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

with(DifferentialGeometry): with(Tensor):

 

Example 1.

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

E > 

DGsetup([x1, x2, x3, x4, x5], M1):

M1 > 

g := evalDG(dx1 &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 > 

HodgeStar(g, dx1);

dx2dx3dx4dx5

(1.2)
M1 > 

HodgeStar(g, dx2);

dx1dx3dx4dx5

(1.3)
M1 > 

HodgeStar(g, dx2 &w dx3);

dx1dx4dx5

(1.4)
M1 > 

HodgeStar(g, dx2 &w dx4);

dx1dx3dx5

(1.5)
M1 > 

HodgeStar(g, 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 > 

DGsetup([x, y], M2):

M2 > 

g := evalDG(a*dx &t dx + b*(dx &t dy + dy &t dx) + c*dy &t dy);

g:=adxdx+bdxdy+bdydx+cdydy

(1.7)
M2 > 

HodgeStar(g, dx);

1acb2bdx+1acb2cdy

(1.8)
M2 > 

HodgeStar(g, dy);

1acb2adx1acb2bdy

(1.9)
M2 > 

f := HodgeStar(g, dx &w dy);

f:=1acb2

(1.10)
M2 > 

HodgeStar(g, 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 > 

DGsetup([r, theta], M3):

M3 > 

g := evalDG(dr &t dr + r^2*dtheta &t dtheta);

g:=drdr+r2dthetadtheta

(1.12)

 

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

M3 > 

Hodge := f -> (HodgeStar(g, f) assuming r > 0);

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[declare](phi(r, 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 @ ExteriorDerivative @ Hodge @ ExteriorDerivative)(phi(r, 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 > 

DGsetup([x, y], [u, v, w], E);

frame name: E

(1.16)
E > 

g := evalDG(du &t du + dv &t dv + dw &t dw);

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

(1.17)
E > 

HodgeStar(g, du &w dv - 3*du &w dw + 2*dv &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 > 

DGsetup([x1, x2, x3, x4], M5):

M1 > 

g := evalDG(dx1 &t dx1 + dx2 &t dx2 + dx3 &t dx3 - dx4 &t dx4);

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

(1.19)

 

 

M1 > 

HodgeStar(g, dx1, detmetric = -1);

dx2dx3dx4

(1.20)
M5 > 

HodgeStar(g, dx3 &w dx4, detmetric = -1);

dx1dx2

(1.21)

Description

 

See Also

DifferentialGeometry

Tensor

DGinfo

ExteriorDerivative

MetricDensity

PermutationSymbol

RaiseLowerIndices