atlas[`&^`] - exterior product operator

Calling Sequence:

     `&^`(F1,F2, ..., Fn)
     F1 &^ F2

Parameters:

        F1, F2, ..., Fn - forms.

Description:

•  The &^ procedure calculates the exterior product of given forms. The main syntax is as follows: omega[1]&^omega[2] i.e.  where  and  are forms. To calculate exterior product for forms  use the following `&^`(omega[1],omega[2], ...omega[k]) i.e.
• Exterior product is linear operation with respect to its arguments. Thus if  are 0-forms then:
• Let  be p-form and  be q-form then the following formula for exterior product takes place:  
• Particularly for 1-forms  and  we have:

Examples:
restart:
with(atlas):

Declare p-forms:
Forms(sigma=1,omega=1,omega[1]=p,omega[2]=q,omega[3]=l);

Declare vectors:
Vectors(X,Y,Z);

Using &^ - procedure:

Exterior product is linear operation with respect to its arguments
'`&^`(omega[1],alpha*omega[2]+beta*omega[3])' = `&^`(omega[1],alpha*omega[2]+beta*omega[3]);

As  is p-form and  is q-form then under main rule for exterior product we have:
'omega[2]&^omega[1]'=omega[2]&^omega[1];

Particularly for 1-forms  and  we have:
'sigma&^omega'=sigma&^omega;

Some more examples:
'`&^`(omega,sigma,omega[3],omega[2],omega[1])' = `&^`(omega,sigma,omega[3],omega[2],omega[1]);

And with Lie derivative:
'L[X]'(omega&^sigma)=L[X](omega&^sigma);

And with exterior derivative:
'd'(omega&^sigma)=d(omega&^sigma);

And again
'L[X]'(omega[1])=L[X](omega[1]);

And finally
'omega[3]&^(L[Z](omega)&^sigma)'=omega[3]&^(L[Z](omega)&^sigma);

See Also:

atlas , atlas[L] , atlas[d] , atlas[`&.`] , atlas[Forms] .