atlas[`&/`] - Pullback 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 allows one to calculate pullback of a covariant tensor field under a mapping. The pullback 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 pullback 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 completely define the linear pullback operator &/.

 

• According to the definition it is necessary to calculate the pullbacks on the domain M. Use atlas[Domain] procedure to jump on M manifold if needed.

 

Examples: 

The following example shows how the pullback 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 pullback operator &/:
Metric(G = g &/ F); 

(2.14)

 

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

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

 

	(2.15)

 

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

 

	(2.16)

 

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

(2.17)

 

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

(2.18)

 

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

(2.19)

 

Some more examples 

Declare abstract mapping between and : 

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

 

 

	(2.20)

 

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

(2.21)

 

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

 

	(2.22)

 

Who is who?
Who(); 

(2.23)

 

>

 

See Also:  

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