Coordinate system changing 

Parabolic coordinate system on a plane 

Problem: 

Find metric, connection and Laplace operator on a plane in parabolic coordinate systems:
  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 is equal to zero of course). 

>	Domain(R^2);

 

(2.1.1)

 

>	Forms(e[k]=1);

 

(2.1.2)

 

>	Vectors(E[j]);

 

(2.1.3)

 

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

 

(2.1.4)

 

>	Frame(E[k]);

 

(2.1.5)

 

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

 

(2.1.6)

 

>	Connection(omega);

 

(2.1.7)

 

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 

First of all we plot "graph paper" of the coordinate system: 

>	plots[coordplot](parabolic);

 

 

Define new domain:
Domain(P); 

(2.2.1)

 

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

(2.2.2)

 

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

(2.2.3)

 

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

(2.2.4)

 

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

(2.2.5)

 

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

 

	(2.2.6)

 

Now we can calculate metric induced on the domain by the mapping.  

>	Metric(G = g &/ pi);

 

(2.2.7)

 

Calculate connection:
Connection(Gamma); 

(2.2.8)

 

>	eval(Gamma);

 

(2.2.9)

 

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

 

(2.2.10)

 

To calculate Laplace operator one can use grad and div operators.   

>	Delta(h)=div(grad(h));

 

(2.2.11)

 

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