atlas[`&^`] - exterior product operator 

Calling Sequence: 

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

Parameters: 

       F1, F2, ..., Fn - forms. 

Description: 

• The &^ - procedure allows one to calculate the exterior product of given forms. The main syntax is as follows: omega[1]&^omega[2] i.e. where and are forms. To calculate the exterior product for forms use the following `&^`(omega[1],omega[2], ...omega[k]) i.e.

 

• The exterior product is a 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 defines the exterior product:

 

• 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);

 

(2.1)

 

Declare vectors:
Vectors(X,Y,Z); 

(2.2)

 

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]); 

(2.3)

 

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]; 

(2.4)

 

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

(2.5)

 

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

(2.6)

 

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

(2.7)

 

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

(2.8)

 

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

(2.9)

 

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

(2.10)

 

>

 

See Also:  

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