atlas[`&/`] - restriction of a [0, k] tensor field under a mapping

Calling Sequence:

     Expr &/ MapId

Parameters:

      MapId - variable  - the mapping identifier  i.e. MapId : dom1 ---> dom2
      Expr - expression - a tensor expression which has to be restricted

Description:

• The &/ procedure calculates restriction of a tensor field under a mapping. The restriction is linear  operation defined on [0,k] tensors only. The definition is as follows.
• Let M and N be manifolds of dimensions . Let F be mapping between the manifolds: F:  defined by functions:
  
  
  
  where  are local coordinates on M and  are local coordinates on N (in some domains).
• For any [0,1] tensor field T on N the restriction of  T  under F is tensor field  on M with components  in local coordinates.
• For tensor product of any [0, k] tensor fields   on N the following formula takes place:
• The formulas considered above defined linear restriction operator &/ completely.
• According to the definition it is necessary to calculate the restrictions on the domain M. Use atlas[Domain]  procedure to jump on M manifold if needed.

Examples:

The following example shows how the restriction operator can be used.
Let M be 2-dimentional sphere  and N be 3-dimensional Euclidean space . Let :   be standard embedding of sphere  into .

restart:
with(atlas):

This procedure is presented just for appropriate simplification (see atlas[simp] ).
`atlas/simp`:=proc(a) normal(a);subs({cos(theta)^2=1-sin(theta)^2,cos(phi)^2=1-sin(phi)^2},%);normal(%); factor(%) end;

Declare 1-forms e[j] and u[k] for corresponding coframes:
Forms(e[j]=1,u[k]=1);

Declare vectors for corresponding frames:
Vectors(E[j],U[k]);

Declare Euclidean space - :
Domain(R^3);

Declare coframe on :
Coframe(e[1]=d(x),e[2]=d(y),e[3]=d(z));

Declare frame on :
Frame(E[j]);

Declare metric on  (standard flat metric):
Metric(g=d(x)&.d(x)+d(y)&.d(y)+d(z)&.d(z));

Declare sphere - :
Domain(S^2);

Declare coframe on :
Coframe(u[1]=d(theta),u[2]=d(phi));

Declare frame on :
Frame(U[j]);

Declare definite mapping : :
Mapping(F,S^2,R^3,
        x=sin(theta)*cos(phi),
        y=sin(theta)*sin(phi),
        z=cos(theta));

Who(F);

Verify that we are on the sphere:
Domain();

Calculate metric induced on the sphere using &/ operator:
Metric(G = g &/ F);

One can calculate restriction of any [0,k] tensor field on  under the mapping:

Restriction of coframe 1-forms:
'e[1]&/F'=e[1]&/F;
'e[2]&/F'=e[2]&/F;

Restriction of 0-forms (scalars):
'(x^2+y^2)&/F'=(x^2+y^2)&/F;
'(y/x)&/F'=(y/x)&/F;

Restriction of "rotation" 1-form:
'(x*d(y)-y*d(x))&/F'=(x*d(y)-y*d(x))&/F;

Restriction of tensor product d(x)&.d(z):
'(d(x)&.d(z))&/F'=(d(x)&.d(z))&/F;

Restriction of exterior product d(x)&^d(y):
'(d(x)&^d(y))&/F'=(d(x)&^d(y))&/F;

Some more examples

Declare abstract mapping between  and :
Mapping(Phi,S^2,R^3);

Restriction of exterior product d(x)&^d(y) under abstract mapping :
'(d(x)&^d(y))&/Phi'=(d(x)&^d(y))&/Phi;

Restriction of coframe 1-forms
'e[1]&/Phi'=e[1]&/Phi;
'e[2]&/Phi'=e[2]&/Phi;

Who is who?
Who();

See Also:

atlas , atlas[Mapping] , atlas[Domain] , atlas[Metric] .