The function exterior_prod(A, B) computes the exterior product of the covariant antisymmetric tensors A and B and returns it as a tensor_type of rank equal to $\mathrm{rank}\left(A\right)+\mathrm{rank}\left(B\right)$.  The result is totally anti-symmetric and uses the antisymmetric indexing function.

The input tensors A and B must be of covariant character and must be completely antisymmetric.  The routine first checks for the use of Maple's antisymmetric indexing function.  If this indexing function has not been used for input tensors of rank greater than one, the antisymmetry of the input is determined by comparing the input tensor components with those of the complete antisymmetrization of the input tensor.

The result is computed first by computing the components of the outer product of A and B and then antisymmetrizing them.

Simplification:  This routine uses the `tensor/lin_com/simp` and `tensor/prod/simp` routines for simplification purposes.  The simplification routines are used internally by the prod and antisymmetrize routines as they are called by exterior_prod.  By default, `tensor/lin_com/simp` and `tensor/prod/simp` are initialized to the `tensor/simp` routine. It is recommended that these routines be customized to suit the needs of the particular problem.

This command is part of the tensor package, so it can be used in the form exterior_prod(..) only after executing the command with(tensor). However, it can always be accessed through the long from of the command by using tensor[exterior_prod](..).

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

$V≔\mathrm{create}\left(\left[-1\right]\,\mathrm{array}\left(\left[\mathrm{v1}\,\mathrm{v2}\,\mathrm{v3}\right]\right)\right)$

$V≔table\left(\left[\mathrm{index\_char}=\left[\mathrm{-1}\right]\,\mathrm{compts}=\left[\begin{array}{ccc}\mathrm{v1}& \mathrm{v2}& \mathrm{v3}\end{array}\right]\right]\right)$

$\mathrm{Uc}≔\mathrm{array}\left(1..3\,1..3\,\mathrm{antisymmetric}\right)\:$

$$\begin{gathered} \mathbf{for}i\mathbf{to}3\mathbf{do} \\ \mathbf{for}j\mathbf{from}i+1\mathbf{to}3\mathbf{do} \\ \mathrm{Uc}\left[i,j\right]≔\mathrm{cat}\left('U'\,i\,j\right) \\ \mathbf{end}\mathbf{do} \\ \mathbf{end}\mathbf{do}\; \end{gathered}$$$U≔\mathrm{create}\left(\left[-1\,-1\right]\,\mathrm{eval}\left(\mathrm{Uc}\right)\right)$

$U≔table\left(\left[\mathrm{index\_char}=\left[\mathrm{-1}\,\mathrm{-1}\right]\,\mathrm{compts}=\left[\begin{array}{ccc}0& \mathrm{U12}& \mathrm{U13}\\ -\mathrm{U12}& 0& \mathrm{U23}\\ -\mathrm{U13}& -\mathrm{U23}& 0\end{array}\right]\right]\right)$

$\mathrm{ex\_prod}≔\mathrm{exterior\_prod}\left(U\,V\right)$

$\mathrm{ex\_prod}≔table\left(\left[\mathrm{index\_char}=\left[\mathrm{-1}\,\mathrm{-1}\,\mathrm{-1}\right]\,\mathrm{compts}=array\left(\mathrm{antisymmetric}\,1..3\,1..3\,1..3\,\left[\left(1\,1\,1\right)=0\,\left(1\,1\,2\right)=0\,\left(1\,1\,3\right)=0\,\left(1\,2\,1\right)=0\,\left(1\,2\,2\right)=0\,\left(1\,2\,3\right)=\mathrm{U12}\mathrm{v3}-\mathrm{U13}\mathrm{v2}+\mathrm{U23}\mathrm{v1}\,\left(1\,3\,1\right)=0\,\left(1\,3\,2\right)=-\mathrm{U12}\mathrm{v3}+\mathrm{U13}\mathrm{v2}-\mathrm{U23}\mathrm{v1}\,\left(1\,3\,3\right)=0\,\left(2\,1\,1\right)=0\,\left(2\,1\,2\right)=0\,\left(2\,1\,3\right)=-\mathrm{U12}\mathrm{v3}+\mathrm{U13}\mathrm{v2}-\mathrm{U23}\mathrm{v1}\,\left(2\,2\,1\right)=0\,\left(2\,2\,2\right)=0\,\left(2\,2\,3\right)=0\,\left(2\,3\,1\right)=\mathrm{U12}\mathrm{v3}-\mathrm{U13}\mathrm{v2}+\mathrm{U23}\mathrm{v1}\,\left(2\,3\,2\right)=0\,\left(2\,3\,3\right)=0\,\left(3\,1\,1\right)=0\,\left(3\,1\,2\right)=\mathrm{U12}\mathrm{v3}-\mathrm{U13}\mathrm{v2}+\mathrm{U23}\mathrm{v1}\,\left(3\,1\,3\right)=0\,\left(3\,2\,1\right)=-\mathrm{U12}\mathrm{v3}+\mathrm{U13}\mathrm{v2}-\mathrm{U23}\mathrm{v1}\,\left(3\,2\,2\right)=0\,\left(3\,2\,3\right)=0\,\left(3\,3\,1\right)=0\,\left(3\,3\,2\right)=0\,\left(3\,3\,3\right)=0\right]\right)\right]\right)$

$\mathrm{op}\left(1\,\mathrm{get\_compts}\left(\mathrm{ex\_prod}\right)\right)$

$\mathrm{antisymmetric}$