atlas[iota] - interior product operator

Calling Sequence:

     iota[V1, V2, ..., Vn](expr)
     iota(V1, V2, ..., Vn, expr)
     atlas[iota](V1, V2, ..., Vn, expr)

Parameters:

         expr - any expression (on which interior product operator is defined) .
        V1, V2, ..., Vn - vector fields.

Description:

•  The iota procedure calculates the interior product of given expression and vector fields. The main syntax is iota[X](alpha) i.e.  where X is a vector and  is a p-form.
• Let X be a vector and  be n-form in some k-dimensional manifold then under definition:    
• Multiple iota operator defined as follows:  
•  It is well known that iota operator is anti-differentiation for p-forms. Thus if  is p-form then:
   

Examples:
restart:
with(atlas):

Declare constants:
Constants(alpha);

Declare functions:
Functions(F=F(x,y));

Declare p-forms:
Forms(e[i]=1,omega=2,omega[1]=p,omega[2]=q);

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

Using iota - procedure:

Just definition for "long" iota operator:
'iota[X,Y,Z](omega[1])'=iota[X,Y,Z](omega[1]);
'iota(X,Y,Z,omega[1])'=iota(X,Y,Z,omega[1]);
'iota[X](Y,Z,omega[1])'=iota[X](Y,Z,omega[1]);

As  is p-form and  is q-form then under main rule for interior product we have:
 'iota[X](omega[1]&^omega[2])'=simplify(iota[X](omega[1]&^omega[2]));

It is obvious that  (see atlas[`&^`] ).

Interior product on any 0-form equals zero:
'iota[X]'(h)=iota[X](h);

Interior product is linear with respect to any argument:
'iota[alpha*X+F*Y+x*Z]'(e[j])=iota[alpha*X+F*Y+x*Z](e[j]);

And
'iota[X]'(F*e[j]+alpha*e[k]+x*e[l])=iota[X](F*e[j]+alpha*e[k]+x*e[l]);

Iota operator reduces covariance:
iota[X](e[j]);
kind(%);

Calculate on 2-form:
iota[X](omega);
kind(%);

Calculate on p-form:
iota[X](omega[1]);
kind(%);

Calculate triple iota on p-form:
iota[X,Y,Z](omega[1]);
kind(%);

Verify the main rule for interior product:
'iota[X](e[i]&^e[j])'=iota[X](e[i]&^e[j]);

But for tensor product we have:
'iota[X]'(e[j]&.e[i])=iota[X](e[j]&.e[i]);

And then:
'iota[Y,X]'(e[i]&.e[j])=iota[Y,X](e[i]&.e[j]);

See Also:

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