The ExteriorDerivative[mu] command computes the exterior derivative of an expression A (possibly tensorial, according to the standard definition) as the exterior product of the covariant operator D_[mu], and a covariant, totally antisymmetric expression A, which can also be a vector in tensor notation with only one covariant index, or a scalar. In this sense, ExteriorDerivative performs, in a tensorial expression and using tensor notation, the operation that DifferentialGeometry[ExteriorDerivative] performs in the equivalent expression using differential forms.

NOTE: if the covariant index passed, mu, is also present in A as a free covariant index, the output is the contracted product D_[~mu](A[.., mu, ..]).

The antisymmetry of A with respect to permutations of its free indices is verified before proceeding using Library[IsTensorialAntisymmetric]. Note that both the antisymmetrization of A as well as the exterior product between two arbitrary tensorial expressions can always be computed passing the expression or product to Antisymmetrize.

The index mu indexing ExteriorDerivative in ExteriorDerivative[mu](A) is expected to be a spacetime index and is used to index the covariant derivative operator D_. When the spacetime is Galilean, so all the Christoffel symbols are zero, the operator d_ is used instead. When the spacetime is non-Galilean, due to the symmetry of the Christoffel symbols under permutation of their 2nd and 3rd indices, as well as the antisymmetric property of the result of ExteriorDerivative[mu](A), all the terms involving Christoffel symbols cancel so that a mathematically equivalent result can be obtained replacing D_ by d_. To obtain a result directly expressed using d_, pass d_ as the last argument.

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

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

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

$\mathrm{Setup}\left(\mathrm{coordinates}=X\right)$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{Define}\left(F\,\mathrm{antisymmetric}\,\mathrm{quiet}\right)\:$$\mathrm{Define}\left(H\,\mathrm{symmetric}\right)$

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

$\left\{F\,H\,{▿}_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,R_{\mathrm{\mu },\mathrm{\nu }}\,R_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,C_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\gamma }}_{i,j}\,{\mathrm{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }}\,G_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\right\}$

$\mathrm{PDEtools}:-\mathrm{declare}\left(\left(F\,H\right)\left(X\right)\right)$

$F\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\mathrm{will\; now\; be\; displayed\; as}F$

$H\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\mathrm{will\; now\; be\; displayed\; as}H$

$\mathrm{ExteriorDerivative}\left[\mathrm{\mu }\right]\left(F\left[\mathrm{\alpha },\mathrm{\beta }\right]\left(X\right)\right)$

$\frac{{▿}_{\mathrm{\mu }}\left(F_{\mathrm{\alpha },\mathrm{\beta }}\right)}{3}+\frac{{▿}_{\mathrm{\alpha }}\left(F_{\mathrm{\beta },\mathrm{\mu }}\right)}{3}-\frac{{▿}_{\mathrm{\beta }}\left(F_{\mathrm{\alpha },\mathrm{\mu }}\right)}{3}$

$\mathrm{ExteriorDerivative}\left[\mathrm{\mu }\right]\left(F\left[\mathrm{\alpha },\mathrm{\beta }\right]\left(X\right)\,\mathrm{d\_}\right)$

$\frac{{\partial }_{\mathrm{\mu }}\left(F_{\mathrm{\alpha },\mathrm{\beta }}\right)}{3}+\frac{{\partial }_{\mathrm{\alpha }}\left(F_{\mathrm{\beta },\mathrm{\mu }}\right)}{3}-\frac{{\partial }_{\mathrm{\beta }}\left(F_{\mathrm{\alpha },\mathrm{\mu }}\right)}{3}$

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

$\frac{{\partial }_{\mathrm{\mu }}\left(F_{\mathrm{\alpha },\mathrm{\beta }}\right)}{3}-\frac{{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\mu }}^{\phantom{}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\mu }}}{F}_{\mathrm{\nu },\mathrm{\beta }}}{3}+\frac{{\partial }_{\mathrm{\alpha }}\left(F_{\mathrm{\beta },\mathrm{\mu }}\right)}{3}-\frac{{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\mu }}^{\phantom{}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\mu }}}{F}_{\mathrm{\beta },\mathrm{\nu }}}{3}-\frac{{\partial }_{\mathrm{\beta }}\left(F_{\mathrm{\alpha },\mathrm{\mu }}\right)}{3}$

