atlas[dual] - dual operator 

Calling Sequence: 

    dual(expr) 

Parameters: 

    expr - any tensor expression. 

Description: 

• The dual - procedure allows one to calculate the dual operator on an expression (just rise and lower indexes in local coordinates). The operator has the following properties.

 

• The dual operator is linear.

 

• For any vector field X in local coordinates we have: .

 

• For any 1-form in local coordinates we have:   .

 

• For any tensor fields and the following rule for dual operator takes place: .

 

Examples: 

>	restart: with(atlas):

 

Declare constants:  

>	Constants(Lambda);

 

(2.1)

 

Declare functions:  

>	Functions(f=f(x,y),h=h(z));

 

(2.2)

 

Declare p-forms:  

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

 

(2.3)

 

Declare vectors:  

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

 

(2.4)

 

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

(2.5)

 

Declare frame:
Frame(E[k]); 

(2.6)

 

Declare metric:Metric(g=4*(d(x)&.d(x)+d(y)&.d(y))/(1+Lambda*(x^2+y^2))^2); 

(2.7)

 

Using dual - procedure: 

For frame vectors:
'dual(E[j])'=dual(E[j]); 

(2.8)

 

For coframe 1-forms:
'dual(e[j])'=dual(e[j]); 

(2.9)

 

For exact 1-form:
'dual(d(f))'=dual(d(f)); 

(2.10)

 

With ToBasis procedure:
dual(X)=dual(ToBasis(X)); 

(2.11)

 

For 1-form:
dual(xi)=dual(ToBasis(xi)); 

(2.12)

 

For [1,1] tensor:
'dual(&.(E[i],e[j]))'=dual(&.(E[i],e[j])); 

(2.13)

 

For metric tensor:
'dual(g)'=simplify(dual(g)); 

(2.14)

 

>

 

See Also:  

atlas, atlas[d], atlas[cov], atlas[`&.`], atlas[`&^`], atlas[iota].