Define is used to define a name or a function as a tensor in the context of the Physics package. After this definition:

the display of contracted indices, one covariant and the other one contravariant;

a check of the number of indices against the number of indices in its definition;

differentiation and simplification of contracted indices in products using Einstein's summation convention for repeated indices;

simplification taking into account the symmetry properties that may have been indicated during the definition of the tensor;

will happen automatically or when using Simplify, diff and other differential operators of Physics, and will happen only for these names or functions defined using Define. In brief, the Physics package will compute using the rules of tensor algebra only with these objects.

The command Define() with no argument returns the tensors defined in that moment.

The typical use of Define, for instance to define a tensor, say A[mu] with one index, is as in Define(A) or Define(A[mu]), Define(A[mu](X)). The second and third manners also tell the system that A has only one index, otherwise the number of indices will be automatically derived and recorded the first time A appears in an expression handled by commands of Physics.

A different, also typical, situation happens when defining a tensor in terms of an arbitrary tensorial expression, say as in $F_{}^{\mathrm{\mu },\mathrm{\nu }}={\partial }_{}^{\mathrm{\mu }}\left(A_{}^{\mathrm{\nu }}\right)-{\partial }_{}^{\mathrm{\nu }}\left(A_{}^{\mathrm{\mu }}\right)$. For this purpose, as shown in the Examples section, use Define(A = tensorial expression), where the free indices of the left-hand side are expected to be the same as the free indices of the right-hand side, that can include, in addition, other repeated (contracted) indices. The right-hand side can also be a procedure that, when applied to the indices of the left-hand side, returns a tensorial expression with the same free indices, or just a Matrix or Array with the same dimension as the number of indices of the left-hand side (the tensor being defined).

When defining a tensor using a defining equation you can also pass to Define additional information indicating symmetry properties as explained in the Options section below related to the simpler definitions as in Define(A[mu]).

Defining tensors using tensorial equations has the following advantages, none of which exist when defining tensors the other ways:

the natural symmetry properties of the tensorial expression in the right-hand side, even when not indicated explicitly, will automatically be taken into account when A enters expressions, as it happens for all the tensors predefined in Physics

when A has numerical indices, covariant or contravariant (numbers prefixed with ~), it will automatically return the expression used in its definition with the values of these indices.

the keywords matrix, nonzero and others explained in g_ for the spacetime metric and other tensors predefined in Physics are all automatically keywords of the newly defined tensor A as well.

If you define an object already defined, Define performs a check for consistency between the previous and the given definitions, and interrupts with an error if there are inconsistencies. To clear previous definitions, use the optional argument clear; if given alone, it will clear all the previous definitions, otherwise it will clear only the definitions of the tensors specified. To clear and redefine a previously defined tensor (that is, define it ignoring previous definitions of it), use the optional argument redo, together with the new definition.

When the dimension of the spacetime is changed in the middle of a session, all definitions previously made by using Define are automatically cleared.

antisymmetric and symmetric

To define symmetry properties for some or all of the indices of an object being defined with the Define command, use the following guidelines:

To define the indices as totally symmetric or antisymmetric with respect to permutations, add the keyword symmetric or antisymmetric,respectively, to the calling sequence. For example, Define(A[mu, nu, rho, tau], symmetric), or just Define(A, symmetric).

If many objects are being defined at once with any of these two keywords, the symmetry properties are assumed to hold for all of them.

