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DifferentialGeometry[Tools]

 DGsimplify

 Calling Sequence DGsimplify(X)

Parameters

 X - a vector field, differential form or tensor

Description

 • This command is typically used to simplify a vector field, differential form or tensor, defined on a manifold M, after a substitution or evaluation of its coefficients at point on M.
 • This command is part of the DifferentialGeometry:-Tools package, and so can be used in the form DGsimplify(...) only after executing the commands with(DifferentialGeometry) and with(Tools) in that order.  It can always be used in the long form DifferentialGeometry:-Tools:-DGsimplify.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tools}\right):$

Example 1.

Define a manifold M with coordinates [x, y, z].

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$

Define a tensor on M and evaluate it at the point [x = 0, y = 2, z = 3].

 > $T≔\mathrm{evalDG}\left({x}^{2}\mathrm{D_x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}+xy\mathrm{D_y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}+\left(z\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_z}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{x}}^{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{x}{}{y}\right]{,}\left[\left[{3}{,}{2}\right]{,}{z}\right]\right]\right]\right)$ (1)
 > $S≔\mathrm{eval}\left(T,\left[x=0,y=2,z=3\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]{,}\left[\left[{2}{,}{3}\right]{,}{0}\right]{,}\left[\left[{3}{,}{2}\right]{,}{3}\right]\right]\right]\right)$ (2)

The terms 0*D_x and 0*D_y can be eliminated with a call to DGsimplify.

 > $\mathrm{DGsimplify}\left(S\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{2}\right]{,}{3}\right]\right]\right]\right)$ (3)
 M >