The d_[mu] command is a computational representation for the differential operator $\mathrm{\∂\_\_mu}$ , defined by

where $x^{\mathrm{\mu }}$ represents the contravariant spacetime vector of a coordinate system.

d_ can also be used without an index, as in d_(A) displayed as $\partial \left(A\right)$, in which case it represents the total differential, and the output comes automatically expanded as ${\partial }_{\mathrm{\mu }}\left(A\right)\partial \left(x^{\mathrm{\mu }}\right)$ (assuming sum rule for repeated indices); unless A is already a coordinate of any system of coordinates defined using Setup or Coordinates, in which case d_(A) returns unevaluated, echoing the input, representing the differential of the coordinate A.

In a galilean system (Euclidean or Minkowski), both $x^{\mathrm{\mu }}$ and the differential of the coordinates $\partial x_{}^{\mathrm{\mu }}$ are vectors (tensors with 1 index), and so $\partial A_{}^{\mathrm{\nu }}=\left({\partial }_{\mathrm{\mu }}{A}_{}^{\mathrm{\nu }}\right)\partial x_{}^{\mathrm{\mu }}$ and ${\partial }_{}^{\mathrm{\mu }}$ are also 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, and so neither the total differential of a vector $\partial A_{}^{\mathrm{\nu }}$ nor ${\partial }_{}^{\mathrm{\mu }}$ are tensors (for a generalization of $\mathrm{\∂\_\_mu}$ to curvilinear coordinates see D_) 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

Likewise, 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 ${\partial }_{\mathrm{\mu }}\left(x_{\mathrm{\nu }}\right)={\partial }_{\mathrm{\mu }}\left(g_{\mathrm{\alpha },\mathrm{\nu }}\right)x_{}^{\mathrm{\alpha }}+g_{\mathrm{\mu },\mathrm{\nu }}$

Computations performed with the Physics package commands take into account Einstein's sum rule for repeated indices - see `.` and Simplify. The distinction between covariant and contravariant indices in the input of tensors is done by prefixing contravariant ones with ~, say as in ~mu; in the output, contravariant indices are displayed as superscripts. For contracted indices, you can enter them one covariant and one contravariant. Note however that - provided that the spacetime metric is galilean (Euclidean or Minkowski), or the object is a tensor also in curvilinear coordinates - this distinction in the input is not relevant, and so contracted indices can be entered as both covariant or both contravariant, in which case they will be automatically rewritten as one covariant and one contravariant. Tensors can have spacetime and space indices at the same time. To change the type of letter used to represent spacetime or space indices see Setup.

When only one argument is given to d_, as in d_[mu](A) or d_(A), the differentiation variables are the current default values, typically $\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}$ ($\mathrm{x0}$ is automatically assigned to $\mathrm{x4}$), represented by $X$ (see Coordinates). The default for the differentiation variables is set with the Setup command. When two arguments are given to d_, the second argument is expected to be a list with the differentiation variables to which the index in ${\mathrm{d\_}}_{\mathrm{\mu }}$ refers or the total differential is to be taken, and so this list should have as many symbols as the dimension of spacetime, which by default is 4 but can be set to any value by the Setup command.

In the particular case where d_ is not indexed and is called with one argument, and this argument is a coordinate of any coordinate system defined using Setup or Coordinates, d_ returns unevaluated, echoing the input. This permits using d_ to represent the differential of coordinates that are not the default differentiation coordinates, useful when working with transformations of coordinates (see TransformCoordinates).

Some automatic checking and simplifications are carried out each time an operation such as d_[mu](A) is executed. The checking is concerned with possible syntax errors. Regarding the automatic simplifications performed by d_, a summary of them is as follows:

- If $A$ does not depend on the differentiation variables, then 0 is returned.

- If $A$ is a sum, product, power, or known function, then the differentiation is expanded accordingly.

- If $A=\mathrm{xn}$, and $\mathrm{xn}$ belongs to the differentiation variables, then ${\mathrm{d\_}}_{\mathrm{\mu }}\left(A\right)$ returns ${\mathrm{g\_}}_{\mathrm{\mu },n}$.

- If $A={\mathrm{d\_}}_{\mathrm{\mu }}\left(B\right)$, then $\mathrm{dAlembertian}\left(B\right)$ (the d'Alembertian) is returned.

- If $A$ is an unknown function (the rule for its derivative is unknown), a Dirac delta function, or a derivative, then the result is returned unevaluated as ${\mathrm{d\_}}_{\mathrm{\mu }}\left(A\right)$.

Sometimes it is convenient to rewrite tensorial expressions involving the covariant derivative D_ or the Christoffel symbols in terms of d_ or its application to the metric g_. For this purpose you can use convert(expression, d_) - see the Examples section.

To perform the differentiation, the d_ command makes us of the diff command of the Physics package, which in turn uses the standard Maple diff command, so that any user-defined differentiation rule, such as for a function foo, of the form `diff/foo`, is automatically taken into account by d_.

The %d_ command is the inert form of d_; that is, it represents the same mathematical operation while displaying the operation unevaluated. To evaluate the operation, use the value command.

