atlas[`&**`] - Hodge operator

Calling Sequence:

     &**(expr)

Parameters:

      expr - any expression containing p-forms

Description:

The &** procedure calculates Hodge operator of a p-form expression.  In standard mathematical notation &** is * - just Hodge asterisk. If a metric is presented then the Hodge operator is defined completely by the following.

• Let be vector bundle of p-forms on manifold M of dimension and metric .
• For any integer let us define Hodge operator * as such unique isomorphism of vector bundles
  *:--->which has the following property.
• For any which belong to we have  where is volume form on M induced by metric .
• Let be the number of in the signature of metric (in the atlas package the integer is represented by variable) then the following equations take place:
•  and
• - on vector bundle .  

Examples:
restart:
with(atlas):

Declare forms:
Forms(e[j]=1,xi=1,alpha=p,beta=p);

Declare vectors:
Vectors(X,Y,Z,E[j]);

Example 1

 Sphere -

Declare coframe:
Coframe(e[1]=d(theta),e[2]=d(phi));

Declare frame:
Frame(E[i]);

Declare  metric of    (see atlas[Metric] ):
Metric(g=d(theta)&.d(theta)+sin(theta)^2*d(phi)&.d(phi));

Volume form :
'&**(1)'=&**(1);

Hodge : p-form -> (n-p)-form
&**(alpha);
kind(%);

Double Hodge operator:
'&**(&**(beta))'=&**(&**(beta));

See Also:

atlas , atlas[Frame] , atlas[Coframe] , atlas[Metric] .