In the DifferentialGeometry package the wedge product of 1-forms is defined in terms of the tensor product by $\mathrm{\alpha }\wedge \mathrm{\beta }=\mathrm{\alpha }\otimes \mathrm{\beta }-\mathrm{\beta }\otimes \mathrm{\alpha }$.

When using these commands together within a single Maple expression, it is important to use parentheses to insure that the operations are executed in the correct order.

In an interactive Maple session, it is usually more convenient to use the commands evalDG and DGzip to perform these basic algebraic operations.

Here are the precise lists of admissible arguments for these commands.

A &plus B, A &minus B -- A and B: Maple expressions, vectors, differential forms of the same degree, differential biforms of the same bidegree, tensors with the same index type and density weights. A and B must be defined on the same frame.

A &mult B -- A: a Maple expression; B: a Maple expression, vector, differential form, differential biform, tensor. A and B must be defined on the same frame.

A &wedge B -- A and B: Maple expressions or differential forms, differential biforms.  If A and B are forms, then the sum of their degrees cannot exceed the dimension of the frame on which they are defined. If A and B are bi-forms, then the sum of their horizontal degrees cannot exceed the dimension of the base manifold on which they are defined.  A and B must be defined on the same frame.

A &tensor B -- A and B: Maple expressions, vectors, differential 1-forms, tensors.  A and B must be defined on the same frame.

These commands are part of the DifferentialGeometry package, and so can be used in the forms given above only after executing the command with(DifferentialGeometry).

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

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

$\mathrm{The\; following\; coordinates\; have\; been\; protected:}$

$\left[x\,y\,z\right]$

$\mathrm{The\; following\; vector\; fields\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{The\; following\; differential\; 1-forms\; have\; been\; defined\; and\; protected:}$

$\left[\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\right]$

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

$\mathrm{X1}≔\mathrm{D\_x}\&plus\mathrm{D\_z}$

$\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[3\right]\,1\right]\right]\right]\right)$

$\mathrm{X2}≔\left(\left(3z\right)\&mult\mathrm{D\_x}\right)\&plus\left(\left(-2y\right)\&mult\mathrm{D\_y}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,3z\right]\,\left[\left[2\right]\,-2y\right]\right]\right]\right)$

$\mathrm{X3}≔\mathrm{X2}\&minus\left(\left(3z\right)\&mult\mathrm{X1}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,-2y\right]\,\left[\left[3\right]\,-3z\right]\right]\right]\right)$

$\mathrm{\α1}≔\left(\sin \left(z\right)\&mult\mathrm{dx}\right)\&minus\left(\cos \left(y\right)\&mult\mathrm{dz}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,\sin \left(z\right)\right]\,\left[\left[3\right]\,-\cos \left(y\right)\right]\right]\right]\right)$

$\mathrm{\α2}≔\left(\cos \left(x\right)\&mult\mathrm{dy}\right)\&plus\left(\cos \left(z\right)\&mult\mathrm{dz}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[2\right]\,\cos \left(x\right)\right]\,\left[\left[3\right]\,\cos \left(z\right)\right]\right]\right]\right)$

$\mathrm{\α3}≔\left(2\&mult\left(\mathrm{dx}\&wedge\mathrm{dy}\right)\right)\&plus\left(5\&mult\left(\mathrm{dy}\&wedge\mathrm{dz}\right)\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,2\right]\,\left[\left[2\,3\right]\,5\right]\right]\right]\right)$

$\mathrm{\α4}≔\mathrm{\α1}\&wedge\mathrm{\α2}$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,\sin \left(z\right)\cos \left(x\right)\right]\,\left[\left[1\,3\right]\,\sin \left(z\right)\cos \left(z\right)\right]\,\left[\left[2\,3\right]\,\cos \left(y\right)\cos \left(x\right)\right]\right]\right]\right)$

$\mathrm{\α5}≔\left(\mathrm{\α1}\&wedge\mathrm{\α2}\right)\&minus\mathrm{\α3}$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,-2\+\sin \left(z\right)\cos \left(x\right)\right]\,\left[\left[1\,3\right]\,\sin \left(z\right)\cos \left(z\right)\right]\,\left[\left[2\,3\right]\,-5\+\cos \left(y\right)\cos \left(x\right)\right]\right]\right]\right)$

