2D Coordinate system changing

  Parabolic coordinate system on a plane

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Problem:

Find metric, connection and Laplace operator on a plane in parabolic coordinate system:
   x = (u^2-v^2)/2
   y = u*v

Solution:

Load atlas package:
restart:
with(atlas):

Plane

First of all we have to describe the space we are working in. The space is 2-dimensional Euclidean (flat) space i.e. a plane. To define the space we declare domain, forms, vectors, coframe, frame, flat metric and calculate connection (it equals to zero of cause).
Domain(R^2);

Forms(e[k]=1);

Vectors(E[j]);

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

Frame(E[k]);

Metric(g=d(x)&.d(x)+d(y)&.d(y));

Connection(omega);

Now the working space is defined completely and we can start to solve the problem.

Redefine `atlas/simp` procedure to simplify the results:
`atlas/simp`:=proc(a) factor(simplify(a)) end:

Parabolic

"Graph paper" of the coordinate system:
plots[coordplot](parabolic);

Define new domain:
Domain(P);

Declare 1-form for the domain coframe
Forms(phi[i]=1);

Declare vectors for the domain frame:
Vectors(Phi[k]);

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

Declare frame of the domain:
Frame(Phi[j]);

Declare mapping of the domain into :
Mapping(pi,P,R^2,
                 x = (u^2-v^2)/2,
                 y = u*v);

Now we can calculate metric induced on the domain by the mapping.
Metric(G = g &/ pi);

Calculate connection:
Connection(Gamma);

eval(Gamma);

Functions(h=h(u,v));

To calculate Laplace operator one can use grad and div operators.
Delta(h)=div(grad(h));