The function raise(con_met, A, 2, 3) computes the tensor A with indices 2 and 3 raised using the contravariant metric con_met.

The function lower(cov_met, A, 1, 4) computes the tensor A with indices 1 and 4 lowered using the covariant metric cov_met.

Each index in the call to raise must be a valid covariant index of A.

Each index in the call to lower must be a valid contravariant index of A.

There must be at least 1 index given and the number of indices cannot exceed the rank of A.

Simplification:  These routines use the `tensor/prod/simp` routine for simplification purposes.  The simplification routine is applied to each component of the result after it is computed.  By default, `tensor/prod/simp` is initialized to the `tensor/simp` routine. It is recommended that the `tensor/prod/simp` routine be customized to suit the needs of the particular problem.

These functions are part of the tensor package, and so can be used in the form raise(..) / lower(..) only after performing the command with(tensor), or with(tensor, raise) / with(tensor, lower).  These functions can always be accessed in the long form tensor[raise](..) / tensor[lower](..).

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

$a≔\mathrm{create}\left(\left[-1\,-1\right]\,\mathrm{array}\left(\left[\left[1\,0\,0\right]\,\left[0\,r^2\,0\right]\,\left[0\,0\,r^2{\sin \left(\mathrm{\theta }\right)}^2\right]\right]\right)\right)$

$a≔table\left(\left[\mathrm{compts}=\left[\begin{array}{ccc}1& 0& 0\\ 0& r^2& 0\\ 0& 0& r^2{\sin \left(\mathrm{\theta }\right)}^2\end{array}\right]\,\mathrm{index\_char}=\left[\mathrm{-1}\,\mathrm{-1}\right]\right]\right)$

$A≔\mathrm{create}\left(\left[1\,1\right]\,\mathrm{array}\left(\left[\left[1\,0\,0\right]\,\left[0\,\frac{1}{r^2}\,0\right]\,\left[0\,0\,\frac{1}{r^2{\sin \left(\mathrm{\theta }\right)}^2}\right]\right]\right)\right)$

$A≔table\left(\left[\mathrm{compts}=\left[\begin{array}{ccc}1& 0& 0\\ 0& \frac{1}{r^2}& 0\\ 0& 0& \frac{1}{r^2{\sin \left(\mathrm{\theta }\right)}^2}\end{array}\right]\,\mathrm{index\_char}=\left[1\,1\right]\right]\right)$

$T≔\mathrm{create}\left(\left[1\,-1\right]\,\mathrm{array}\left(\left[\left[w\,x\,0\right]\,\left[y\,z\,0\right]\,\left[0\,y^2\,xyw\right]\right]\right)\right)$

$T≔table\left(\left[\mathrm{compts}=\left[\begin{array}{ccc}w& x& 0\\ y& z& 0\\ 0& y^2& xyw\end{array}\right]\,\mathrm{index\_char}=\left[1\,\mathrm{-1}\right]\right]\right)$

$\mathrm{raise}\left(A\,T\,2\right)$

$table\left(\left[\mathrm{compts}=\left[\begin{array}{ccc}w& \frac{x}{r^2}& 0\\ y& \frac{z}{r^2}& 0\\ 0& \frac{y^2}{r^2}& \frac{xyw{\csc \left(\mathrm{\theta }\right)}^2}{r^2}\end{array}\right]\,\mathrm{index\_char}=\left[1\,1\right]\right]\right)$

$\mathrm{lower}\left(a\,T\,1\right)$

$table\left(\left[\mathrm{compts}=\left[\begin{array}{ccc}w& x& 0\\ r^{2}y& r^{2}z& 0\\ 0& r^2{\sin \left(\mathrm{\theta }\right)}^2{y}^2& r^2{\sin \left(\mathrm{\theta }\right)}^{2}xyw\end{array}\right]\,\mathrm{index\_char}=\left[\mathrm{-1}\,\mathrm{-1}\right]\right]\right)$