The dAlembertian command is a computational representation for the d'Alembert differential operator, displayed as a square box $\square$. The definition used is

where the differential operator $\mathrm{\∂\_\_mu}$, represented in Maple by d_, is defined by

and where $x^{\mathrm{\mu }}$ represents the contravariant spacetime vector of a coordinate system defined using Coordinates or Setup.

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 ${\partial }_{}^{\mathrm{\mu }}$ is a vector representing

where $x_{\mathrm{\mu }}=g_{\mathrm{\mu },\mathrm{\nu }}{x}_{}^{\mathrm{\nu }}$ is the covariant spacetime vector, and so ${\partial }_{}^{\mathrm{\mu }}\left({\partial }_{\mathrm{\mu }}\left(A\right)\right)={\partial }_{\mathrm{\mu }}\left({\partial }_{}^{\mathrm{\mu }}\left(A\right)\right)$.

Unlike the galilean case, in a curvilinear system of coordinates, neither $\mathrm{x\_\_mu}$ nor ${\partial }_{}^{\mathrm{\mu }}$ are vectors 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

Hence in a non-galilean spacetime ${\partial }_{}^{\mathrm{\mu }}\left({\partial }_{\mathrm{\mu }}\left(A\right)\right)\ne {\partial }_{\mathrm{\mu }}\left({\partial }_{}^{\mathrm{\mu }}\left(A\right)\right)$, because the metric $g{}_{}^{\mathrm{\mu },\mathrm{\nu }}$ depends on the coordinates and so the two factors in the right-hand-side above do not commute, and the formulas relating the second application of d_[mu] to the dAlembertian are

reflecting the nontensorial character in curvilinear coordinates of ${\partial }_{}^{\mathrm{\mu }}$  and of the dAlembertian.

The %dAlembertian command is the inert form of dAlembertian; that is, it represents the same mathematical operation while displaying the operation unevaluated. To evaluate the operation, use the value command. To obtain the form of dAlembertian or its inert representation as a sum of diff constructions, use convert/diff.

As in the case of d_[mu], when only one argument is given to dAlembertian, the differentiation variables are the ones set by the Setup command, typically $\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}$ ($\mathrm{x0}$ is automatically assigned to $\mathrm{x4}$), represented by $X$ (see Coordinates). To change these default differentiation variables, see Setup.

Regardless of the existence of default differentiation variables, you can always call dAlembertian with two arguments, where the second argument is a list with the differentiation variables you want; in this case, the list should have as many symbols as the spacetime dimension, which by default is 4 but can be set to any value by the Setup command.

Some automatic simplifications are carried out each time dAlembertian(A) is called, as follows:

- If $A$ does not depend on the differentiation variables, then 0 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 dAlembertian(A).

- If $A$ is of the form ${\mathrm{d\_}}_{\mathrm{\mu }}\left(B\right)$, then d_[mu](dAlembertian(B)) is returned.

- Otherwise, dAlembertian(A) is computed calling d_ as in ${\partial }_{}^{\mathrm{\mu }}\left({\partial }_{\mathrm{\mu }}\left(A\right)\right)$.

In general, to accomplish differentiation, dAlembertian calls d_, which in turn makes calls to the Physics/diff command, which in turn uses the standard Maple diff command with appropriate arguments. In this way, any user-defined differentiation rule in the library or that you created, such as for a function foo of the form `diff/foo`, is automatically taken into account by dAlembertian.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{\square }\left(f\right)$

$\mathrm{convert}\left(\,\mathrm{diff}\right)$

$-f_{\mathrm{x1},\mathrm{x1}}-f_{\mathrm{x2},\mathrm{x2}}-f_{\mathrm{x3},\mathrm{x3}}+f_{\mathrm{x4},\mathrm{x4}}$

$\mathrm{d\_}\left[\mathrm{\mu }\right]\left(f\left(X\right)A\left[\mathrm{\mu }\right]\left(X\right)+B\left[\mathrm{\mu }\right]\left(X\right)\right)$

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

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

${\partial }_{\mathrm{\mu }}\left(\mathrm{\square }\left(f\right)\right)A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}+2{\partial }_{\mathrm{\mu }}\left({\partial }_{\mathrm{\nu }}\left(f\right)\right){\partial }_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)+{\partial }_{\mathrm{\mu }}\left(f\right)\mathrm{\square }\left(A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)+\mathrm{\square }\left(f\right){\partial }_{\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)+2{\partial }_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(f\right){\partial }_{\mathrm{\mu }}\left({\partial }_{\mathrm{\nu }}\left(A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)\right)+f{\partial }_{\mathrm{\mu }}\left(\mathrm{\square }\left(A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)\right)+{\partial }_{\mathrm{\mu }}\left(\mathrm{\square }\left(B_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)\right)$

