Let S1 and  S2 be two lists of vectors, differential forms, or tensors. If every element of S1 is in the span of S2 and every element of S1 is in the span of S2, then DGequal(S1, S2) returns true and otherwise false.

If the two transformations Phi1 and Phi2 have the same domain frame, range frame, and the same coordinate expressions, then DGequal(Phi1, Phi2) returns true and otherwise false.  The command DGequal(Phi1, Phi2) computes the differences between the Jacobian matrices and the coordinate equations for the two transformations Phi1 and Phi2 and tests if these differences are zero.

This command is part of the DifferentialGeometry:-Tools package, and so can be used in the form DGequal(...) only after executing the commands with(DifferentialGeometry) and with(Tools) in that order.  It can always be used in the long form DifferentialGeometry:-Tools:-DGequal.

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

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

$\mathrm{S1}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_x}+\mathrm{D\_y}\,\mathrm{D\_x}+\mathrm{D\_y}+\mathrm{D\_z}\right]\right)$

$\mathrm{S1}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_x}+\mathrm{D\_y}\,\mathrm{D\_x}+\mathrm{D\_y}+\mathrm{D\_z}\right]$

$\mathrm{S2}≔\mathrm{evalDG}\left(\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]\right)$

$\mathrm{S2}≔\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]$

$\mathrm{DGequal}\left(\mathrm{S1}\,\mathrm{S2}\right)$

$\mathrm{true}$

$\mathrm{S3}≔\mathrm{evalDG}\left(\left[\mathrm{dx}\&w\mathrm{dy}\,\mathrm{dx}\&w\mathrm{dz}\right]\right)$

$\mathrm{S3}≔\left[\mathrm{dx}\bigwedge \mathrm{dy}\,\mathrm{dx}\bigwedge \mathrm{dz}\right]$

$\mathrm{S4}≔\mathrm{evalDG}\left(\left[\mathrm{dx}\&w\mathrm{dy}\,\mathrm{dy}\&w\mathrm{dz}\right]\right)$

$\mathrm{S4}≔\left[\mathrm{dx}\bigwedge \mathrm{dy}\,\mathrm{dy}\bigwedge \mathrm{dz}\right]$

$\mathrm{DGequal}\left(\mathrm{S3}\,\mathrm{S4}\right)$

$\mathrm{false}$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,M\right)\:$$\mathrm{DGsetup}\left(\left[u\,v\right]\,N\right)\:$

$\mathrm{\Φ1}≔\mathrm{Transformation}\left(M\,N\,\left[u=\frac{1}{x}+\frac{1}{y}\,v=xy\right]\right)$

$\mathrm{\Φ1}≔\left[u=\frac{1}{x}+\frac{1}{y}\,v=xy\right]$

$\mathrm{\Φ2}≔\mathrm{Transformation}\left(M\,N\,\left[u=\frac{x+y}{xy}\,v=xy\right]\right)$

$\mathrm{\Φ2}≔\left[u=\frac{x+y}{xy}\,v=xy\right]$

$\mathrm{DGequal}\left(\mathrm{\Φ1}\,\mathrm{\Φ2}\right)$

$\mathrm{true}$

$\mathrm{\Φ3}≔\mathrm{Transformation}\left(M\,N\,\left[u=\mathrm{sqrt}\left(x^2\right)\,v=0\right]\right)$

$\mathrm{\Φ3}≔\left[u=\sqrt{x^2}\,v=0\right]$

$\mathrm{\Φ4}≔\mathrm{Transformation}\left(M\,N\,\left[u=x\,v=0\right]\right)$

$\mathrm{\Φ4}≔\left[u=x\,v=0\right]$

$\mathrm{DGequal}\left(\mathrm{\Φ3}\,\mathrm{\Φ4}\right)$

$\mathrm{false}$

$\mathrm{DGequal}\left(\mathrm{\&Phi;3}\&comma;\mathrm{\&Phi;4}\right)\mathbf{assuming}0<x$

$\mathrm{true}$

$A≔\left[\mathrm{Matrix}\left(\left[\left[1\&comma;0\right]\&comma;\left[0\&comma;-1\right]\right]\right)\&comma;\mathrm{Matrix}\left(\left[\left[0\&comma;1\right]\&comma;\left[0\&comma;0\right]\right]\right)\&comma;\mathrm{Matrix}\left(\left[\left[0\&comma;0\right]\&comma;\left[1\&comma;0\right]\right]\right)\right]$

$A≔\left[\left[\begin{array}{cc}1& 0\\ 0& \mathrm{-1}\end{array}\right]\&comma;\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\&comma;\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]\right]$