$\mathrm{\α6}≔\mathrm{\α1}\&wedge\mathrm{\α3}$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,3\right]\,\left[\left[\left[1\,2\,3\right]\,-2\cos \left(y\right)\+5\sin \left(z\right)\right]\right]\right]\right)$

$\mathrm{T1}≔\mathrm{X1}\&tensor\mathrm{X1}$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,1\right]\,\left[\left[1\,3\right]\,1\right]\,\left[\left[3\,1\right]\,1\right]\,\left[\left[3\,3\right]\,1\right]\right]\right]\right)$

$\mathrm{T2}≔\mathrm{X1}\&tensor\mathrm{\α1}$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,\sin \left(z\right)\right]\,\left[\left[1\,3\right]\,-\cos \left(y\right)\right]\,\left[\left[3\,1\right]\,\sin \left(z\right)\right]\,\left[\left[3\,3\right]\,-\cos \left(y\right)\right]\right]\right]\right)$

$\mathrm{T3}≔1\&tensor\left(\mathrm{dx}\&wedge\mathrm{dy}\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,1\right]\,\left[\left[2\,1\right]\,-1\right]\right]\right]\right)$

$\mathrm{T4}≔\left(\left(\left(\mathrm{dx}\&tensor\mathrm{dx}\right)\&tensor\mathrm{D\_y}\right)\&tensor\mathrm{D\_z}\right)\&tensor\mathrm{dz}$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\,''con\_bas''\,''con\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,2\,3\,3\right]\,1\right]\right]\right]\right)$

$\mathrm{T5}≔\left(\frac{1}{y^2}\right)\&mult\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,\frac{1}{y^2}\right]\,\left[\left[2\,2\right]\,\frac{1}{y^2}\right]\right]\right]\right)$

$\mathrm{V1}≔\left(2\&mult\left(\mathrm{D\_x}\&wedge\mathrm{D\_y}\right)\right)\&plus\left(3\&mult\left(\mathrm{D\_y}\&wedge\mathrm{D\_z}\right)\right)$

$\mathrm{\_DG}\left(\left[\left[''multivector''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,2\right]\,\left[\left[2\,3\right]\,3\right]\right]\right]\right)$

$\mathrm{LA}≔\mathrm{AlgebraLibraryData}\left(''Quaternions''\,Q\right)$

$\mathrm{LA}≔\left[{\mathrm{e1}}^2=\mathrm{e1}\,\mathrm{e1}·\mathrm{e2}=\mathrm{e2}\,\mathrm{e1}·\mathrm{e3}=\mathrm{e3}\,\mathrm{e1}·\mathrm{e4}=\mathrm{e4}\,\mathrm{e2}·\mathrm{e1}=\mathrm{e2}\,{\mathrm{e2}}^2=-\mathrm{e1}\,\mathrm{e2}·\mathrm{e3}=\mathrm{e4}\,\mathrm{e2}·\mathrm{e4}=-\mathrm{e3}\,\mathrm{e3}·\mathrm{e1}=\mathrm{e3}\,\mathrm{e3}·\mathrm{e2}=-\mathrm{e4}\,{\mathrm{e3}}^2=-\mathrm{e1}\,\mathrm{e3}·\mathrm{e4}=\mathrm{e2}\,\mathrm{e4}·\mathrm{e1}=\mathrm{e4}\,\mathrm{e4}·\mathrm{e2}=\mathrm{e3}\,\mathrm{e4}·\mathrm{e3}=-\mathrm{e2}\,{\mathrm{e4}}^2=-\mathrm{e1}\right]$

$\mathrm{DGsetup}\left(\mathrm{LA}\,\left[e\,i\,j\,k\right]\,\left[\mathrm{\theta }\right]\right)$

$\mathrm{algebra\; name:\; Q}$

$\mathrm{Q1}≔i\&algmultj$

$\mathrm{\_DG}\left(\left[\left[''vector''\,Q\,\left[\right]\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)$

$\mathrm{Q2}≔\left(\left(\left(e\&plusi\right)\&plusj\right)\&plusk\right)\&algmult\left(e\&minus\left(\left(i\&plusj\right)\&plusk\right)\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,Q\,\left[\right]\right]\,\left[\left[\left[1\right]\,4\right]\right]\right]\right)$