compute the inner product of two forms with respect to a given metric tensor - Maple Help

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Tensor[FormInnerProduct] - compute the inner product of two forms with respect to a given metric tensor

Calling Sequences

     FormInnerProduct(g, α, β,keyword)

     FormInnerProduct(g, g1, α1, β1, keyword)

Parameters

   g         - a covariant metric tensor on a manifold or on a Lie algebra with frame name, e.g., M

   α, β        - two forms (of the same degree) on M, or lists of such

   α1, β1     - two forms (of the same degree) on M, or lists of such, where M is a Lie algebra with coefficients in a representation space V

   g1                - a covariant metric tensor on the representation space V

      keyword    - the keyword argument inversemetric = h, where h is the inverse of the metric g.

 

Description

Examples

Description

• 

 Let  g = gij dxi dxj and let h = hijxixj be the inverse metric. If α= ai dxi and β= bj dxj are 1-forms, then their inner product is α, β = hijai bj. For monomial p-forms  α1α2  ...  αp and β1β2  ...  βp , the inner product is given by

 α1α2  ...  αp , β1β2  ...  βp = detαr βs.

This formula is extended by bi-linearity to give the general formula for the inner product of a pair of pforms.

• 

In the special case of forms defined on a Lie algebra with coefficients x and y in a representation, the inner product formula for monomials becomes

x α1α2  ...  αp , y β1β2  ...  βp=  gVx,y det αr βs

where x, y ϵ V and gV is the inner product on V. 

Examples

withDifferentialGeometry:withTensor:withLieAlgebras:

 

First define a manifold M with local coordinates x,y,z and define a metric on M.

DGsetupx,y,z,M:

M > 

gevalDGadx &t dx+bdy &t dy+cdz &t dz

g:=adxdx+bdydy+cdzdz

(2.1)

 

Example 1.

Compute the inner product of two 1-forms

M > 

α1evalDGa1dx+a2dy+a3dz

α1:=a1dx+a2dy+a3dz

(2.2)
M > 

β1evalDGb1dx+b2dy+b3dz

β1:=b1dx+b2dy+b3dz

(2.3)
M > 

FormInnerProductg,α1,β1

a1b1a+a2b2b+a3b3c

(2.4)

 

Example 2.

Compute the inner products of a list of monomial 2-forms.

M > 

g2evalDGadx &t dx+bdy &t dy+cdz &t dz

g2:=adxdx+bdydy+cdzdz

(2.5)
M > 

ΩevalDGdx &w dy,dx &w dz,dy &w dz

Ω:=dxdy,dxdz,dydz

(2.6)
M > 

FormInnerProductg2,Ω,Ω

1ab0001ac0001bc

(2.7)

 

Compute the inner product of a pair of 2-forms.

M > 

α2evalDG2dx &w dy+dy &w dz

α2:=2dxdy+dydz

(2.8)
M > 

β2evalDG3dx &w dz+4dy &w dz

β2:=3dxdz+4dydz

(2.9)
M > 

FormInnerProductg2,α2,α2

4ab+1bc

(2.10)

 

Example 3.

In this example we compute the inner products of forms defined on a Lie algebra with coefficients in a representation.

M > 

LDSimpleLieAlgebraDataso(4),so4

LD:=e1,e2=e4,e1,e3=e5,e1,e4=e2,e1,e5=e3,e2,e3=e6,e2,e4=e1,e2,e6=e3,e3,e5=e1,e3,e6=e2,e4,e5=e6,e4,e6=e5,e5,e6=e4

(2.11)
M > 

DGsetupLD

Lie algebra: so4

(2.12)
so4 > 

DGsetupx1,x2,x3,x4,V

frame name: V

(2.13)
so4 > 

ρStandardRepresentationso4,representationspace=V

ρ:=e1,0−100100000000000,e2,00−10000010000000,e3,000−1000000001000,e4,000000−1001000000,e5,0000000−100000100,e6,00000000000−10010

(2.14)
V > 

DGsetupρ,so4V,O,o

Lie algebra with coefficients: so4V

(2.15)
so4V > 

gKillingFormso4V

g:=4o1o14o2o24o3o34o4o44o5o54o6o6

(2.16)
so4V > 

hInverseMetricg

h:=14O1O114O2O214O3O314O4O414O5O514O6O6

(2.17)
so4V > 

gVevalDGdx1 &t dx1+dx2 &t dx2+dx3 &t dx3+dx4 &t dx4

gV:=dx1dx1+dx2dx2+dx3dx3+dx4dx4

(2.18)

 

Compute the inner product of a pair of zero forms.

V > 

FormInnerProductg,gV,ax1+bx2,cx1+dx2

ac+bd

(2.19)

 

Compute the inner product of a pair of 1-forms.

V > 

FormInnerProductg,gV,x1o1,x1o3

0

(2.20)
so4V > 

FormInnerProductg,gV,x2o1,x1o1

0

(2.21)
so4V > 

FormInnerProductg,gV,x2o2,x2o2

14

(2.22)
V > 

FormInnerProductg,gV,x2o1 &w o2,x2o1 &w o2

116

(2.23)

 

Compute the the length of a 2-form.

V > 

α3evalDGax2o1 &w o2+bx4o1 &w o3+cxo2 &w o5

α3:=ax2o1o2+bx4o1o3+cxo2o5

(2.24)
so4V > 

FormInnerProductg,gV,α3,α3

14a2+b2

(2.25)

See Also

DifferentialGeometry

Tensor

ContractIndices

InverseMetric

RaiseLowerIndices

SpinorInnerProduct

TensorInnerProduct

 


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