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

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

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

$\mathrm{Define}\left(A\,B\right)$

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

$\left\{A\,B\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

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

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

$\left\{X\right\}$

$\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(\left(f\,A\,h\right)\left(X\right)\right)$

$f\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\mathrm{will\; now\; be\; displayed\; as}f$

$A\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\mathrm{will\; now\; be\; displayed\; as}A$

$h\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\mathrm{will\; now\; be\; displayed\; as}h$

$\mathrm{ee}≔f\left(X\right)A\left[\mathrm{\mu }\right]\left(X\right)\cos \left(h\left(X\right)\right)+B\left[\mathrm{\mu },\mathrm{\nu }\right]A\left[\mathrm{\nu }\right]\left(X\right)\mathrm{Dirac}\left(\left[X\right]\right)$

$\mathrm{ee}≔f{A}_{\mathrm{\mu }}\cos \left(h\right)+{\mathrm{\δ}}^{\left(4\right)}\left(\left[X\right]\right)A_{\mathrm{\nu }}{B}_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}$

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

${\partial }_{\mathrm{\rho }}\left(f\right)A_{\mathrm{\mu }}\cos \left(h\right)+f{\partial }_{\mathrm{\rho }}\left(A_{\mathrm{\mu }}\right)\cos \left(h\right)-f{A}_{\mathrm{\mu }}{\partial }_{\mathrm{\rho }}\left(h\right)\sin \left(h\right)+B_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}\left({\partial }_{\mathrm{\rho }}\left({\mathrm{\δ}}^{\left(4\right)}\left(\left[X\right]\right)\right)A_{\mathrm{\nu }}+{\mathrm{\δ}}^{\left(4\right)}\left(\left[X\right]\right){\partial }_{\mathrm{\rho }}\left(A_{\mathrm{\nu }}\right)\right)$

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

$\mathrm{\square }\left(f\right)A_{\mathrm{\mu }}\cos \left(h\right)+2\cos \left(h\right){\partial }_{\mathrm{\rho }}\left(A_{\mathrm{\mu }}\right){\partial }_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(f\right)-2A_{\mathrm{\mu }}\sin \left(h\right){\partial }_{\mathrm{\rho }}\left(f\right){\partial }_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(h\right)+f\mathrm{\square }\left(A_{\mathrm{\mu }}\right)\cos \left(h\right)-2f\sin \left(h\right){\partial }_{\mathrm{\rho }}\left(A_{\mathrm{\mu }}\right){\partial }_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(h\right)-f{A}_{\mathrm{\mu }}\mathrm{\square }\left(h\right)\sin \left(h\right)-f{A}_{\mathrm{\mu }}\cos \left(h\right){\partial }_{\mathrm{\rho }}\left(h\right){\partial }_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(h\right)+B_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}\left(\mathrm{\square }\left({\mathrm{\δ}}^{\left(4\right)}\left(\left[X\right]\right)\right)A_{\mathrm{\nu }}+2{\partial }_{\mathrm{\rho }}\left({\mathrm{\δ}}^{\left(4\right)}\left(\left[X\right]\right)\right){\partial }_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(A_{\mathrm{\nu }}\right)+{\mathrm{\δ}}^{\left(4\right)}\left(\left[X\right]\right)\mathrm{\square }\left(A_{\mathrm{\nu }}\right)\right)$

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

$\mathrm{The\; repeated\; and\; free\; indices\; in\; the\; given\; expression\; check\; ok.}$

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

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

$\left[\varnothing \,\left\{\mathrm{\rho }\right\}\,\left\{\mathrm{\rho }\right\}\,\varnothing \,\left\{\mathrm{\rho }\right\}\,\varnothing \,\left\{\mathrm{\rho }\right\}\,\left\{\mathrm{\nu }\,\mathrm{\rho }\right\}\right],\left\{\mathrm{\mu }\right\}$

$\mathrm{d\_}\left(A\left(X\right)\right)$

${\partial }_{\mathrm{\mu }}\left(A\right)\mathbf{\ⅆ}\left(X_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)$

$\mathrm{d\_}\left(\mathrm{x1}\right)$

$\mathbf{\ⅆ}\left(\mathrm{x1}\right)$

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

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

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

$\mathrm{Setup}\left(\mathrm{diff}\right)$

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

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

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

$X\left[1\right]$

$\mathrm{x1}$

$\mathrm{d\_}\left(X\left[1\right]\right)$

$\mathbf{\ⅆ}\left(\mathrm{x1}\right)$

$Y\left[1\right]$

$\mathrm{y1}$

$\mathrm{d\_}\left(Y\left[1\right]\right)$

$\mathbf{\ⅆ}\left(\mathrm{y1}\right)$

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

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

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\,Y=\left(\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right)\right\}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right\}$

$\mathrm{Setting}\mathrm{lowercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{space}\mathrm{indices}$

$\mathrm{The\; Schwarzschild\; metric\; in\; coordinates}\left[r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]$