To define more detailed symmetry properties (for one Tensor at a time) use, for example, Define(A[mu, nu, rho, tau], symmetric = {{mu,nu}, {rho,tau}}),

so that: $A_{\mathrm{\mu },\mathrm{\nu },\mathrm{\rho },\mathrm{\tau }}$ is symmetric with respect to permuting the indices $\left\{\mathrm{\mu }\,\mathrm{\nu }\right\}$, and also permuting the indices $\left\{\mathrm{\rho }\,\mathrm{\tau }\right\}$. In the same way, you can simultaneously define symmetric and antisymmetric properties of different kinds. For example, Define(A[mu,nu,rho,tau], symmetric = {mu,nu}, antisymmetric = {rho,tau}), so that for the commands in the Physics package, $A_{\mathrm{\mu },\mathrm{\nu },\mathrm{\rho },\mathrm{\tau }}=A_{\mathrm{\nu },\mathrm{\mu },\mathrm{\rho },\mathrm{\tau }}$ $=-A_{\mathrm{\nu },\mathrm{\mu },\mathrm{\tau },\mathrm{\rho }}$.

clear

Clear previous definitions. If given with no additional information, clear removes all definitions in the current session.

If given with tensors  $A,B,\dots$, clear removes the specified definitions.

query

Query about previous definitions in the current session. For example, Define(query), where query is any of query, history, indices, functionality, and structure.

quiet

Proceed without displaying messages on the screen.

redo

To re-define an object that has already been defined, where the new definition is not the same as the old, use either the clear option and then define the object again, or do both in one go using the redo option.

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

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

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

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

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

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

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

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

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

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

$\mathrm{Tensor\; structured\; as}\left[\mathrm{name}\,\mathrm{indices}\,\mathrm{variables}\right]:$

$\left[B\,\left[0\,0\,0\right]\,0\right]$

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

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

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

$\mathrm{Tensor\; structured\; as}\left[\mathrm{name}\,\mathrm{indices}\,\mathrm{variables}\right]:$

$\left[B\,\left[0\,0\,0\right]\,0\right]$

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

$0$

$\mathrm{diff}\left(B\left[\mathrm{\mu }\right]\,B\left[\mathrm{~nu}\right]\right)$

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

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

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

$\left\{X\right\}$

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

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

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

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

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

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

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

$\left\{A_{\mathrm{\mu }}\,B_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,F_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\right\}$

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

$-F_{\mathrm{\mu },\mathrm{\nu }}$

$F\left[\mathrm{\mu },\mathrm{\nu }\right]+F\left[\mathrm{\nu },\mathrm{\mu }\right]$

$0$

$A\left[\mathrm{\mu }\right]A\left[\mathrm{\nu }\right]F\left[\mathrm{\mu },\mathrm{\nu }\right]$

$F_{\mathrm{\mu },\mathrm{\nu }}{A}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}$

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

$0$

$F\left[\right]$

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

$F\left[\mathrm{nonzero}\right]$

$F_{\mathrm{\mu },\mathrm{\nu }}=\left\{\left(1\,2\right)={\left(A_2\right)}_{\mathrm{x1}}-{\left(A_1\right)}_{\mathrm{x2}}\,\left(1\,3\right)={\left(A_3\right)}_{\mathrm{x1}}-{\left(A_1\right)}_{\mathrm{x3}}\,\left(1\,4\right)={\left(A_4\right)}_{\mathrm{x1}}-{\left(A_1\right)}_{\mathrm{x4}}\,\left(2\,1\right)={\left(A_1\right)}_{\mathrm{x2}}-{\left(A_2\right)}_{\mathrm{x1}}\,\left(2\,3\right)={\left(A_3\right)}_{\mathrm{x2}}-{\left(A_2\right)}_{\mathrm{x3}}\,\left(2\,4\right)={\left(A_4\right)}_{\mathrm{x2}}-{\left(A_2\right)}_{\mathrm{x4}}\,\left(3\,1\right)={\left(A_1\right)}_{\mathrm{x3}}-{\left(A_3\right)}_{\mathrm{x1}}\,\left(3\,2\right)={\left(A_2\right)}_{\mathrm{x3}}-{\left(A_3\right)}_{\mathrm{x2}}\,\left(3\,4\right)={\left(A_4\right)}_{\mathrm{x3}}-{\left(A_3\right)}_{\mathrm{x4}}\,\left(4\,1\right)={\left(A_1\right)}_{\mathrm{x4}}-{\left(A_4\right)}_{\mathrm{x1}}\,\left(4\,2\right)={\left(A_2\right)}_{\mathrm{x4}}-{\left(A_4\right)}_{\mathrm{x2}}\,\left(4\,3\right)={\left(A_3\right)}_{\mathrm{x4}}-{\left(A_4\right)}_{\mathrm{x3}}\right\}$

