atlas[`&D`] - Pushforward (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 allows one to calculate the pushforward - mapping differential of a tensor field. The mapping differential is a linearly defined operation on [k,0] tensors at 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 the mapping of corresponding tangent spaces :  at 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 completely define the linear restriction operator &D.

 

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; 

(2.1)

 

Declare 1-forms e[j] and u[k] for corresponding coframes: 

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

 

(2.2)

 

Declare vectors for corresponding frames: 

>	Vectors(E[j],U[k]);

 

(2.3)

 

Declare Euclidean space - :  

>	Domain(R^3);

 

(2.4)

 

Declare coframe on :  

>	Coframe(e[1]=d(x),e[2]=d(y),e[3]=d(z));

 

(2.5)

 

Declare frame on :
Frame(E[j]); 

(2.6)

 

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

(2.7)

 

Declare sphere - : 

>	Domain(S^2);

 

(2.8)

 

Declare coframe on :  

>	Coframe(u[1]=d(theta),u[2]=d(phi));

 

(2.9)

 

Declare frame on :
Frame(U[j]); 

(2.10)

 

Declare definite mapping :: 

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

 

 

	(2.11)

 

>

Who(F);

 

 

F: mapping
	(2.12)

 

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

(2.13)

 

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

(2.14)

 

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; 

 

	(2.15)

 

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

(2.16)

 

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

 

	(2.17)

 

>

 

Some more examples 

Declare abstract mapping between and : 

>	Mapping(Phi,S^2,R^3);

 

 

	(2.18)

 

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

(2.19)

 

Who is who?
Who(); 

(2.20)

 

>

 

See Also:  

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