Specifically,

The resultant tensor_type, Cf1 say, of this routine is the Christoffel symbols of the first kind, indexed as shown below:

where $\left[\mathrm{ij}\,k\right]$ is in conventional notation.

D1g, the partials of the covariant metric should be obtained using the function tensor[d1metric] once the metric itself is known.

Indexing Function: Because of the symmetry in the first two indices of the Christoffel symbols of the first kind, the array of the calculated symbols use the `index/cf1` indexing function.  This function indexes an array of rank 3 so that it is automatically symmetric in its first two indices.  Use of this indexing function decreases the number of symbols that must be assigned and stored to the number of independent symbols.

Simplification: This routine uses the `tensor/Christoffel1/simp` routine to carry out simplification of each independent Christoffel symbol of the first kind.  By default, it is initialized to the `tensor/simp` function.  It is recommended that the `tensor/Christoffel1/simp` routine be customized to suit the particular needs of the problem at hand.

This function is part of the tensor package, and so can be used in the form Christoffel1(..) only after performing the command with(tensor) or with(tensor, Christoffel1).  The function can always be accessed in the long form tensor[Christoffel1](..).

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

$\mathrm{coord}≔\left[t\,r\,\mathrm{th}\,\mathrm{ph}\right]\:$

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

$\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{th}\right)}^2\:$

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

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

$\mathrm{D1g}≔\mathrm{d1metric}\left(g\,\mathrm{coord}\right)\:$

$\mathrm{Cf1}≔\mathrm{Christoffel1}\left(\mathrm{D1g}\right)$

$\mathrm{Cf1}≔table\left(\left[\mathrm{index\_char}=\left[\mathrm{-1}\,\mathrm{-1}\,\mathrm{-1}\right]\,\mathrm{compts}=array\left(\mathrm{cf1}\,1..4\,1..4\,1..4\,\left[\left(1\,1\,1\right)=0\,\left(1\,1\,2\right)=-\frac{m}{r^2}\,\left(1\,1\,3\right)=0\,\left(1\,1\,4\right)=0\,\left(1\,2\,1\right)=\frac{m}{r^2}\,\left(1\,2\,2\right)=0\,\left(1\,2\,3\right)=0\,\left(1\,2\,4\right)=0\,\left(1\,3\,1\right)=0\,\left(1\,3\,2\right)=0\,\left(1\,3\,3\right)=0\,\left(1\,3\,4\right)=0\,\left(1\,4\,1\right)=0\,\left(1\,4\,2\right)=0\,\left(1\,4\,3\right)=0\,\left(1\,4\,4\right)=0\,\left(2\,1\,1\right)=\frac{m}{r^2}\,\left(2\,1\,2\right)=0\,\left(2\,1\,3\right)=0\,\left(2\,1\,4\right)=0\,\left(2\,2\,1\right)=0\,\left(2\,2\,2\right)=\frac{m}{{\left(-r+2m\right)}^2}\,\left(2\,2\,3\right)=0\,\left(2\,2\,4\right)=0\,\left(2\,3\,1\right)=0\,\left(2\,3\,2\right)=0\,\left(2\,3\,3\right)=-r\,\left(2\,3\,4\right)=0\,\left(2\,4\,1\right)=0\,\left(2\,4\,2\right)=0\,\left(2\,4\,3\right)=0\,\left(2\,4\,4\right)=-r{\sin \left(\mathrm{th}\right)}^2\,\left(3\,1\,1\right)=0\,\left(3\,1\,2\right)=0\,\left(3\,1\,3\right)=0\,\left(3\,1\,4\right)=0\,\left(3\,2\,1\right)=0\,\left(3\,2\,2\right)=0\,\left(3\,2\,3\right)=-r\,\left(3\,2\,4\right)=0\,\left(3\,3\,1\right)=0\,\left(3\,3\,2\right)=r\,\left(3\,3\,3\right)=0\,\left(3\,3\,4\right)=0\,\left(3\,4\,1\right)=0\,\left(3\,4\,2\right)=0\,\left(3\,4\,3\right)=0\,\left(3\,4\,4\right)=-\frac{\sin \left(2\mathrm{th}\right)r^2}{2}\,\left(4\,1\,1\right)=0\,\left(4\,1\,2\right)=0\,\left(4\,1\,3\right)=0\,\left(4\,1\,4\right)=0\,\left(4\,2\,1\right)=0\,\left(4\,2\,2\right)=0\,\left(4\,2\,3\right)=0\,\left(4\,2\,4\right)=-r{\sin \left(\mathrm{th}\right)}^2\,\left(4\,3\,1\right)=0\,\left(4\,3\,2\right)=0\,\left(4\,3\,3\right)=0\,\left(4\,3\,4\right)=-\frac{\sin \left(2\mathrm{th}\right)r^2}{2}\,\left(4\,4\,1\right)=0\,\left(4\,4\,2\right)=r{\sin \left(\mathrm{th}\right)}^2\,\left(4\,4\,3\right)=\frac{\sin \left(2\mathrm{th}\right)r^2}{2}\,\left(4\,4\,4\right)=0\right]\right)\right]\right)$