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tensor

  

cov_diff

  

covariant derivative of a tensor_type

 

Calling Sequence

Parameters

Description

Examples

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:

coordt,r,θ,φ

coord:=t,r,θ,φ

(1)

g_comptsarraysymmetric,sparse,1..4,1..4:

g_compts1,112mr:g_compts2,21g_compts1,1:

g_compts3,3r2:g_compts4,4r2sinθ2:

gcreate1,1,evalg_compts

g:=tablecompts=12mr0000112mr0000r20000r2sinθ2,index_char=1,1

(2)

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

d1gd1metricg,coord:

g_inverseinvertg,detg:

Cf1Christoffel1d1g:

Cf2Christoffel2g_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_gcov_diffg,coord,Cf2:

entriesget_comptscd_g

0

(3)

Now compute the Riemann tensor and find its covariant derivatives:

d2gd2metricd1g,coord:

RmRiemanng_inverse,d2g,Cf1:

cd_Rmcov_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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