The D_[mu] command is a computational representation for ${▿}_{\mathrm{\mu }}$, the covariant differential operator, a tensor, the generalization of ${\partial }_{\mathrm{\mu }}$ to curvilinear coordinates.

D_ can also be used without an index, as in D_(A) displayed as $▿\left(A\right)$, in which case it represents the total differential in curvilinear coordinates, and the output comes automatically expanded as D_[mu](A) * d_(X[~mu]), displayed as $▿\left(A\right)\mathbf{\ⅆ}\left(X^{\mathrm{\mu }}\right)$ where X is a coordinates system and X[~mu] is the corresponding spacetime vector.

The covariant D_[mu] = ${▿}_{\mathrm{\mu }}$ is defined in terms of the Christoffel symbols as

when acting on a covariant tensor as $A_{\mathrm{\nu }}$, and as

when acting on a contravariant tensor. For tensors of higher rank, for each covariant index there is one negative Christoffel term as in the formula for $A_{\mathrm{\nu }}$ and for each contravariant index there is one positive Christoffel term as in the formula for $A^{\mathrm{\nu }}$.

To express ${▿}_{\mathrm{\mu }}\left(A_{\mathrm{\nu }}\right)$ using this definition in terms of Christoffel symbols use convert to d_ or expand which also know about other identities for ${▿}_{\mathrm{\mu }}$.

The contravariant $g_{}^{\mathrm{\mu },\mathrm{\nu }}{▿}_{\mathrm{\nu }}={▿}_{}^{\mathrm{\mu }}$ is entered as D_[~mu], so using the same D_ just with a contravariant index.

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_ or D_[mu], say as in D_[mu](A), the differentiation variables are the default ones, as indicated by Setup(differentiationvariables). Note you can set various coordinate systems and choose which one is to be considered the differentiation variables for D_ using Setup.

You can also override the default differentiation variables by passing two arguments to D_ or D_[mu], in which case the second argument is expected to be a list with the differentiation variables, as in D_[mu](A, [x1, x2, ...]), 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 with the Setup command.

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, a summary is as follows:

If $A$ does not depend on the differentiation variables or is equal to the metric g_, then 0 is returned.

If $A$ is a scalar then d_[mu](A) is returned.

If $A$ belongs to the differentiation variables, then the definition of D_[mu](A) is applied and the resulting value returned.

If $A$ is a sum, product, power, or known function, then the differentiation is distributed accordingly, the same way it is done by d_.

A number of additional identities for covariant derivatives are returned when using expand.

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_, so it represents the same mathematical operation but without performing it. To perform the operation, use value.

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

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

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

$\mathrm{Setup}\left(\mathrm{coordinatesystems}\=\mathrm{cartesian}\right)$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(x\,y\,z\,t\right)\right\}$

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

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

${\mathrm{g\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\=\left(\left[\begin{array}{rrrr}-1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 1\end{array}\right]\right)$

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

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

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

${\mathrm{Christoffel}}_{\mathrm{nonzero}}$

${\mathrm{\Gamma }}_{\mathrm{\alpha },\mathrm{\mu },\mathrm{\nu }}=\varnothing$

$\mathrm{D\_}\left(X_{\mathrm{`~nu`}}\right)$

$\mathbf{\ⅆ}\left(X_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)$

${\mathrm{D\_}}_{\mathrm{\μ}}\left(A_{\mathrm{`~nu`}}\left(X\right)\right)$

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

$\mathrm{Setup}\left(\mathrm{metric}\=\mathrm{arbitrary}\right)$

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

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

$\mathrm{The\; arbitrary\; metric\; in\; coordinates}\left[x\,y\,z\,t\right]$

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

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

${\mathrm{g\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\=\left(\left[\begin{array}{cccc}\mathrm{\_F1}\left(x\,y\,z\,t\right)& \mathrm{\_F2}\left(x\,y\,z\,t\right)& \mathrm{\_F3}\left(x\,y\,z\,t\right)& \mathrm{\_F4}\left(x\,y\,z\,t\right)\\ \mathrm{\_F2}\left(x\,y\,z\,t\right)& \mathrm{\_F5}\left(x\,y\,z\,t\right)& \mathrm{\_F6}\left(x\,y\,z\,t\right)& \mathrm{\_F7}\left(x\,y\,z\,t\right)\\ \mathrm{\_F3}\left(x\,y\,z\,t\right)& \mathrm{\_F6}\left(x\,y\,z\,t\right)& \mathrm{\_F8}\left(x\,y\,z\,t\right)& \mathrm{\_F9}\left(x\,y\,z\,t\right)\\ \mathrm{\_F4}\left(x\,y\,z\,t\right)& \mathrm{\_F7}\left(x\,y\,z\,t\right)& \mathrm{\_F9}\left(x\,y\,z\,t\right)& \mathrm{\_F10}\left(x\,y\,z\,t\right)\end{array}\right]\right)$

$\left[\mathrm{metric}=\left\{\left(1\,1\right)=\mathrm{\_F1}\left(X\right)\,\left(1\,2\right)=\mathrm{\_F2}\left(X\right)\,\left(1\,3\right)=\mathrm{\_F3}\left(X\right)\,\left(1\,4\right)=\mathrm{\_F4}\left(X\right)\,\left(2\,2\right)=\mathrm{\_F5}\left(X\right)\,\left(2\,3\right)=\mathrm{\_F6}\left(X\right)\,\left(2\,4\right)=\mathrm{\_F7}\left(X\right)\,\left(3\,3\right)=\mathrm{\_F8}\left(X\right)\,\left(3\,4\right)=\mathrm{\_F9}\left(X\right)\,\left(4\,4\right)=\mathrm{\_F10}\left(X\right)\right\}\,\mathrm{spaceindices}=\mathrm{lowercaselatin\_is}\right]$