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

$\frac{{\partial }_{\mathrm{\mu }}\left(F_{\mathrm{\alpha },\mathrm{\beta }}\right)}{3}+\frac{{\partial }_{\mathrm{\alpha }}\left(F_{\mathrm{\beta },\mathrm{\mu }}\right)}{3}-\frac{{\partial }_{\mathrm{\beta }}\left(F_{\mathrm{\alpha },\mathrm{\mu }}\right)}{3}$

$\mathrm{Define}\left(T\left[\mathrm{\mu },\mathrm{\alpha },\mathrm{\beta }\right]=\right)$

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

$\left\{H\,{▿}_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,F_{\mathrm{\alpha },\mathrm{\beta }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,R_{\mathrm{\mu },\mathrm{\nu }}\,R_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,T_{\mathrm{\mu },\mathrm{\alpha },\mathrm{\beta }}\,C_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\gamma }}_{i,j}\,{\mathrm{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }}\,G_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\right\}$

$T\left[1,1,1\right]$

$0$

$T\left[1,2,3\right]$

$\frac{{\left(F_{2,3}\right)}_r}{3}-\frac{{\left(F_{1,3}\right)}_{\mathrm{\theta }}}{3}+\frac{{\left(F_{1,2}\right)}_{\mathrm{\phi }}}{3}$

$T\left[\mathrm{~1},\mathrm{~2},\mathrm{~3}\right]$

$\frac{{\csc \left(\mathrm{\theta }\right)}^2\left(2m-r\right)\left({\left(F_{2,3}\right)}_r+{\left(F_{3,1}\right)}_{\mathrm{\theta }}-{\left(F_{2,1}\right)}_{\mathrm{\phi }}\right)}{3r^5}$

$T\left[1,2,3\right]+T\left[2,1,3\right]$

$0$

$\mathrm{Riemann}\left[\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }\right]H\left[\mathrm{\alpha },\mathrm{\beta }\right]\left(X\right)$

$H_{\mathrm{\alpha },\mathrm{\beta }}{R}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}\mathrm{\mu },\mathrm{\nu }}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }\phantom{\mathrm{\mu },\mathrm{\nu }}}$

$\mathrm{Library}:-\mathrm{IsTensorialAntisymmetric}\left(\right)$

$\mathrm{true}$

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

$\frac{{▿}_{\mathrm{\mu }}\left(H_{\mathrm{\alpha },\mathrm{\beta }}\right)R_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\beta }}}{3}+\frac{H_{\mathrm{\alpha },\mathrm{\beta }}{▿}_{\mathrm{\mu }}\left(R_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\beta }}\right)}{3}-\frac{2{▿}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(H_{\mathrm{\alpha },\mathrm{\beta }}\right)R_{\mathrm{\mu },\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{\mathrm{\mu }}\phantom{,\mathrm{\nu }}\mathrm{\alpha },\mathrm{\beta }}}{3}-\frac{2H_{\mathrm{\alpha },\mathrm{\beta }}{▿}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(R_{\mathrm{\mu },\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{\mathrm{\mu }}\phantom{,\mathrm{\nu }}\mathrm{\alpha },\mathrm{\beta }}\right)}{3}$

