tensor/cov_diff

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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

Specifically,

Cf2[compts[i,j,k]]:={[[i],[j k]]}

•	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 .

•

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

Define the coordinate variables and the Schwarzchild covariant metric tensor:

(1)

g := TABLE([compts = matrix([[1-2*m/r, 0, 0, 0], [0, -1/(1-2*m/r), 0, 0], [0, 0, -r^2, 0], [0, 0, 0, -r^2*sin(theta)^2]]), index_char = [-1, -1]])

(2)

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

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.

>	${entries(get_compts(cd_g))}$

(3)

Now compute the Riemann tensor and find its covariant derivatives:

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

(4)

	See Also
	Physics[Christoffel], Physics[D_], Physics[d_], Physics[Einstein], Physics[g_], Physics[LeviCivita], Physics[Ricci], Physics[Riemann], Physics[Weyl], tensor, tensor/partial_diff, tensor[Christoffel2], tensor[indexing], tensor[simp]