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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

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

  

Specifically,

Cf2comptsi,j,kijk

• 

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](..).

Examples

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

withtensor:

Define the coordinate variables and the Schwarzchild covariant metric tensor:

coord:=t,r,θ,φ

coord:=t,r,θ,φ

(1)

g_compts:=arraysymmetric,sparse,1..4,1..4:

g_compts1,1:=12mr:g_compts2,2:=1g_compts1,1:

g_compts3,3:=r2:g_compts4,4:=r2sinθ2:

g:=create1,1,evalg_compts

g:=tableindex_char=1,1,compts=12mr0000112mr0000r20000r2sinθ2

(2)

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

d1g:=d1metricg,coord:

g_inverse:=invertg,detg:

Cf1:=Christoffel1d1g:

Cf2:=Christoffel2g_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_diffg,coord,Cf2:

entriesget_comptscd_g

0

(3)

Now compute the Riemann tensor and find its covariant derivatives:

d2g:=d2metricd1g,coord:

Rm:=Riemanng_inverse,d2g,Cf1:

cd_Rm:=cov_diffRm,coord,Cf2:

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

cd_Rmcompts1,2,1,2,2

6mr4

(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]


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