Given the coordinate variables, coord, and any tensor_type U, partial_diff(U, coord) constructs the partial derivatives of U, which will be a new tensor_type (not necessarily a tensor, of course) of rank one higher than that of U.

The extra index due to differentiation is of covariant character, by convention. Thus the index_char field of the result is $\left[U_{\mathrm{index\_char}}\,\mathrm{-1}\right]$.

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

When computing the first and second partial derivatives of the covariant metric tensor components, it is suggested that the tensor[d1metric] and tensor[d2metric] routines be used instead of the partial_diff routine so that the symmetries of the first and second partials be implemented using the tensor package indexing functions. The partial_diff routine does not preserve any symmetric properties that the indices of its input may have.

This function is part of the tensor package, and so can be used in the form partial_diff(..) only after performing the command with(tensor) or with(tensor, partial_diff).  The function can always be accessed in the long form tensor[partial_diff](..).

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

$\mathrm{coord}≔\left[r\,\mathrm{\theta }\,\mathrm{\psi }\right]\:$

$A≔\mathrm{array}\left(1..3\,\left[f\left(r\right)\,g\left(\mathrm{\theta }\right)\,h\left(\mathrm{\psi }\right)\right]\right)\:$

$U≔\mathrm{create}\left(\left[1\right]\,\mathrm{op}\left(A\right)\right)$

$U≔table\left(\left[\mathrm{index\_char}=\left[1\right]\,\mathrm{compts}=\left[\begin{array}{ccc}f\left(r\right)& g\left(\mathrm{\theta }\right)& h\left(\mathrm{\psi }\right)\end{array}\right]\right]\right)$

$\mathrm{part\_U}≔\mathrm{partial\_diff}\left(U\,\mathrm{coord}\right)$

$\mathrm{part\_U}≔table\left(\left[\mathrm{index\_char}=\left[1\,\mathrm{-1}\right]\,\mathrm{compts}=\left[\begin{array}{ccc}\frac{\ⅆ}{\ⅆr}f\left(r\right)& 0& 0\\ 0& \frac{\ⅆ}{\ⅆ\mathrm{\theta }}g\left(\mathrm{\theta }\right)& 0\\ 0& 0& \frac{\ⅆ}{\ⅆ\mathrm{\psi }}h\left(\mathrm{\psi }\right)\end{array}\right]\right]\right)$

$V≔\mathrm{create}\left(\left[\right]\,H\left(r\,\mathrm{\theta }\,\mathrm{\psi }\right)\right)$

$V≔table\left(\left[\mathrm{index\_char}=\left[\right]\,\mathrm{compts}=H\left(r\,\mathrm{\theta }\,\mathrm{\psi }\right)\right]\right)$

$\mathrm{part\_V}≔\mathrm{partial\_diff}\left(V\,\mathrm{coord}\right)$

$\mathrm{part\_V}≔table\left(\left[\mathrm{index\_char}=\left[\mathrm{-1}\right]\,\mathrm{compts}=\left[\begin{array}{ccc}\frac{\partial }{\partial r}H\left(r\,\mathrm{\theta }\,\mathrm{\psi }\right)& \frac{\partial }{\partial \mathrm{\theta }}H\left(r\,\mathrm{\theta }\,\mathrm{\psi }\right)& \frac{\partial }{\partial \mathrm{\psi }}H\left(r\,\mathrm{\theta }\,\mathrm{\psi }\right)\end{array}\right]\right]\right)$