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

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

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

${\partial }_{\mathrm{\mu }}\left(f\right)\mathrm{\square }\left(A_{\mathrm{\nu }}\right)+2{\partial }_{\mathrm{\alpha }}\left({\partial }_{\mathrm{\mu }}\left(f\right)\right){\partial }_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(A_{\mathrm{\nu }}\right)+{\partial }_{\mathrm{\mu }}\left(\mathrm{\square }\left(f\right)\right)A_{\mathrm{\nu }}$

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

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

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

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

$\mathrm{\square }\left(f\right)$

$\mathrm{dAlembertian}\left(f\left(Y\right)\,\left[Y\right]\right)$

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

$\mathrm{Setup}\left(\mathrm{differentiationvariables}=Y\right)$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{Y=\left(\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right)\right\}$

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

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

$0$

$\mathrm{dAlembertian}\left(f\left(X\right)\,\left[X\right]\right)$

$\mathrm{\square }\left(f\,\left[X\right]\right)$

$\mathrm{dAlembertian}\left(f\left(Y\right)\,\left[Y\right]\right)$

$\mathrm{\square }\left(f\left(Y\right)\right)$

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

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

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

$F\left[\mathrm{\mu },\mathrm{\nu }\right]≔\mathrm{d\_}\left[\mathrm{\mu }\right]\left(A\left[\mathrm{\nu }\right]\left(X\right)\right)-\mathrm{d\_}\left[\mathrm{\nu }\right]\left(A\left[\mathrm{\mu }\right]\left(X\right)\right)$

$F_{\mathrm{\mu },\mathrm{\nu }}≔{\partial }_{\mathrm{\mu }}\left(A_{\mathrm{\nu }}\right)-{\partial }_{\mathrm{\nu }}\left(A_{\mathrm{\mu }}\right)$

$'\mathrm{Fundiff}'\left(\mathrm{Intc}\left({F\left[\mathrm{\mu },\mathrm{\nu }\right]}^2\,X\right)\,A\left[\mathrm{\rho }\right]\left(Y\right)\right)$

$\left(\frac{\mathrm{\delta }}{\mathrm{\delta }{A}_{\mathrm{\rho }}\left(Y\right)}\right){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left({\partial }_{\mathrm{\mu }}\left(A_{\mathrm{\nu }}\right)-{\partial }_{\mathrm{\nu }}\left(A_{\mathrm{\mu }}\right)\right)}^2\ⅆ\mathrm{x1}\right)\ⅆ\mathrm{x2}\right)\ⅆ\mathrm{x3}\right)\ⅆ\mathrm{x4}$

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

$-2\left(-{\partial }_{\mathrm{\mu }}\left({\partial }_{\mathrm{\nu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)\right)+\mathrm{\square }\left(A_{\mathrm{\mu }}\right)\right)g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\rho }}}^{\phantom{}\mathrm{\mu },\mathrm{\rho }}+2\left(-\mathrm{\square }\left(A_{\mathrm{\nu }}\right)+{\partial }_{\mathrm{\mu }}\left({\partial }_{\mathrm{\nu }}\left(A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)\right)\right)g_{\phantom{}\phantom{\mathrm{\nu },\mathrm{\rho }}}^{\phantom{}\mathrm{\nu },\mathrm{\rho }}$

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

$4{\partial }_{\mathrm{\mu }}\left({\partial }_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\left(A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)\right)-4\mathrm{\square }\left(A_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\right)$

$L≔\frac{1}{2}\mathrm{d\_}\left[\mathrm{\mu }\right]\left(\mathrm{\Phi }\left(X\right)\right)\mathrm{d\_}\left[\mathrm{\mu }\right]\left(\mathrm{\Phi }\left(X\right)\right)-\frac{m^2}{2}{\mathrm{\Phi }\left(X\right)}^2+\frac{\mathrm{\lambda }}{4}{\mathrm{\Phi }\left(X\right)}^4$

$L≔\frac{{\partial }_{\mathrm{\mu }}\left(\mathrm{\Phi }\right){\partial }_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(\mathrm{\Phi }\right)}{2}-\frac{m^2{\mathrm{\Phi }}^2}{2}+\frac{\mathrm{\lambda }{\mathrm{\Phi }}^4}{4}$

$'\mathrm{Fundiff}'\left(\mathrm{Intc}\left(L\,X\right)\,\mathrm{\Phi }\left(Y\right)\right)$

$\left(\frac{\mathrm{\delta }}{\mathrm{\delta }\mathrm{\Phi }\left(Y\right)}\right){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{{\partial }_{\mathrm{\mu }}\left(\mathrm{\Phi }\right){\partial }_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(\mathrm{\Phi }\right)}{2}-\frac{m^2{\mathrm{\Phi }}^2}{2}+\frac{\mathrm{\lambda }{\mathrm{\Phi }}^4}{4}\right)\ⅆ\mathrm{x1}\right)\ⅆ\mathrm{x2}\right)\ⅆ\mathrm{x3}\right)\ⅆ\mathrm{x4}$

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

${\mathrm{\Phi }}^3\mathrm{\lambda }-\mathrm{\Phi }{m}^2-\mathrm{\square }\left(\mathrm{\Phi }\right)$