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

$0$

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

$\frac{\left({\partial }_{\mathrm{\mu }}\left(H_{\mathrm{\alpha },\mathrm{\beta }}\right)-{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\kappa }}\mathrm{\alpha },\mathrm{\mu }}^{\phantom{}\mathrm{\kappa }\phantom{\mathrm{\alpha },\mathrm{\mu }}}{H}_{\mathrm{\kappa },\mathrm{\beta }}-{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\kappa }}\mathrm{\beta },\mathrm{\mu }}^{\phantom{}\mathrm{\kappa }\phantom{\mathrm{\beta },\mathrm{\mu }}}{H}_{\mathrm{\alpha },\mathrm{\kappa }}\right)R_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\beta }}}{3}+\frac{H_{\mathrm{\alpha },\mathrm{\beta }}\left({\partial }_{\mathrm{\mu }}\left(R_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\beta }}\right)-{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\kappa }}\mathrm{\mu },\mathrm{\nu }}^{\phantom{}\mathrm{\kappa }\phantom{\mathrm{\mu },\mathrm{\nu }}}{R}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\kappa }\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\kappa }}\mathrm{\alpha },\mathrm{\beta }}-{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\alpha }}\mathrm{\kappa },\mathrm{\mu }}^{\phantom{}\mathrm{\alpha }\phantom{\mathrm{\kappa },\mathrm{\mu }}}{R}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\nu }\phantom{\mathrm{\beta },\mathrm{\kappa }}}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\nu }}\mathrm{\beta },\mathrm{\kappa }}+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\beta }}\mathrm{\kappa },\mathrm{\mu }}^{\phantom{}\mathrm{\beta }\phantom{\mathrm{\kappa },\mathrm{\mu }}}{R}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\kappa }}}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\kappa }}+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\kappa },\mathrm{\mu }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\kappa },\mathrm{\mu }}}{R}_{\phantom{}\phantom{\mathrm{\kappa }}\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\kappa }\phantom{\mathrm{\nu }}\mathrm{\alpha },\mathrm{\beta }}\right)}{3}-\frac{2g_{\phantom{}\phantom{\mathrm{\lambda },\mathrm{\mu }}}^{\phantom{}\mathrm{\lambda },\mathrm{\mu }}\left({\partial }_{\mathrm{\lambda }}\left(H_{\mathrm{\alpha },\mathrm{\beta }}\right)-{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\kappa }}\mathrm{\alpha },\mathrm{\lambda }}^{\phantom{}\mathrm{\kappa }\phantom{\mathrm{\alpha },\mathrm{\lambda }}}{H}_{\mathrm{\kappa },\mathrm{\beta }}-{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\kappa }}\mathrm{\beta },\mathrm{\lambda }}^{\phantom{}\mathrm{\kappa }\phantom{\mathrm{\beta },\mathrm{\lambda }}}{H}_{\mathrm{\alpha },\mathrm{\kappa }}\right)R_{\mathrm{\mu },\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{\mathrm{\mu }}\phantom{,\mathrm{\nu }}\mathrm{\alpha },\mathrm{\beta }}}{3}-\frac{2H_{\mathrm{\alpha },\mathrm{\beta }}{g}_{\phantom{}\phantom{\mathrm{\lambda },\mathrm{\mu }}}^{\phantom{}\mathrm{\lambda },\mathrm{\mu }}\left({\partial }_{\mathrm{\lambda }}\left(R_{\mathrm{\mu },\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{\mathrm{\mu }}\phantom{,\mathrm{\nu }}\mathrm{\alpha },\mathrm{\beta }}\right)-{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\kappa }}\mathrm{\lambda },\mathrm{\mu }}^{\phantom{}\mathrm{\kappa }\phantom{\mathrm{\lambda },\mathrm{\mu }}}{R}_{\mathrm{\kappa },\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{\mathrm{\kappa }}\phantom{,\mathrm{\nu }}\mathrm{\alpha },\mathrm{\beta }}+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\kappa }}\mathrm{\lambda },\mathrm{\nu }}^{\phantom{}\mathrm{\kappa }\phantom{\mathrm{\lambda },\mathrm{\nu }}}{R}_{\mathrm{\kappa },\mathrm{\mu }\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{\mathrm{\kappa }}\phantom{,\mathrm{\mu }}\mathrm{\alpha },\mathrm{\beta }}-{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\alpha }}\mathrm{\kappa },\mathrm{\lambda }}^{\phantom{}\mathrm{\alpha }\phantom{\mathrm{\kappa },\mathrm{\lambda }}}{R}_{\mathrm{\mu },\mathrm{\nu }\phantom{\mathrm{\beta },\mathrm{\kappa }}}^{\phantom{\mathrm{\mu }}\phantom{,\mathrm{\nu }}\mathrm{\beta },\mathrm{\kappa }}+{\mathrm{\Gamma }}_{\phantom{}\phantom{\mathrm{\beta }}\mathrm{\kappa },\mathrm{\lambda }}^{\phantom{}\mathrm{\beta }\phantom{\mathrm{\kappa },\mathrm{\lambda }}}{R}_{\mathrm{\mu },\mathrm{\nu }\phantom{\mathrm{\alpha },\mathrm{\kappa }}}^{\phantom{\mathrm{\mu }}\phantom{,\mathrm{\nu }}\mathrm{\alpha },\mathrm{\kappa }}\right)}{3}$

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

$0$

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

Weinberg, S. Gravitation and Cosmology: Principles and Applications of The General Theory of Relativity, John Wiley & Sons, Inc, 1972.

The Physics[ExteriorDerivative] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.