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

$0$

$F\left[1,2\right]$

${\left(A_2\right)}_{\mathrm{x1}}-{\left(A_1\right)}_{\mathrm{x2}}$

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

${\left(A_{\phantom{}\phantom{2}}^{\phantom{}2}\right)}_{\mathrm{x1}}+{\left(A_1\right)}_{\mathrm{x2}}$

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

$-{\left(A_2\right)}_{\mathrm{x1}}-{\left(A_{\phantom{}\phantom{1}}^{\phantom{}1}\right)}_{\mathrm{x2}}$

$F\left[\mathrm{~mu},\mathrm{~nu}\right]=\mathrm{Matrix}\left(4\,4\,\left\{\left(1\,2\right)=\mathrm{E\_\_1}\,\left(1\,3\right)=\mathrm{E\_\_2}\,\left(1\,4\right)=\mathrm{E\_\_3}\,\left(2\,3\right)=\mathrm{H\_\_3}\,\left(2\,4\right)=-\mathrm{H\_\_2}\,\left(3\,4\right)=\mathrm{H\_\_1}\right\}\,\mathrm{shape}=\mathrm{antisymmetric}\right)$

$F_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

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

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

$\left\{A_{\mathrm{\mu }}\,B_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,F_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\right\}$

$F\left[1,2\right]$

$\mathrm{E\_\_1}$

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

$-\mathrm{E\_\_1}$

$F\left[\mathrm{\mu },\mathrm{\nu },\mathrm{matrix}\right]$

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

$F\left[\mathrm{~mu},\mathrm{\nu },\mathrm{matrix}\right]$

$F_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\nu }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\nu }}}=\left[\right]$

$\mathrm{Define}\left(\mathrm{redo}\,A\left[\mathrm{\mu },\mathrm{\nu },\mathrm{\rho },\mathrm{\sigma }\right]\left(x\,y\,z\right)\,\mathrm{symmetric}=\left\{\mathrm{\mu }\,\mathrm{\rho }\right\}\,\mathrm{antisymmetric}=\left\{\mathrm{\nu }\,\mathrm{\sigma }\right\}\right)$

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

$\left\{A_{\mathrm{\mu },\mathrm{\nu },\mathrm{\rho },\mathrm{\sigma }}\,B_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,F_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\right\}$

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

$\mathrm{Tensor\; with\; symmetry\; properties\; defined\; as:}{A}_{1,2,3,4}\mathrm{,\; where}\left\{\mathrm{antisymmetric}=\left\{\left[2\,4\right]\right\}\,\mathrm{symmetric}=\left\{\left[1\,3\right]\right\}\right\}\mathrm{,\; and\; structured\; as}\left[\mathrm{name}\,\mathrm{indices}\,\mathrm{variables}\right]:$

$\left[A\,\left[4\,0\,0\right]\,3\right]$

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

$\mathrm{Used\; as\; indices\; in\; tensors:}$

$\left\{\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\mu }\,\mathrm{\nu }\,\mathrm{\rho }\,\mathrm{\sigma }\right\}$

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

$\mathrm{Used\; as\; variables\; in\; tensor\; functions:}$

$\left\{x\,\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\,y\,z\right\}$

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

$\mathrm{History\; of\; definitions\; of\; tensors:}$

$\left\{B\,A_{\mathrm{\mu },\mathrm{\nu },\mathrm{\rho },\mathrm{\sigma }}\,B_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,F_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,B_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\,B_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\right\}$