$\mathrm{Parameters:}\left[m\right]$

$\mathrm{Signature:}\left(\mathrm{-\; -\; -\; +}\right)$

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

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

$\mathrm{Define}\left(A\right)$

$\mathrm{Defined\; as\; tensors}$

$\left\{A_{\mathrm{\nu }}\,B_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}\,{▿}_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,R_{\mathrm{\mu },\mathrm{\nu }}\,R_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,C_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\gamma }}_{i,j}\,{\mathrm{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }}\,G_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\,Y_{\mathrm{\mu }}\right\}$

$\mathrm{D\_}\left[\mathrm{\mu }\right]\left(A\left[\mathrm{~nu}\right]\left(X\right)\right)$

${▿}_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)\right)$

$\mathrm{convert}\left(\,\mathrm{d\_}\right)$

${\partial }_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)\right)+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\mu }}^{\phantom{}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\mu }}}{A}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(X\right)$

$\mathrm{convert}\left(\,\mathrm{g\_}\right)$

${\partial }_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(X\right)\right)+\frac{g_{\phantom{}\phantom{\mathrm{\beta },\mathrm{\nu }}}^{\phantom{}\mathrm{\beta },\mathrm{\nu }}\left({\partial }_{\mathrm{\mu }}\left(g_{\mathrm{\alpha },\mathrm{\beta }}\right)+{\partial }_{\mathrm{\alpha }}\left(g_{\mathrm{\beta },\mathrm{\mu }}\right)-{\partial }_{\mathrm{\beta }}\left(g_{\mathrm{\alpha },\mathrm{\mu }}\right)\right)A_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(X\right)}{2}$

$X\left[\mathrm{\nu }\right]=\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\nu }\right]X\left[\mathrm{~alpha}\right]$

$X_{\mathrm{\nu }}=g_{\mathrm{\alpha },\mathrm{\nu }}{X}_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}$

$\mathrm{\%d\_}\left[\mathrm{\mu }\right]\left(\mathrm{lhs}\left(\right)\right)=\mathrm{d\_}\left[\mathrm{\mu }\right]\left(\mathrm{rhs}\left(\right)\right)$

${\partial }_{\mathrm{\mu }}\left(X_{\mathrm{\nu }}\right)={\partial }_{\mathrm{\mu }}\left(g_{\mathrm{\alpha },\mathrm{\nu }}\right)X_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}+g_{\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\nu }\right]\mathrm{g\_}\left[\mathrm{~nu},\mathrm{~beta}\right]$

$g_{\mathrm{\alpha },\mathrm{\nu }}{g}_{\phantom{}\phantom{\mathrm{\beta },\mathrm{\nu }}}^{\phantom{}\mathrm{\beta },\mathrm{\nu }}$

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

$g_{\mathrm{\alpha },\mathrm{\nu }}{g}_{\phantom{}\phantom{\mathrm{\beta },\mathrm{\nu }}}^{\phantom{}\mathrm{\beta },\mathrm{\nu }}={\mathrm{\delta }}_{\mathrm{\alpha }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\alpha }}\mathrm{\beta }}$

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

${\partial }_{\mathrm{\mu }}\left(g_{\mathrm{\alpha },\mathrm{\nu }}\right)g_{\phantom{}\phantom{\mathrm{\beta },\mathrm{\nu }}}^{\phantom{}\mathrm{\beta },\mathrm{\nu }}+g_{\mathrm{\alpha },\mathrm{\nu }}{\partial }_{\mathrm{\mu }}\left(g_{\phantom{}\phantom{\mathrm{\beta },\mathrm{\nu }}}^{\phantom{}\mathrm{\beta },\mathrm{\nu }}\right)=0$

$\mathrm{g\_}\left[\mathrm{~alpha},\mathrm{~rho}\right]·$

$g_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\rho }}}^{\phantom{}\mathrm{\alpha },\mathrm{\rho }}{g}_{\phantom{}\phantom{\mathrm{\beta },\mathrm{\nu }}}^{\phantom{}\mathrm{\beta },\mathrm{\nu }}{\partial }_{\mathrm{\mu }}\left(g_{\mathrm{\alpha },\mathrm{\nu }}\right)+{\partial }_{\mathrm{\mu }}\left(g_{\phantom{}\phantom{\mathrm{\beta },\mathrm{\rho }}}^{\phantom{}\mathrm{\beta },\mathrm{\rho }}\right)=0$

$\mathrm{isolate}\left(\,\mathrm{d\_}\left[\mathrm{\mu }\right]\left(\mathrm{g\_}\left[\mathrm{~beta},\mathrm{~rho}\right]\right)\right)$

${\partial }_{\mathrm{\mu }}\left(g_{\phantom{}\phantom{\mathrm{\beta },\mathrm{\rho }}}^{\phantom{}\mathrm{\beta },\mathrm{\rho }}\right)=-g_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\rho }}}^{\phantom{}\mathrm{\alpha },\mathrm{\rho }}{g}_{\phantom{}\phantom{\mathrm{\beta },\mathrm{\nu }}}^{\phantom{}\mathrm{\beta },\mathrm{\nu }}{\partial }_{\mathrm{\mu }}\left(g_{\mathrm{\alpha },\mathrm{\nu }}\right)$