atlas[d] - exterior differentiation

Calling Sequence:

     d(expr)

Parameters:

      expr - any expression .

Description:

• The d procedure calculates exterior derivative of a p-form expression.
• The exterior derivative is operator  d :  -> where is p-form and is (p+1)-form. The operator has the following properties.
• For any 0-form we have:
• For any p-forms and constants we have:
• If is p-form and is q-form then exterior derivative of their exterior product is as follows:
• For any p-form Poincare's lemma takes place:

Examples:
restart:
with(atlas):

Declare constants:
Constants(alpha,beta,p,q);

Declare functions:
Functions(F=F(x[1],x[2],x[3]),f=f(x[k]),G=G(x,z),z=z(phi),x[k]=x[k](y[j]));

Declare p-forms:
Forms(omega=n,sigma=m,Omega=P,omega[1]=p,omega[2]=q);

Using d- procedure:

f- declared as a function with indefinite number of variables:
'd'(f)=d(f);

F- declared as function on three variables:
'd'(F)=d(F);

G- declared as function G=G(x,z) and z=z(phi) thus:
'd'(G)=d(G);

As  declared as constant thus:
'd'(alpha)=d(alpha);

Exterior derivative is linear operation:
'd'(alpha*omega+beta*sigma)=d(alpha*omega+beta*sigma);

There were not any declarations about x and y so they are 0-forms by defaults:
'd'(x*y)=d(x*y);

Verify that double-d gives 0 (Poincare's lemma):
'd'(d(Omega))=d(d(Omega));

As y is 0-form (by defaults) and  has been declared as P-form then:
'd'(y*Omega)=d(y*Omega);

and  - declared as p- and q-forms respectively:
'd'(omega[1]&^omega[2])=d(omega[1]&^omega[2]);

See Also:

atlas , atlas[Constants] , atlas[Functions] , atlas[Forms] , atlas[`&^`] .