tensor(deprecated)/cov_diff - Maple Help

tensor

 cov_diff
 covariant derivative of a tensor_type

 Calling Sequence cov_diff( U, coord, Cf2)

Parameters

 U - tensor_type whose covariant derivative is to be computed coord - list of names of the coordinate variables Cf2 - rank three tensor_type of character [1,-1,-1] representing the Christoffel symbols of the second kind

Description

Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[Tensor][CovariantDerivative] and Physics[D_] instead.

 Specifically,

${\mathrm{Cf2}}_{{\mathrm{compts}}_{i,j,k}}≔\left\{\begin{array}{c}i\\ jk\end{array}\right\}$

 • 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 $\left[{U}_{\mathrm{index_char}},-1\right]$.
 • 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](..).

Examples

Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[Tensor][CovariantDerivative] and Physics[D_] instead.

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

Define the coordinate variables and the Schwarzchild covariant metric tensor:

 > $\mathrm{coord}≔\left[t,r,\mathrm{\theta },\mathrm{\phi }\right]$
 ${\mathrm{coord}}{≔}\left[{t}{,}{r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right]$ (1)
 > $\mathrm{g_compts}≔\mathrm{array}\left(\mathrm{symmetric},\mathrm{sparse},1..4,1..4\right):$
 > $\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}{\mathrm{sin}\left(\mathrm{\theta }\right)}^{2}:$
 > $g≔\mathrm{create}\left(\left[-1,-1\right],\mathrm{eval}\left(\mathrm{g_compts}\right)\right)$
 ${g}{≔}{table}{}\left(\left[{\mathrm{index_char}}{=}\left[{-1}{,}{-1}\right]{,}{\mathrm{compts}}{=}\left[\begin{array}{cccc}{1}{-}\frac{{2}{}{m}}{{r}}& {0}& {0}& {0}\\ {0}& {-}\frac{{1}}{{1}{-}\frac{{2}{}{m}}{{r}}}& {0}& {0}\\ {0}& {0}& {-}{{r}}^{{2}}& {0}\\ {0}& {0}& {0}& {-}{{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}\end{array}\right]\right]\right)$ (2)

Compute the Christoffel symbols of the second kind using the appropriate routines:

 > $\mathrm{d1g}≔\mathrm{d1metric}\left(g,\mathrm{coord}\right):$
 > $\mathrm{g_inverse}≔\mathrm{invert}\left(g,\mathrm{detg}\right):$
 > $\mathrm{Cf1}≔\mathrm{Christoffel1}\left(\mathrm{d1g}\right):$
 > $\mathrm{Cf2}≔\mathrm{Christoffel2}\left(\mathrm{g_inverse},\mathrm{Cf1}\right):$

Now given a tensor, you can compute its covariant derivatives using cov_diff. First, compute the covariant derivatives of the metric.  Expect to get zero.

 > $\mathrm{cd_g}≔\mathrm{cov_diff}\left(g,\mathrm{coord},\mathrm{Cf2}\right):$
 > $\left\{\mathrm{entries}\left(\mathrm{get_compts}\left(\mathrm{cd_g}\right)\right)\right\}$
 $\left\{\left[{0}\right]\right\}$ (3)

Now compute the Riemann tensor and find its covariant derivatives:

 > $\mathrm{d2g}≔\mathrm{d2metric}\left(\mathrm{d1g},\mathrm{coord}\right):$
 > $\mathrm{Rm}≔\mathrm{Riemann}\left(\mathrm{g_inverse},\mathrm{d2g},\mathrm{Cf1}\right):$
 > $\mathrm{cd_Rm}≔\mathrm{cov_diff}\left(\mathrm{Rm},\mathrm{coord},\mathrm{Cf2}\right):$

Show the covariant derivative of the 1212 component with respect to x2:

 > $\mathrm{cd_Rm}\left[\mathrm{compts}\right]\left[1,2,1,2,2\right]$
 ${-}\frac{{6}{}{m}}{{{r}}^{{4}}}$ (4)