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tensor

  

Killing_eqns

  

compute component expressions for Killings equations

 

Calling Sequence

Parameters

Description

Examples

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.

• 

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.

• 

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.

• 

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).

• 

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

• 

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.

withtensor:

Generate the Killing equation expressions for an arbitrary vector in the geometry of Euclidean 3-space using polar coordinates: First, compute the Christoffel symbols of the second kind:

coordr,θ,φ:

g_comptsarraysymmetric,sparse,1..3,1..3,1,1=1,2,2=r2,3,3=r2sinθ2:

gcreate1,1,evalg_compts

g:=tablecompts=1000r2000r2sinθ2,index_char=1,1

(1)

ginvinvertg,'detg':

d1gd1metricg,coord:d2gd2metricd1g,coord:

Cf1Christoffel1d1g:

Cf2Christoffel2ginv,Cf1:

Next, define the arbitrary vector field:

Vcreate1,arrayv1r,θ,φ,v2r,θ,φ,v3r,θ,φ

V:=tablecompts=v1r,θ,φv2r,θ,φv3r,θ,φ,index_char=1

(2)

Now compute the Killing equation expressions:

KVKilling_eqnsV,coord,Cf2

KV:=tablecompts=rv1r,θ,φ12rv2r,θ,φr+θv1r,θ,φr2v2r,θ,φr12rv3r,θ,φr+φv1r,θ,φr2v3r,θ,φr12rv2r,θ,φr+θv1r,θ,φr2v2r,θ,φrθv2r,θ,φ+rv1r,θ,φ12θv3r,θ,φsinθ+φv2r,θ,φsinθ2cosθv3r,θ,φsinθ12rv3r,θ,φr+φv1r,θ,φr2v3r,θ,φr12θv3r,θ,φsinθ+φv2r,θ,φsinθ2cosθv3r,θ,φsinθv1r,θ,φcosθ2r+sinθcosθv2r,θ,φ+rv1r,θ,φ+φv3r,θ,φ,index_char=1,1

(3)

Now try it for an arbitrary symmetric 0, 2-tensor:

tarraysymmetric,1..3,1..3:

forito3doforjfromito3doti,jcat't',i,jr,θ,φend doend do;Tcreate1,1,evalt

t33r,θ,φ

T:=tablecompts=t11r,θ,φt12r,θ,φt13r,θ,φt12r,θ,φt22r,θ,φt23r,θ,φt13r,θ,φt23r,θ,φt33r,θ,φ,index_char=1,1

(4)

KTKilling_eqnsT,coord,Cf2

KT:=tablecompts=ARRAYsymmetric,1..3,1..3,1..3,1,1,1=rt11r,θ,φ,1,1,2=132rt12r,θ,φr+θt11r,θ,φr4t12r,θ,φr,1,1,3=132rt13r,θ,φr+φt11r,θ,φr4t13r,θ,φr,1,2,1=132rt12r,θ,φr+θt11r,θ,φr4t12r,θ,φr,1,2,2=132r2t11r,θ,φ+rt22r,θ,φr+2θt12r,θ,φr4t22r,θ,φr,1,2,3=13rt23r,θ,φsinθr+θt13r,θ,φrsinθ+φt12r,θ,φrsinθ2cosθt13r,θ,φr4t23r,θ,φsinθrsinθ,1,3,1=132rt13r,θ,φr+φt11r,θ,φr4t13r,θ,φr,1,3,2=13rt23r,θ,φsinθr+θt13r,θ,φrsinθ+φt12r,θ,φrsinθ2cosθt13r,θ,φr4t23r,θ,φsinθrsinθ,1,3,3=132t11r,θ,φcosθ2r2+2sinθcosθt12r,θ,φr+2r2t11r,θ,φ+rt33r,θ,φr+2φt13r,θ,φr4t33r,θ,φr,2,1,1=132rt12r,θ,φr+θt11r,θ,φr4t12r,θ,φr,2,1,2=132r2t11r,θ,φ+rt22r,θ,φr+2θt12r,θ,φr4t22r,θ,φr,2,1,3=13rt23r,θ,φsinθr+θt13r,θ,φrsinθ+φt12r,θ,φrsinθ2cosθt13r,θ,φr4t23r,θ,φsinθrsinθ,2,2,1=132r2t11r,θ,φ+rt22r,θ,φr+2θt12r,θ,φr4t22r,θ,φr,2,2,2=θt22r,θ,φ+2rt12r,θ,φ,2,2,3=132t13r,θ,φrsinθ+2θt23r,θ,φsinθ+φt22r,θ,φsinθ4cosθt23r,θ,φsinθ,2,3,1=13rt23r,θ,φsinθr+θt13r,θ,φrsinθ+φt12r,θ,φrsinθ2cosθt13r,θ,φr4t23r,θ,φsinθrsinθ,2,3,2=132t13r,θ,φrsinθ+2θt23r,θ,φsinθ+φt22r,θ,φsinθ4cosθt23r,θ,φsinθ,2,3,3=132t12r,θ,φcosθ2sinθr2t22r,θ,φcosθ3+2rt12r,θ,φsinθ+θt33r,θ,φsinθ+2φt23r,θ,φsinθ4cosθt33r,θ,φ+2cosθt22r,θ,φsinθ,3,1,1=132rt13r,θ,φr+φt11r,θ,φr4t13r,θ,φr,3,1,2=13rt23r,θ,φsinθr+θt13r,θ,φrsinθ+φt12r,θ,φrsinθ2cosθt13r,θ,φr4t23r,θ,φsinθrsinθ,3,1,3=132t11r,θ,φcosθ2r2+2sinθcosθt12r,θ,φr+2r2t11r,θ,φ+rt33r,θ,φr+2φt13r,θ,φr4t33r,θ,φr,3,2,1=13rt23r,θ,φsinθr+θt13r,θ,φrsinθ+φt12r,θ,φrsinθ2cosθt13r,θ,φr4t23r,θ,φsinθrsinθ,3,2,2=132t13r,θ,φrsinθ+2θt23r,θ,φsinθ+φt22r,θ,φsinθ4cosθt23r,θ,φsinθ,3,2,3=132t12r,θ,φcosθ2sinθr2t22r,θ,φcosθ3+2rt12r,θ,φsinθ+θt33r,θ,φsinθ+2φt23r,θ,φsinθ4cosθt33r,θ,φ+2cosθt22r,θ,φsinθ,3,3,1=132t11r,θ,φcosθ2r2+2sinθcosθt12r,θ,φr+2r2t11r,θ,φ+rt33r,θ,φr+2φt13r,θ,φr4t33r,θ,φr,3,3,2=132t12r,θ,φcosθ2sinθr2t22r,θ,φcosθ3+2rt12r,θ,φsinθ+θt33r,θ,φsinθ+2φt23r,θ,φsinθ4cosθt33r,θ,φ+2cosθt22r,θ,φsinθ,3,3,3=2t13r,θ,φcosθ2r+2t23r,θ,φcosθsinθ+2rt13r,θ,φ+φt33r,θ,φ,index_char=1,1,1

(5)

See Also

tensor(deprecated)

tensor(deprecated)/cov_diff

tensor(deprecated)[Christoffel2]

tensor(deprecated)[simp]

tensor(deprecated)[symmetrize]

 


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