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

${\mathrm{g\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\=\left(\left[\begin{array}{cccc}\mathrm{\_F1}\left(x\,y\,z\,t\right)& \mathrm{\_F2}\left(x\,y\,z\,t\right)& \mathrm{\_F3}\left(x\,y\,z\,t\right)& \mathrm{\_F4}\left(x\,y\,z\,t\right)\\ \mathrm{\_F2}\left(x\,y\,z\,t\right)& \mathrm{\_F5}\left(x\,y\,z\,t\right)& \mathrm{\_F6}\left(x\,y\,z\,t\right)& \mathrm{\_F7}\left(x\,y\,z\,t\right)\\ \mathrm{\_F3}\left(x\,y\,z\,t\right)& \mathrm{\_F6}\left(x\,y\,z\,t\right)& \mathrm{\_F8}\left(x\,y\,z\,t\right)& \mathrm{\_F9}\left(x\,y\,z\,t\right)\\ \mathrm{\_F4}\left(x\,y\,z\,t\right)& \mathrm{\_F7}\left(x\,y\,z\,t\right)& \mathrm{\_F9}\left(x\,y\,z\,t\right)& \mathrm{\_F10}\left(x\,y\,z\,t\right)\end{array}\right]\right)$

$\mathrm{PDEtools}:-\mathrm{declare}\left(\right)$

$\mathrm{\_F1}\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_F1}$

$\mathrm{\_F10}\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_F10}$

$\mathrm{\_F2}\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_F2}$

$\mathrm{\_F3}\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_F3}$

$\mathrm{\_F4}\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_F4}$

$\mathrm{\_F5}\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_F5}$

$\mathrm{\_F6}\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_F6}$

$\mathrm{\_F7}\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_F7}$

$\mathrm{\_F8}\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_F8}$

$\mathrm{\_F9}\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\_F9}$

${\mathrm{Christoffel}}_{1\,1\,1}$

$\frac{1}{2}\mathrm{diff}\left(\mathrm{\_F1}\left(X\right)\,x\right)$

$\mathrm{D\_}\left(A_{\mathrm{`~nu`}}\left(X\right)\right)$

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

${\mathrm{D\_}}_{\mathrm{\μ}}\left(A_{\mathrm{`~nu`}}\left(X\right)\right)$

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

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

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

${\mathrm{\%D\_}}_{\mathrm{\μ}}\left({\mathrm{g\_}}_{\mathrm{\α}\,\mathrm{\β}}\right)$

${\mathrm{\%D\_}}_{\mathrm{\μ}}\left({\mathrm{g\_}}_{\mathrm{\α}\,\mathrm{\β}}\right)$

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

$0$

${\mathrm{D\_}}_{\mathrm{\μ}}\left(\mathrm{\Φ}\left(X\right)\right)$

${\partial }_{\mathrm{\mu }}\left(\mathrm{\Phi }\left(X\right)\right)$

$\mathrm{D\_}\left(\mathrm{\Φ}\left(X\right)\right)$

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

$A_{\mathrm{\μ}}^2$

$A_{\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}$

${\mathrm{D\_}}_{\mathrm{\ν}}\left({A_{\mathrm{\μ}}\left(X\right)}^2\right)$

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

$\mathrm{ds2}≔x^2{\mathrm{dx}}^2\+y^2{\mathrm{dy}}^2\+z^2{\mathrm{dz}}^2\+xy\mathrm{dx}\mathrm{dy}-{\mathrm{dt}}^2$

$\mathrm{ds2}≔x^2{\mathrm{dx}}^2+xy\mathrm{dx}\mathrm{dy}+y^2{\mathrm{dy}}^2+z^2{\mathrm{dz}}^2-{\mathrm{dt}}^2$

$\mathrm{Setup}\left(\mathrm{metric}\=\mathrm{ds2}\right)$

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

$\mathrm{Coordinates:}\left[x\,y\,z\,t\right]\mathrm{.\; Signature:}\left(\mathrm{-\; -\; -\; +}\right)$

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

${\mathrm{g\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\=\left(\left[\begin{array}{cccc}{x}^2& \frac{1}{2}xy& 0& 0\\ \frac{1}{2}xy& y^2& 0& 0\\ 0& 0& z^2& 0\\ 0& 0& 0& -1\end{array}\right]\right)$

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

$\left[\mathrm{metric}=\left\{\left(1\,1\right)=x^2\,\left(1\,2\right)=\frac{xy}{2}\,\left(2\,2\right)=y^2\,\left(3\,3\right)=z^2\,\left(4\,4\right)=\mathrm{-1}\right\}\right]$

${\mathrm{D\_}}_{\mathrm{\μ}}\left(A_{\mathrm{`~mu`}}\left(X\right)\right)$

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

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

$\frac{A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right){\partial }_{\mathrm{\mu }}\left(x\right)}{x}+\frac{A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right){\partial }_{\mathrm{\mu }}\left(y\right)}{y}+\frac{A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right){\partial }_{\mathrm{\mu }}\left(z\right)}{z}+{\partial }_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(X\right)\right)$

Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.

The Physics[D_] command was introduced in Maple 16.

For more information on Maple 16 changes, see Updates in Maple 16.