Specifically,

Given the coordinate variables, coord, and the Christoffel symbols of the second kind, Cf2, and any tensor_type U, cov_diff( U, coord, Cf2 ) constructs the covariant derivative of U, which will be a new tensor_type of rank one higher than that of U.

The extra index due to the covariant derivative is of covariant character, as one would expect.  Thus, the index_char field of the resultant tensor_type is $\left[U_{\mathrm{index\_char}}\,\mathrm{-1}\right]$.

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

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

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

$\mathrm{coord}≔\left[t\,r\,\mathrm{\theta }\,\mathrm{\phi }\right]$

$\mathrm{coord}≔\left[t\,r\,\mathrm{\theta }\,\mathrm{\phi }\right]$

$\mathrm{g\_compts}≔\mathrm{array}\left(\mathrm{symmetric}\,\mathrm{sparse}\,1..4\,1..4\right)\:$

$\mathrm{g\_compts}\left[1,1\right]≔1-\frac{2m}{r}\:$$\mathrm{g\_compts}\left[2,2\right]≔-\frac{1}{\mathrm{g\_compts}\left[1,1\right]}\:$

$\mathrm{g\_compts}\left[3,3\right]≔-r^2\:$$\mathrm{g\_compts}\left[4,4\right]≔-r^2{\sin \left(\mathrm{\theta }\right)}^2\:$

$g≔\mathrm{create}\left(\left[-1\,-1\right]\,\mathrm{eval}\left(\mathrm{g\_compts}\right)\right)$

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

$\mathrm{d1g}≔\mathrm{d1metric}\left(g\,\mathrm{coord}\right)\:$

$\mathrm{g\_inverse}≔\mathrm{invert}\left(g\,\mathrm{detg}\right)\:$

$\mathrm{Cf1}≔\mathrm{Christoffel1}\left(\mathrm{d1g}\right)\:$

$\mathrm{Cf2}≔\mathrm{Christoffel2}\left(\mathrm{g\_inverse}\,\mathrm{Cf1}\right)\:$

$\mathrm{cd\_g}≔\mathrm{cov\_diff}\left(g\,\mathrm{coord}\,\mathrm{Cf2}\right)\:$

$\left\{\mathrm{entries}\left(\mathrm{get\_compts}\left(\mathrm{cd\_g}\right)\right)\right\}$

$\left\{\left[0\right]\right\}$

$\mathrm{d2g}≔\mathrm{d2metric}\left(\mathrm{d1g}\,\mathrm{coord}\right)\:$

$\mathrm{Rm}≔\mathrm{Riemann}\left(\mathrm{g\_inverse}\,\mathrm{d2g}\,\mathrm{Cf1}\right)\:$

$\mathrm{cd\_Rm}≔\mathrm{cov\_diff}\left(\mathrm{Rm}\,\mathrm{coord}\,\mathrm{Cf2}\right)\:$

$\mathrm{cd\_Rm}\left[\mathrm{compts}\right]\left[1,2,1,2,2\right]$

$-\frac{6m}{r^4}$