## How change from one to another coordinate system

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.

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 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: $$J=\left(\begin{array}{cc}\frac{\partial x}{\partial r}&\frac{\partial x}{\partial t}\\\frac{\partial y}{\partial r}&\frac{\partial y}{\partial t}\end{array}\right)$$
 Thanks for answer. Is parameter t in equation x = r cos(t) angle phi in the picture?

## How change from one to another coordinate system

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.

 Ok. Thank you