How change from one to another coordinate system

[soks]

Hello!

I have a problem. How can I convert a left part from picture which is in coordinate system
(r, s) to coordinate system (x, y) and then to coordinate system (ζ, η) (right part). I need Jacobian matrix because of integration some function above this region.

Any helpful links or answers are welcome.

http://img442.imageshack.us/img442/3074/20121122230303.jpg

 

[meldraft]

Given a parametrization for your new coordinates (e.g. x=r cos(t), y=r sin(t), the new coordinates are:

$$\left(\begin{array}{cc}x'\\y'\end{array}\right)=det(Jacobian(x,y)) \left(\begin{array}{cc}x\\y\end{array}\right)$$

Where the Jacobian is merely:

$$J=\left(\begin{array}{cc}\frac{\partial x}{\partial r}&\frac{\partial x}{\partial t}\\\frac{\partial y}{\partial r}&\frac{\partial y}{\partial t}\end{array}\right)$$

 

[soks]

Thanks for answer.

Is parameter t in equation x = r cos(t) angle phi in the picture?

 

[meldraft]

Yes, but this was the general formula. In your case, you are going from r,phi to x,y so you should take the inverse equations, i.e.

$$r=\sqrt(x^2+y^2) , t=atan(y/x)$$

You would then have r,t on the left hand side of the equation and the new Jacobian times x,y on the other one.

 

[soks]

Ok. Thank you