tensor[Killing_eqns]  compute component expressions for Killings equations

Calling Sequence


Killing_eqns( T, coord, Cf2)


Parameters


T



symmetric covariant tensor

coord



list of coordinate names

Cf2



Christoffel symbols of the second kind





Description


Important: The tensor package has been deprecated. Use the superseding packages DifferentialGeometry and Physics instead.
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The function Killing_eqns(T, coord, Cf2 ) computes the expressions for Killing's equations for each component of the totally symmetric covariant tensor T. Specifically, the symmetric part of the covariant derivative of T is computed and returned as a tensor_type. The components of T satisfy Killing's equations if all of the components of the result are zero. Note that the rank of the result is one more than that of T.

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This routine is useful in two ways: first, as a means of verifying that a tensor satisfies Killing's equations, and second, as a way of generating the differential equations for any unknown components of a symmetric tensor which is to satisfy Killing's equations.

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T must be of rank 1 or greater. If T is of second rank or more, the component array of T must use Maple's symmetric indexing function (since T must be symmetric).

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Cf2 should be indexed using the cf2 indexing function provided by the tensor package. It can be computed using the Christoffel2 routine.

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



Examples


Important: The tensor package has been deprecated. Use the superseding packages DifferentialGeometry and Physics instead.
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Generate the Killing equation expressions for an arbitrary vector in the geometry of Euclidean 3space using polar coordinates: First, compute the Christoffel symbols of the second kind:
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 (1) 
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Next, define the arbitrary vector field:
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 (2) 
Now compute the Killing equation expressions:
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 (3) 
Now try it for an arbitrary symmetric 0, 2tensor:
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 (4) 
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 (5) 


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