apply the Hodge star operator to a differential form - Maple Programming Help

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Tensor[HodgeStar] - apply the Hodge star operator to a differential form

Calling Sequences

     HodgeStar(g, omega)

Parameters

   g      - a metric tensor

   omega  - a differential form

   option - (optional) the keyword argument detmetric

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

First create a 5-dimensional manifold M and define a metric tensor g on the tangent space of M.

E > 

DGsetupx1,x2,x3,x4,x5,M1:

M1 > 

gevalDGdx1 &t dx1+dx2 &t dx2+dx3 &t dx3+dx4 &t dx4+dx5 &t dx5

g:=dx1dx1+dx2dx2+dx3dx3+dx4dx4+dx5dx5

(1.1)

 

The standard basis dx1,dx2, ...,dx5 is an orthonormal basis for g and therefore the Hodge star is easily computed.

M1 > 

HodgeStarg,dx1

dx2dx3dx4dx5

(1.2)
M1 > 

HodgeStarg,dx2

dx1dx3dx4dx5

(1.3)
M1 > 

HodgeStarg,dx2 &w dx3

dx1dx4dx5

(1.4)
M1 > 

HodgeStarg,dx2 &w dx4

dx1dx3dx5

(1.5)
M1 > 

HodgeStarg,dx2 &w dx3 &w dx4

dx1dx5

(1.6)

 

Example 2.

To show the dependence of the Hodge star upon the metric, we consider a general metric g on a 2-dimensional manifold.

M1 > 

DGsetupx,y,M2:

M2 > 

gevalDGadx &t dx+bdx &t dy+dy &t dx+cdy &t dy

g:=adxdx+bdxdy+bdydx+cdydy

(1.7)
M2 > 

HodgeStarg,dx

1acb2bdx+1acb2cdy

(1.8)
M2 > 

HodgeStarg,dy

1acb2adx1acb2bdy

(1.9)
M2 > 

fHodgeStarg,dx &w dy

f:=1acb2

(1.10)
M2 > 

HodgeStarg,f

dxdy

(1.11)

 

Example 3.

The Laplacian of a function with respect to a metric g can be calculated using the exterior derivative operation and the Hodge star operator.

To illustrate this result, we use the Euclidean metric in polar coordinates r,ϑ.

M2 > 

DGsetupr,θ,M3:

M3 > 

gevalDGdr &t dr+r2dtheta &t dtheta

g:=drdr+r2dthetadtheta

(1.12)

 

To simplify the definition of the Laplacian, we define the Hodge operator with g fixed.

M3 > 

Hodgef&rarr;HodgeStarg&comma;fassuming0<r

Hodge:=f&rarr;DifferentialGeometry:-Tensor:-HodgeStarg&comma;fassuming0<r

(1.13)

 

To display the Laplacian of &phi; in compact form we invoke the PDEtools[declare] command.

M3 > 

PDEtools&lsqb;declare&rsqb;&phi;r&comma;&theta;

&phi;r&comma;&theta;will now be displayed as&phi;

(1.14)

 

Here is the formula for the Laplacian in terms of HodgeStar and ExteriorDerivative.  Recall that @ is the composition of functions.

M3 > 

&Delta;Hodge&commat;ExteriorDerivative&commat;Hodge&commat;ExteriorDerivative&phi;r&comma;&theta;

&Delta;:=r&phi;r&plus;r2&phi;r&comma;r&plus;&phi;&theta;&comma;&theta;r2

(1.15)

 

Example 4.

The HodgeStar program also works in the more general context of a vector bundle EM.

 

M3 > 

DGsetupx&comma;y&comma;u&comma;v&comma;w&comma;E

frame name: E

(1.16)
E > 

gevalDGdu &t du&plus;dv &t dv&plus;dw &t dw

g:=dudu&plus;dvdv&plus;dwdw

(1.17)
E > 

HodgeStarg&comma;du &w dv3du &w dw&plus;2dv &w dw

2du&plus;3dv&plus;dw

(1.18)

Example 5.

The HodgeStar operation can also be performed using an indefinite metric. The keyword argument detmetric = -1 must be used when the metric has negative determinant.

 

E > 

DGsetupx1&comma;x2&comma;x3&comma;x4&comma;M5&colon;

M1 > 

gevalDGdx1 &t dx1&plus;dx2 &t dx2&plus;dx3 &t dx3dx4 &t dx4

g:=dx1dx1&plus;dx2dx2&plus;dx3dx3dx4dx4

(1.19)

 

 

M1 > 

HodgeStarg&comma;dx1&comma;detmetric&equals;1

dx2dx3dx4

(1.20)
M5 > 

HodgeStarg&comma;dx3 &w dx4&comma;detmetric&equals;1

dx1dx2

(1.21)

Description

 

See Also

DifferentialGeometry

Tensor

DGinfo

ExteriorDerivative

MetricDensity

PermutationSymbol

RaiseLowerIndices

 


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