atlas[dual] - dual operator

Calling Sequence:

     dual(expr)

Parameters:

      expr - any tensor expression.

Description:

• The dual  procedure calculates 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);

Declare functions:
Functions(f=f(x,y),h=h(z));

Declare p-forms:
Forms(e[k]=1,xi=1);

Declare vectors:
Vectors(E[j],X,Y,Z);

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

Declare frame:
Frame(E[k]);

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

Using dual  - procedure:

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

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

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

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

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

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

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

See Also:

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