$\mathrm{LD1}≔\mathrm{LieAlgebraData}\left(A\&comma;\mathrm{alg1}\right)$

$\mathrm{LD1}≔\left[\left[\mathrm{e1}\&comma;\mathrm{e2}\right]=2\mathrm{e2}\&comma;\left[\mathrm{e1}\&comma;\mathrm{e3}\right]=-2\mathrm{e3}\&comma;\left[\mathrm{e2}\&comma;\mathrm{e3}\right]=\mathrm{e1}\right]$

$\mathrm{LD2}≔\mathrm{LieAlgebraData}\left(\left[\left[\mathrm{v1}\&comma;\mathrm{v2}\right]=2\mathrm{v2}\&comma;\left[\mathrm{v1}\&comma;\mathrm{v3}\right]=-2\mathrm{v3}\&comma;\left[\mathrm{v2}\&comma;\mathrm{v3}\right]=\mathrm{v1}\right]\&comma;\left[\mathrm{v1}\&comma;\mathrm{v2}\&comma;\mathrm{v3}\right]\&comma;\mathrm{alg1}\right)$

$\mathrm{LD2}≔\left[\left[\mathrm{e1}\&comma;\mathrm{e2}\right]=2\mathrm{e2}\&comma;\left[\mathrm{e1}\&comma;\mathrm{e3}\right]=-2\mathrm{e3}\&comma;\left[\mathrm{e2}\&comma;\mathrm{e3}\right]=\mathrm{e1}\right]$

$\mathrm{DGequal}\left(\mathrm{LD1}\&comma;\mathrm{LD2}\right)$

$\mathrm{true}$

$L≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\&comma;\mathrm{A1}\&comma;\left[3\right]\right]\&comma;\left[\left[\left[2\&comma;3\&comma;1\right]\&comma;1\right]\&comma;\left[\left[1\&comma;3\&comma;2\right]\&comma;-1\right]\&comma;\left[\left[1\&comma;2\&comma;3\right]\&comma;1\right]\right]\right]\right)$

$L≔\left[\left[\mathrm{e1}\&comma;\mathrm{e2}\right]=\mathrm{e3}\&comma;\left[\mathrm{e1}\&comma;\mathrm{e3}\right]=-\mathrm{e2}\&comma;\left[\mathrm{e2}\&comma;\mathrm{e3}\right]=\mathrm{e1}\right]$

$\mathrm{DGsetup}\left(L\right)$

$\mathrm{Lie\; algebra:\; A1}$

$\mathrm{DGsetup}\left(\left[\mathrm{x1}\&comma;\mathrm{x2}\&comma;\mathrm{x3}\right]\&comma;V\right)$

$\mathrm{frame\; name:\; V}$

$\mathrm{\&rho;1}≔\mathrm{Representation}\left(\mathrm{A1}\&comma;V\&comma;\mathrm{Adjoint}\left(\mathrm{A1}\right)\right)$

$\mathrm{\&rho;1}≔\left[\left[\mathrm{e1}\&comma;\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& \mathrm{-1}\\ 0& 1& 0\end{array}\right]\right]\&comma;\left[\mathrm{e2}\&comma;\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ \mathrm{-1}& 0& 0\end{array}\right]\right]\&comma;\left[\mathrm{e3}\&comma;\left[\begin{array}{ccc}0& \mathrm{-1}& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]\right]\right]$

$\mathrm{DGequal}\left(\mathrm{\&rho;1}\&comma;\mathrm{\&rho;1}\right)$

$\mathrm{true}$

$\mathrm{\&rho;2}≔\mathrm{ChangeBasis}\left(\mathrm{\&rho;1}\&comma;\left[\mathrm{D\_x1}\&comma;\mathrm{D\_x2}\&comma;\mathrm{D\_x3}\right]\&comma;\mathrm{V1}\right)$

$\mathrm{\&rho;2}≔\mathrm{ChangeBasis}\left(\left[\left[\mathrm{e1}\&comma;\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& \mathrm{-1}\\ 0& 1& 0\end{array}\right]\right]\&comma;\left[\mathrm{e2}\&comma;\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ \mathrm{-1}& 0& 0\end{array}\right]\right]\&comma;\left[\mathrm{e3}\&comma;\left[\begin{array}{ccc}0& \mathrm{-1}& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]\right]\right]\&comma;\left[\mathrm{D\_x1}\&comma;\mathrm{D\_x2}\&comma;\mathrm{D\_x3}\right]\&comma;\mathrm{V1}\right)$

$\mathrm{DGequal}\left(\mathrm{\&rho;1}\&comma;\mathrm{\&rho;2}\right)$

$\mathrm{false}$