atlas[`&$`] - generalized interior product operator 

Calling Sequence: 

    `&$`(A1, A2, ..., An, expr) 

Parameters: 

       expr - any expression.
       A1, A2, ..., An - vectors or 1-forms. 

Description: 

• The `&$` - procedure allows one to calculate the generalized interior product (see atlas[iota]) of given expressions and vector or 1-forms fields.

 

• Let X be a vector and be a tensor then under the definition:  

 

• Let be 1-form and be a vector then under the definition:  

 

• For any vectors or 1-forms  the multiple `&$` operator is defined as follows:  

 

• For any vector or 1-form and any tensors and the following rule applies:

 

Examples: 

>	restart: with(atlas):

 

Declare constants:  

>	Constants(alpha);

 

(2.1)

 

Declare p-forms:  

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

 

(2.2)

 

Declare vectors:  

>	Vectors(X,Y,Z);

 

(2.3)

 

Using the `&$`- procedure: 

This is just the definition for the "long" operator:
'`&$`(X,Y,Z,omega[1])'=`&$`(X,Y,Z,omega[1]);
'`&$`(e[i],e[j],e[k],`&.`(X,Y,Z,Y))'=`&$`(e[i],e[j],e[k],`&.`(X,Y,Z,Y));
 

 

	(2.4)

 

The generalized interior product is linear with respect to any argument:
'`&$`(alpha*X+F*Y+x*Z,e[j])'=`&$`(alpha*X+F*Y+x*Z,e[j]); 

(2.5)

 

>	'`&$`(alpha*e[1]+F*e[2]+x*e[3],X)'=`&$`(alpha*e[1]+F*e[2]+x*e[3],X);

 

(2.6)

 

And
'`&$`(X,F*e[j]+alpha*e[k]+x*e[l])'=`&$`(X,F*e[j]+alpha*e[k]+x*e[l]); 

(2.7)

 

>	'`&$`(F*e[j]+alpha*e[k]+x*e[l],X)'=`&$`(F*e[j]+alpha*e[k]+x*e[l],X);

 

(2.8)

 

>	'`&$`(e[k],Y,`&.`(X,e[i],e[j]))'=`&$`(e[k],Y,`&.`(X,e[i],e[j]));

 

(2.9)

 

Example 1 

>	restart: with(atlas):

 

Declare forms: 

>	Forms(e[j]=1,xi=1);

 

(2.1.1)

 

Declare vectors: 

>	Vectors(X,Y,Z,E[j]);

 

(2.1.2)

 

Declare the coframe:
Coframe(e[1]=x*d(x)+y*d(y),e[2]=x*d(y)-y*d(x)); 

(2.1.3)

 

Declare the frame:
Frame(E[i]); 

(2.1.4)

 

Connection definition: 

>	omega[1,1]:=x*e[1];

 

(2.1.5)

 

>	omega[2,2]:=y*e[2];

 

(2.1.6)

 

>	omega[1,2]:=y*e[1];

 

(2.1.7)

 

>	omega[2,1]:=-x*e[2];

 

(2.1.8)

 

Connection declaration: 

>	Connection(omega);

 

(2.1.9)

 

Curvature calculation:
Curvature(Omega); 

(2.1.10)

 

Riemann calculation:
Riemann(R); 

(2.1.11)

 

>	`&$`(e[i],R);

 

(2.1.12)

 

>	`&$`(e[i],E[j],R);

 

(2.1.13)

 

>	`&$`(e[i],E[j],E[k],E[l],R);

 

(2.1.14)

 

>

 

See Also:  

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