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DifferentialGeometry[Tools]

 DGequal

 Calling Sequence DGequal(S1, S2)

Parameters

 S1, S2 - two lists of vectors, differential forms, or tensors; two transformations; two Lie algebra data structures; or two representations

Description

 • 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.

Examples

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

Example 1.

First initialize a 4-dimensional manifold M with coordinates [x, y, z, w].

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

Show that the vector subspaces spanned by the lists of vectors S1 and S2 are the same.

 > $\mathrm{S1}≔\mathrm{evalDG}\left(\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_x}+\mathrm{D_y},\mathrm{D_x}+\mathrm{D_y}+\mathrm{D_z}\right]\right)$
 ${\mathrm{S1}}{≔}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_x}}{+}{\mathrm{D_y}}{,}{\mathrm{D_x}}{+}{\mathrm{D_y}}{+}{\mathrm{D_z}}\right]$ (1)
 > $\mathrm{S2}≔\mathrm{evalDG}\left(\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]\right)$
 ${\mathrm{S2}}{≔}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}\right]$ (2)
 > $\mathrm{DGequal}\left(\mathrm{S1},\mathrm{S2}\right)$
 ${\mathrm{true}}$ (3)

Show that the subspaces of differential forms spanned by the lists of 2-forms S3 and S4 are not the same.

 > $\mathrm{S3}≔\mathrm{evalDG}\left(\left[\mathrm{dx}&w\mathrm{dy},\mathrm{dx}&w\mathrm{dz}\right]\right)$
 ${\mathrm{S3}}{≔}\left[{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{,}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dz}}\right]$ (4)
 > $\mathrm{S4}≔\mathrm{evalDG}\left(\left[\mathrm{dx}&w\mathrm{dy},\mathrm{dy}&w\mathrm{dz}\right]\right)$
 ${\mathrm{S4}}{≔}\left[{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{,}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}\right]$ (5)
 > $\mathrm{DGequal}\left(\mathrm{S3},\mathrm{S4}\right)$
 ${\mathrm{false}}$ (6)

Example 2.

First initialize manifolds M and N with coordinates [x, y] and [u, v].

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

Show that the transformations Phi1 and Phi2 are the same.

 > $\mathrm{Φ1}≔\mathrm{Transformation}\left(M,N,\left[u=\frac{1}{x}+\frac{1}{y},v=xy\right]\right)$
 ${\mathrm{Φ1}}{≔}\left[{u}{=}\frac{{1}}{{x}}{+}\frac{{1}}{{y}}{,}{v}{=}{x}{}{y}\right]$ (7)
 > $\mathrm{Φ2}≔\mathrm{Transformation}\left(M,N,\left[u=\frac{x+y}{xy},v=xy\right]\right)$
 ${\mathrm{Φ2}}{≔}\left[{u}{=}\frac{{x}{+}{y}}{{x}{}{y}}{,}{v}{=}{x}{}{y}\right]$ (8)
 > $\mathrm{DGequal}\left(\mathrm{Φ1},\mathrm{Φ2}\right)$
 ${\mathrm{true}}$ (9)

Show that the transformations Phi3 and Phi4 are not the same without assuming that x > 0.

 > $\mathrm{Φ3}≔\mathrm{Transformation}\left(M,N,\left[u=\sqrt{{x}^{2}},v=0\right]\right)$
 ${\mathrm{Φ3}}{≔}\left[{u}{=}\sqrt{{{x}}^{{2}}}{,}{v}{=}{0}\right]$ (10)
 > $\mathrm{Φ4}≔\mathrm{Transformation}\left(M,N,\left[u=x,v=0\right]\right)$
 ${\mathrm{Φ4}}{≔}\left[{u}{=}{x}{,}{v}{=}{0}\right]$ (11)
 > $\mathrm{DGequal}\left(\mathrm{Φ3},\mathrm{Φ4}\right)$
 ${\mathrm{false}}$ (12)
 > $\mathrm{DGequal}\left(\mathrm{Φ3},\mathrm{Φ4}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}0
 ${\mathrm{true}}$ (13)

Example 3.

Define  two Lie algebras data structures. Check that they are equal.

 > $A≔\left[\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,-1\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,1\right],\left[0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0\right],\left[1,0\right]\right]\right)\right]$
 ${A}{≔}\left[\left[\begin{array}{rr}{1}& {0}\\ {0}& {-}{1}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {1}\\ {0}& {0}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {0}\\ {1}& {0}\end{array}\right]\right]$ (14)
 > $\mathrm{LD1}≔\mathrm{LieAlgebraData}\left(A,\mathrm{alg1}\right)$
 ${\mathrm{LD1}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]$ (15)
 > $\mathrm{LD2}≔\mathrm{LieAlgebraData}\left(\left[\left[\mathrm{v1},\mathrm{v2}\right]=2\mathrm{v2},\left[\mathrm{v1},\mathrm{v3}\right]=-2\mathrm{v3},\left[\mathrm{v2},\mathrm{v3}\right]=\mathrm{v1}\right],\left[\mathrm{v1},\mathrm{v2},\mathrm{v3}\right],\mathrm{alg1}\right)$
 ${\mathrm{LD2}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]$ (16)
 > $\mathrm{DGequal}\left(\mathrm{LD1},\mathrm{LD2}\right)$
 ${\mathrm{true}}$ (17)

Example 4.

Define two representations of a Lie algebra and test for equality. First define the Lie algebra.

 > $L≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{A1},\left[3\right]\right],\left[\left[\left[2,3,1\right],1\right],\left[\left[1,3,2\right],-1\right],\left[\left[1,2,3\right],1\right]\right]\right]\right)$
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]$ (18)
 > $\mathrm{DGsetup}\left(L\right)$
 ${\mathrm{Lie algebra: A1}}$ (19)

Define the representation space V.

 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3}\right],V\right)$
 ${\mathrm{frame name: V}}$ (20)
 > $\mathrm{ρ1}≔\mathrm{Representation}\left(\mathrm{A1},V,\mathrm{Adjoint}\left(\mathrm{A1}\right)\right)$
 ${\mathrm{ρ1}}{≔}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{ccc}{0}& {0}& {0}\\ {0}& {0}& {-1}\\ {0}& {1}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{ccc}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {-1}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{ccc}{0}& {-1}& {0}\\ {1}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]\right]$ (21)
 > $\mathrm{DGequal}\left(\mathrm{ρ1},\mathrm{ρ1}\right)$
 ${\mathrm{true}}$ (22)

Make a change of basis in the representation space.

 > $\mathrm{ρ2}≔\mathrm{ChangeBasis}\left(\mathrm{ρ1},\left[\mathrm{D_x1},\mathrm{D_x2},\mathrm{D_x3}\right],\mathrm{V1}\right)$
 ${\mathrm{ρ2}}{≔}{\mathrm{ChangeBasis}}{}\left(\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{ccc}{0}& {0}& {0}\\ {0}& {0}& {-1}\\ {0}& {1}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{ccc}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {-1}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{ccc}{0}& {-1}& {0}\\ {1}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[{\mathrm{D_x1}}{,}{\mathrm{D_x2}}{,}{\mathrm{D_x3}}\right]{,}{\mathrm{V1}}\right)$ (23)

The representations rho1 and rho2 are equivalent but they are not equal.

 > $\mathrm{DGequal}\left(\mathrm{ρ1},\mathrm{ρ2}\right)$
 ${\mathrm{false}}$ (24)
 V >