The Coordinates command is used to set the systems of coordinates used by the other commands of the Physics package, as for instance the differentiation operators d_, D_, dAlembertian, diff and Fundiff, and all the general relativity tensors.

The Coordinates command also sets an alias for a capital letter X to represent a list of coordinates as explained below. These simplified representations for the spacetime coordinates eliminate the redundancy of having to type all the coordinates and in the correct order (setting these aliases does not prevent you from doing so, however), when entering functions of spacetime or using the Physics differentiation commands d_, D_ or the dAlembertian.

After setting a coordinate system, the capital letter that labels it can also be used in algebraic computations to represent the corresponding SpaceTimeVector (tensor with 1 index) - see the Examples section.

When the Coordinates command is called with no arguments, it returns the capital letters currently set to represent coordinate systems.

When Coordinates is called with one argument, a capital letter - say X - an alias is set for X to represent the sequence of n (the spacetime dimension, say 4) coordinates conformed by: $\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}$. Also $\mathrm{x0}$ is assigned the value $\mathrm{x4}$. As a rule, this sequence of spacetime coordinates is built by using the corresponding lowercase letter as a root, concatenated with positive integers from 1 to n. The capital letters that can be used as labels for systems of coordinates are the letters from A to Z, excluding D, E, G, I, O and R.

When the spacetime dimension is changed in the middle of a Maple session, the systems of coordinates previously set are automatically redefined to have the same number of coordinates as the new value for the dimension.

You can also call Coordinates with an equation, having any of the capital letters mentioned in the previous paragraph on the left-hand side, say Y, and a list of variables - all of type name- say [u, v, w, ...] on the right side, so that Y will be aliased to [u, v, w, ...]. In these cases, macros are also set as y1 = u, y2 = v, y3 = w, ... so that you can continue using concatenated lowercase letters y1, y2, ... to refer to each of the individual coordinates, apart from referring to them via Y[1], Y[2], ... or directly as u, v, ...

Coordinates also understands three keywords: cartesian, representing $\left[x\,y\,z\,t\right]$, cylindrical, representing $\left[\mathrm{\rho }\,\mathrm{\phi }\,z\,t\right]$, and spherical, representing $\left[r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]$, that can be on the right-hand side of the equation X = ... passed as first argument, so that the capital letter on the left-hand side will be aliased these coordinates.

The first time Coordinates is called to set a system of coordinates, this system is also set as the default differentiation variables for all of d_, D_ and the dAlembertian as explained in the corresponding help pages.

You can set many systems of coordinates to work with at the same time. This is particularly useful, for instance, when working with functional derivatives or coordinate transformations. To change the default differentiation variables from one system of coordinates to another one use Setup

You can also view the systems of coordinates already set, or set new ones, by launching the setup maplet (enter Setup() at the Maple prompt).

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

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

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

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

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

$\left\{X\right\}$

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

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

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

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

$X$

$X$

$X\left[1\right]$

$\mathrm{x1}$

$X\left[4\right]$

$\mathrm{x4}$

$f\left(X\right)$

$f\left(X\right)$

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

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

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

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

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

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\,Y=\left(\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right)\,Z=\left(\mathrm{z1}\,\mathrm{z2}\,\mathrm{z3}\,\mathrm{z4}\right)\right\}$

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

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

${\partial }_{\mathrm{\mu }}\left(f\left(Y\right)\,\left[Y\right]\right)$

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

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

$f\left(X\right)$

$f\left(X\right)$

$'\mathrm{Fundiff}'\left(\,f\left(Y\right)\right)$

$\left(\frac{\mathrm{\delta }}{\mathrm{\delta }f\left(Y\right)}\right)f\left(X\right)$

${\mathrm{\delta }}^{\left(4\right)}\left(\left[\mathrm{x1}-\mathrm{y1}\,\mathrm{x2}-\mathrm{y2}\,\mathrm{x3}-\mathrm{y3}\,\mathrm{x4}-\mathrm{y4}\right]\right)$

$\mathrm{\%Fundiff}\left(f\left(Y\right)\,f\left(Z\right)\right)$

$\left(\frac{\mathrm{\delta }}{\mathrm{\delta }f\left(Z\right)}\right)f\left(Y\right)$

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

${\mathrm{\delta }}^{\left(4\right)}\left(\left[\mathrm{y1}-\mathrm{z1}\,\mathrm{y2}-\mathrm{z2}\,\mathrm{y3}-\mathrm{z3}\,\mathrm{y4}-\mathrm{z4}\right]\right)$

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

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

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

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

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

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

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

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

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

$0$

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

$0$

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

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

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

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

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

Physics:-SpaceTimeVector[~mu](X)

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\mathrm{-1}& 0& 0& 0\\ 0& \mathrm{-1}& 0& 0\\ 0& 0& \mathrm{-1}& 0\\ 0& 0& 0& 1\end{array}\right]$

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

$\mathrm{x1}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{\%d\_}\left[\mathrm{\mu }\right]\left(X\left[\mathrm{~nu}\right]\right)=\mathrm{d\_}\left[\mathrm{\mu }\right]\left(X\left[\mathrm{~nu}\right]\right)$

${\partial }_{\mathrm{\mu }}\left(X_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)={\mathrm{\delta }}_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}$

$\mathrm{\%d\_}\left[\mathrm{~mu}\right]\left(X\left[\mathrm{~nu}\right]\right)=\mathrm{d\_}\left[\mathrm{~mu}\right]\left(X\left[\mathrm{~nu}\right]\right)$

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

$\mathrm{Coordinates}\left(Y\,Z=\left[a\,b\,c\,d\right]\,A=\mathrm{cartesian}\,B=\mathrm{cylindrical}\,C=\mathrm{spherical}\right)$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{A=\left(x\,y\,z\,t\right)\,B=\left(\mathrm{\rho }\,\mathrm{\phi }\,z\,t\right)\,C=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\,X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\,Y=\left(\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right)\,Z=\left(a\,b\,c\,d\right)\right\}$

$\left\{A\,B\,C\,X\,Y\,Z\right\}$

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

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{A=\left(x\,y\,z\,t\right)\,B=\left(\mathrm{\rho }\,\mathrm{\phi }\,z\,t\right)\,C=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\,X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\,Y=\left(\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right)\,Z=\left(a\,b\,c\,d\right)\right\}$

$\left\{A\,B\,C\,X\,Y\,Z\right\}$

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

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

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

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

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

$0$

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

$g_{\mathrm{\mu },\mathrm{\nu }}$

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

$r,\mathrm{\theta },\mathrm{\phi }$

$C\left[4\right],\mathrm{c4},\mathrm{c0},t$

$t,t,t,t$

Landau, L.D. and Lifshitz, E.M. The Classical Theory of Fields. Vol 2. Elsevier, 1975.