atlas[Mapping] - declaration of a mapping between manifolds (domains) 

Calling Sequence: 

    Mapping(F, M, N)
    Mapping(F, M, N, y1=f1(x1,x2...xm), y2=f2(x1,x2...xm), ..., yn=fn(x1,x2...xm)) 

Parameters: 

    F - variable - the mapping identifier i.e. F : M ---> N
    M - variable - first domain identifier
   N  -  variable - second domain identifier 

Description: 

The Mapping procedure allows one to declare a mapping between manifolds (or its domains). Once a mapping is defined, the user can calculate the pullback of  

any [0,k] tensor field under the mapping (see atlas[`&/`]).  

The Mapping procedure can be used in two ways: 

• Mapping(F, M, N) - declares an abstract mapping between manifolds or domains such that:
  F: .

 

• Mapping(F, M, N, y1=f1(x1,x2...xm), y2=f2(x1,x2...xm), ..., yn=fn(x1,x2...xm)) - declares a mapping between manifolds such that: F: . The mapping defined by functions:
  
  
  
  where  ;   are local coordinates on M and   are local coordinates on N.

 

Examples: 

>	restart: with(atlas):

 

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

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

 

(2.1)

 

Declare vectors: 

>	Vectors(E[i],U[j],X,Y,Z,V[k]);

 

(2.2)

 

Declare domain (sphere):  

>	Domain(S^2);

 

(2.3)

 

Declare coframe on :  

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

 

(2.4)

 

Declare frame  

>	Frame(U[j]);

 

(2.5)

 

Declare domain (plane): 

>	Domain(R^2);

 

(2.6)

 

Declare coframe on :  

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

 

(2.7)

 

Frame declaration: 

>	Frame(E[j]);

 

(2.8)

 

Declare abstract mapping between the sphere and the plane: 

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

 

 

	(2.9)

 

Declare definite mapping between the sphere and the plane: 

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

 

 

	(2.10)

 

Declare another definite mapping between the sphere and the plane: 

>	Mapping(psi,R^2,S^2,        phi=arctan(y/x),        theta=arcsin(sqrt(x^2+y^2)));

 

 

	(2.11)

 

Clarify "who is who". 

>

Who(Phi);

 

 

Phi: mapping
	(2.12)

 

>

Who(phi);

 

 

phi: mapping
	(2.13)

 

>

Who(psi);

 

 

psi: mapping
	(2.14)

 

>

Who();

 

(2.15)

 

One can calculate pullback of any [0,k] tensor field under a mapping (see atlas[&/]): 

pullback of exterior product d(phi)&^d(theta) under mapping psi:
'(d(phi)&^d(theta))&/psi'=(d(phi)&^d(theta))&/psi; 

(2.16)

 

pullback of "rotation" 1-form under mapping psi:
'(d(phi))&/psi'=(d(phi))&/psi; 

(2.17)

 

pullback of "rotation" 1-form under mapping phi
'(y*d(x)-x*d(y))&/phi'=(y*d(x)-x*d(y))&/phi; 

(2.18)

 

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

(2.19)

 

>

 

See Also:  

atlas, atlas[Domain], atlas[`&/`], atlas[Projectors], atlas[Invariants], atlas[Who].