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:
coord ≔ t,r,θ,φ
g_compts ≔ array⁡symmetric,sparse,1..4,1..4:
g_compts1,1 ≔ 1−2⁢mr:g_compts2,2 ≔ −1g_compts1,1:
g_compts3,3 ≔ −r2:g_compts4,4 ≔ −r2⁢sin⁡θ2:
g ≔ create⁡−1,−1,eval⁡g_compts
Compute the Christoffel symbols of the second kind using the appropriate routines:
d1g ≔ d1metric⁡g,coord:
g_inverse ≔ invert⁡g,detg:
Cf1 ≔ Christoffel1⁡d1g:
Cf2 ≔ Christoffel2⁡g_inverse,Cf1:
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.
cd_g ≔ cov_diff⁡g,coord,Cf2:
Now compute the Riemann tensor and find its covariant derivatives:
d2g ≔ d2metric⁡d1g,coord:
Rm ≔ Riemann⁡g_inverse,d2g,Cf1:
cd_Rm ≔ cov_diff⁡Rm,coord,Cf2:
Show the covariant derivative of the 1212 component with respect to x2:
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