{\mathrm{\alpha },\mathrm{\nu }}{X}_{}^{\mathrm{\alpha }}$, in a galilean spacetime ${\partial }_{\mathrm{\mu }}\left(X_{\mathrm{\nu }}\right)=g_{\mathrm{\mu },\mathrm{\nu }}$ while this is not correct in a nongalilean spacetime, where the metric depends on the coordinates, and the correct formula is ${\partial }_{\mathrm{\mu }}\left(X_{\mathrm{\nu }}\right)={\partial }_{\mathrm{\mu }}\left(g_{\mathrm{\alpha },\mathrm{\nu }}\right)X_{}^{\mathrm{\alpha }}+g_{\mathrm{\mu },\mathrm{\nu }}$

Remark: in tensor computations, the distinction between covariant and contravariant indices is important when the spacetime is not Euclidean and the indices assume numerical values. The label of a system of coordinates set with Coordinates represent the contravariant components of the corresponding SpaceTimeVector. To indicate than an index is contravariant you prefix it with ~. On the other hand, in Maple, the selection operation is also performed through indexation. Hence, if X is a label for a system of coordinates, entering X[1] returns x1, the contravariant component of the corresponding SpaceTimeVector, even when the index used is the number 1, the covariant version of the contravariant ~1. So in a context where covariant and contravariant indexation is relevant and where you are going to assign numerical values to the indices, it is recommended to represent the spacetime vector with the SpaceTimeVector function, as in SpaceTimeVector[mu](X), instead of directly using X[mu]. In all other cases it is safe and simpler to use X[mu] and all the Physics commands understand both representations as equivalent.

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

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

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

$\mathrm{Coordinates}\left(X\,Y\right)$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\right\}$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\,Y=\left(\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right)\right\}$

$\left\{X\,Y\right\}$

$\mathrm{Setup}\left(\mathrm{diff}=X\,\mathrm{dimension}=4\right)$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\right\}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{differentiationvariables}=\left[X\right]\,\mathrm{dimension}=4\right]$

$\mathrm{SpaceTimeVector}\left[\mathrm{\mu }\right]\left(X\right)$

$X_{\mathrm{\mu }}$

$\mathrm{diff}\left(F\left(X\right)\,X\left[\mathrm{\mu }\right]\right)$

${\partial }_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(F\left(X\right)\right)$

$\mathrm{diff}\left(F\left(X\right)\,\mathrm{SpaceTimeVector}\left[\mathrm{\mu }\right]\left(X\right)\right)$

${\partial }_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(F\left(X\right)\right)$

$\mathrm{g\_}\left[\mathrm{\mu },\mathrm{\nu }\right]\mathrm{\%diff}\left(F\left(X\right)\,X\left[\mathrm{\mu }\right]\,X\left[\mathrm{\nu }\right]\right)$

$g_{\mathrm{\mu },\mathrm{\nu }}\left(\frac{{\partial }^2}{\partial X_{\mathrm{\mu }}\partial X_{\mathrm{\nu }}}F\left(X\right)\right)$

$\mathrm{value}\left(\right)$

$g_{\mathrm{\mu },\mathrm{\nu }}{\partial }_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left({\partial }_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(F\left(X\right)\right)\right)$

$\mathrm{Simplify}\left(\right)$

$\mathrm{\square }\left(F\left(X\right)\right)$

$\mathrm{diff}\left(F\left(Y\right)\,Y\left[\mathrm{\mu }\right]\,Y\left[\mathrm{\mu }\right]\right)$

$\mathrm{\square }\left(F\left(Y\right)\,\left[Y\right]\right)$

$\mathrm{diff}\left(F\left(Y-X\right)\,X\left[\mathrm{\mu }\right]\,Y\left[\mathrm{\nu }\right]\right)$

${\partial }_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left({\partial }_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(F\left(\mathrm{y1}-\mathrm{x1}\,\mathrm{y2}-\mathrm{x2}\,\mathrm{y3}-\mathrm{x3}\,\mathrm{y4}-\mathrm{x4}\right)\,\left[Y\right]\right)\right)$

$X\left[\mathrm{~alpha}\right]X\left[\mathrm{~beta}\right]$

$X_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{X}_{\phantom{}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\beta }}$

$\mathrm{d\_}\left[\mathrm{\mu }\right]\left(\right)$

$X_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{\mathrm{\delta }}_{\mathrm{\mu }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\mu }}\mathrm{\beta }}+{\mathrm{\delta }}_{\mathrm{\mu }\phantom{\mathrm{\alpha }}}^{\phantom{\mathrm{\mu }}\mathrm{\alpha }}{X}_{\phantom{}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\beta }}$

$\mathrm{dAlembertian}\left(\right)$

$2g_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}$

$\mathrm{LeviCivita}\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }\right]$

${\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}{X}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}{X}_{\phantom{}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\beta }}$

$\mathrm{Simplify}\left(\right)$

$0$

$\mathrm{SpaceTimeVector}\left[\mathrm{~1}\right]\left(X\right)$

$\mathrm{x1}$

$\mathrm{SpaceTimeVector}\left[1\right]\left(X\right)=\mathrm{g\_}\left[1,\mathrm{\mu }\right]X\left[\mathrm{\mu }\right]$

$-\mathrm{x1}=X_{\mathrm{\mu }}{\mathrm{\delta }}_{1\phantom{\mathrm{\mu }}}^{\phantom{1}\mathrm{\mu }}$

$\mathrm{Simplify}\left(\right)$

$-\mathrm{x1}=-\mathrm{x1}$

$\mathrm{g\_}\left[\right]$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{SpaceTimeVector}\left[\mathrm{~mu}\right]\left(X\right)$

$X_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}$

$\mathrm{\%d\_}\left[1\right]\left(\right)=\mathrm{\%diff}\left(\,\mathrm{x1}\right)$

${\partial }_1\left(X_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)=\frac{\partial }{\partial \mathrm{x1}}{X}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}$

$\mathrm{value}\left(\right)$

${\mathrm{\delta }}_{1\phantom{\mathrm{\mu }}}^{\phantom{1}\mathrm{\mu }}={\mathrm{\delta }}_{1\phantom{\mathrm{\mu }}}^{\phantom{1}\mathrm{\mu }}$

$\mathrm{\%diff}\left(\,\mathrm{\%SpaceTimeVector}\left[\mathrm{~1}\right]\left(X\right)\right)=\mathrm{\%diff}\left(\,X\left[\mathrm{~1}\right]\right)$

$\frac{\partial }{\partial X_{\phantom{}\phantom{1}}^{\phantom{}1}}{X}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\frac{\partial }{\partial X_{\phantom{}\phantom{1}}^{\phantom{}1}}{X}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}$

$\mathrm{value}\left(\right)$

${\mathrm{\delta }}_{1\phantom{\mathrm{\mu }}}^{\phantom{1}\mathrm{\mu }}={\mathrm{\delta }}_{1\phantom{\mathrm{\mu }}}^{\phantom{1}\mathrm{\mu }}$