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.
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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.

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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 .

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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.

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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.
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Define the coordinate variables and the Schwarzchild covariant metric tensor:
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 (1) 
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 (2) 
Compute the Christoffel symbols of the second kind using the appropriate routines:
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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.
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 (3) 
Now compute the Riemann tensor and find its covariant derivatives:
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Show the covariant derivative of the 1212 component with respect to x2:
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 (4) 


See Also


DifferentialGeometry[Tensor][CovariantDerivative], Physics[Christoffel], Physics[D_], Physics[d_], Physics[Einstein], Physics[g_], Physics[LeviCivita], Physics[Ricci], Physics[Riemann], Physics[Weyl], tensor(deprecated), tensor(deprecated)/partial_diff, tensor(deprecated)[Christoffel2], tensor(deprecated)[indexing], tensor(deprecated)[simp]

