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 allows one to calculate 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);

 

(2.1)

 

Declare functions:  

>	Functions(F=F(x,y));

 

(2.2)

 

Declare p-forms:  

>	Forms(e[i]=1,omega=2,omega[1]=p,omega[2]=q);

 

(2.3)

 

Declare vectors:  

>	Vectors(X,Y,Z);

 

(2.4)

 

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]); 

 

 

	(2.5)

 

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])); 

(2.6)

 

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

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

(2.7)

 

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]); 

(2.8)

 

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

(2.9)

 

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

 

	(2.10)

 

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

 

	(2.11)

 

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

 

	(2.12)

 

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

 

	(2.13)

 

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

(2.14)

 

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

(2.15)

 

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

(2.16)

 

>

 

See Also:  

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