atlas[`&**`] - Hodge operator 

Calling Sequence: 

    &**(expr) 

Parameters: 

    expr - any expression containing p-forms 

Description: 

The &** procedure allows one to calculate Hodge operator on an expression containing p-forms.  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);

 

(2.1)

 

Declare vectors: 

>	Vectors(X,Y,Z,E[j]);

 

(2.2)

 

Example 1 

Sphere -  

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

(2.1.1)

 

Declare frame:
Frame(E[i]); 

(2.1.2)

 

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

(2.1.3)

 

Volume form :
'&**(1)'=&**(1); 

(2.1.4)

 

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

 

	(2.1.5)

 

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

(2.1.6)

 

>

 

See Also:  

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