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tensor

  

exterior_diff

  

Compute the exterior derivative of a completely anti-symmetric covariant tensor.

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

exterior_diff(T, coord)

Parameters

T

-

covariant anti-symmetric tensor or scalar

coord

-

list of coordinate variable names

Description

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

• 

The function exterior_diff( T, coord ) computes the exterior derivative of the (covariant) components of the anti-symmetric tensor T and returns them as a tensor_type of rank equal to rankT+1.  The result is totally anti-symmetric and uses the antisymmetric indexing function (unless T is a scalar).

• 

If T is not a scalar or vector, it should be indexed using the antisymmetric indexing function.  If it is not indexed using the antisymmetric indexing function and it is not a scalar or vector, then the routine will determine its anti-symmetric part and compare that with the tensor to see if the components are really totally anti-symmetric. If it is found not to be completely anti-symmetric, the routine exits with an error.

• 

The result is computed first by finding the first partials of the tensor components and then antisymmetrizing them.

• 

Simplification:  This routine uses the `tensor/lin_com/simp` and `tensor/partial_diff/simp` routines for simplification purposes.  The simplification routines are used internally by the partial_diff and antisymmetrize routines as they are called by exterior_diff.  By default, `tensor/lin_com/simp` and `tensor/partial_diff/simp` are initialized to the `tensor/simp` routine.  It is recommended that these routines be customized to suit the needs of the particular problem.

• 

This command is part of the tensor package, so it can be used in the form exterior_diff(..) only after executing the command with(tensor). However, it can always be accessed through the long from of the command by using tensor[exterior_diff](..).

Examples

Important: The tensor package has been deprecated. Use the superseding packages DifferentialGeometry and Physics instead.

withtensor:

Define the coordinates and an arbitrary skew-symmetric second rank tensor:

coordx,y,z:

tcarrayantisymmetric,1..3,1..3:

forito3doforjfromi+1to3dotci,jcat't',i,jx,y,zend doend do;Tcreate1,1,evaltc

T:=tablecompts=0t12x,y,zt13x,y,zt12x,y,z0t23x,y,zt13x,y,zt23x,y,z0,index_char=1,1

(1)

Now compute the exterior derivative:

ex_derexterior_diffT,coord

ex_der:=tablecompts=ARRAYantisymmetric,1..3,1..3,1..3,1,1,1=0,1,1,2=0,1,1,3=0,1,2,1=0,1,2,2=0,1,2,3=zt12x,y,zyt13x,y,z+xt23x,y,z,1,3,1=0,1,3,2=zt12x,y,z+yt13x,y,zxt23x,y,z,1,3,3=0,2,1,1=0,2,1,2=0,2,1,3=zt12x,y,z+yt13x,y,zxt23x,y,z,2,2,1=0,2,2,2=0,2,2,3=0,2,3,1=zt12x,y,zyt13x,y,z+xt23x,y,z,2,3,2=0,2,3,3=0,3,1,1=0,3,1,2=zt12x,y,zyt13x,y,z+xt23x,y,z,3,1,3=0,3,2,1=zt12x,y,z+yt13x,y,zxt23x,y,z,3,2,2=0,3,2,3=0,3,3,1=0,3,3,2=0,3,3,3=0,index_char=1,1,1

(2)

Note that the result uses antisymmetric indexing:

op1,get_comptsex_der

antisymmetric

(3)

See Also

tensor(deprecated)

tensor(deprecated)/exterior_prod

tensor(deprecated)/partial_diff

tensor(deprecated)[antisymmetrize]

tensor(deprecated)[simp]

 


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