atlas[`&D`] - differential of a mapping

Calling Sequence:

     Expr &D MapId

Parameters:

      MapId - variable  - the mapping identifier  i.e. MapId : dom1 ---> dom2
      Expr - expression - a tensor expression

Description:

• The &D  procedure calculates mapping differential of a tensor field. The mapping differential is linear  defined operation on [k,0] tensors in a point 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).
• The differential of  F is mapping of corresponding tangent spaces : in a point.
• For any [1,0] tensor field T on M  is tensor field on N with components in local coordinates.
• For tensor product of any [k, 0] tensor fields on M the following formula takes place:
• The formulas considered above defined linear operator &D completely.

Examples:

The following example shows how the 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 [k,0] tensor field on  under the mapping:

For of frame vectors:
'U[1] &D F'=U[1] &D F;
'U[2] &D F'=U[2] &D F;

Restriction of tensor product d(x)&.d(z):
'(U[1]&.U[2]) &D F'=(U[1]&.U[2]) &D F;

For metric tensor:
dual(G);
(% &D F);

Some more examples

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

For abstract mapping :
'(U[1] &. U[2]) &D Phi'=(U[1] &. U[2]) &D Phi;

Who is who?
Who();

See Also:

atlas , atlas[Mapping] , atlas[Domain] , atlas[`&/`] .