Mappings in Maple differential geometry package atlas

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Features List & Examples

atlasWizard - Maplet™

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 declares mapping between manifolds (or its domains). Ones mapping defined the user can calculate restriction 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 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);

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

Declare domain (sphere):
Domain(S^2);

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

Declare frame
Frame(U[j]);

Declare domain (plane):
Domain(R^2);

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

Frame declaration:
Frame(E[j]);

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

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));

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)));

Clarify "who is who".
Who(Phi);

Phi: mapping

TABLE([frame = {U[1] = Diff(x,theta)*E[1]+Diff(y,theta)*E[2], U[2] = Diff(x,phi)*E[1]+Diff(y,phi)*E[2]}, coframe = {e[1] = Diff(x,theta)*u[1]+Diff(x,phi)*u[2], e[2] = Diff(y,theta)*u[1]+Diff(y,phi)*u[2...

Who(phi);

phi: mapping

Who(psi);

psi: mapping

$TABLE([frame = {E[1] = -(1-x^2-y^2)^(1/2)/(x^2+y^2)^(3/2)/(-1+x^2+y^2)*(-y*U[2]*(1-x^2-y^2)^(1/2)*(x^2+y^2)^(1/2)+x^3*U[1]+x*U[1]*y^2), E[2] = -(1-x^2-y^2)^(1/2)/(x^2+y^2)^(3/2)/(-1+x^2+y^2)*(x*U[2]*(1...$
$TABLE([frame = {E[1] = -(1-x^2-y^2)^(1/2)/(x^2+y^2)^(3/2)/(-1+x^2+y^2)*(-y*U[2]*(1-x^2-y^2)^(1/2)*(x^2+y^2)^(1/2)+x^3*U[1]+x*U[1]*y^2), E[2] = -(1-x^2-y^2)^(1/2)/(x^2+y^2)^(3/2)/(-1+x^2+y^2)*(x*U[2]*(1...$
$TABLE([frame = {E[1] = -(1-x^2-y^2)^(1/2)/(x^2+y^2)^(3/2)/(-1+x^2+y^2)*(-y*U[2]*(1-x^2-y^2)^(1/2)*(x^2+y^2)^(1/2)+x^3*U[1]+x*U[1]*y^2), E[2] = -(1-x^2-y^2)^(1/2)/(x^2+y^2)^(3/2)/(-1+x^2+y^2)*(x*U[2]*(1...$
$TABLE([frame = {E[1] = -(1-x^2-y^2)^(1/2)/(x^2+y^2)^(3/2)/(-1+x^2+y^2)*(-y*U[2]*(1-x^2-y^2)^(1/2)*(x^2+y^2)^(1/2)+x^3*U[1]+x*U[1]*y^2), E[2] = -(1-x^2-y^2)^(1/2)/(x^2+y^2)^(3/2)/(-1+x^2+y^2)*(x*U[2]*(1...$
$TABLE([frame = {E[1] = -(1-x^2-y^2)^(1/2)/(x^2+y^2)^(3/2)/(-1+x^2+y^2)*(-y*U[2]*(1-x^2-y^2)^(1/2)*(x^2+y^2)^(1/2)+x^3*U[1]+x*U[1]*y^2), E[2] = -(1-x^2-y^2)^(1/2)/(x^2+y^2)^(3/2)/(-1+x^2+y^2)*(x*U[2]*(1...$
$TABLE([frame = {E[1] = -(1-x^2-y^2)^(1/2)/(x^2+y^2)^(3/2)/(-1+x^2+y^2)*(-y*U[2]*(1-x^2-y^2)^(1/2)*(x^2+y^2)^(1/2)+x^3*U[1]+x*U[1]*y^2), E[2] = -(1-x^2-y^2)^(1/2)/(x^2+y^2)^(3/2)/(-1+x^2+y^2)*(x*U[2]*(1...$
$TABLE([frame = {E[1] = -(1-x^2-y^2)^(1/2)/(x^2+y^2)^(3/2)/(-1+x^2+y^2)*(-y*U[2]*(1-x^2-y^2)^(1/2)*(x^2+y^2)^(1/2)+x^3*U[1]+x*U[1]*y^2), E[2] = -(1-x^2-y^2)^(1/2)/(x^2+y^2)^(3/2)/(-1+x^2+y^2)*(x*U[2]*(1...$

Who();

$PIECEWISE([{S^2, R^2}, Domains],[{phi, Phi, psi}, Mappings],[{X, Y, Z, E[i], U[j], V[k], e[j], u[k]}, Tensors],[{e[j], u[k]}, Forms],[{-I, Catalan, I, _Z, Pi}, Constants],[{}, Functions])$

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

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

`&/`(`&^`(d(phi),d(theta)),psi) = (1-x^2-y^2)^(1/2)/(x^2+y^2)^(1/2)/(-1+x^2+y^2)*`&^`(e[1],e[2])

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

`&/`(d(phi),psi) = -1/(x^2+y^2)*y*e[1]+e[2]*x/(x^2+y^2)

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

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