covariant derivative of a tensor_type
cov_diff( U, coord, Cf2)
tensor_type whose covariant derivative is to be computed
list of names of the coordinate variables
rank three tensor_type of character [1,-1,-1] representing the Christoffel symbols of the second kind
Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[Tensor][CovariantDerivative] and Physics[D_] instead.
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 Uindex_char,−1.
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](..).
Define the coordinate variables and the Schwarzchild covariant metric tensor:
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.
Now compute the Riemann tensor and find its covariant derivatives:
Show the covariant derivative of the 1212 component with respect to x2:
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