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How change from one to another coordinate system

  1. Nov 22, 2012 #1
    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 [Broken]
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Nov 26, 2012 #2
    Given a parametrization for your new coordinates (e.g. x=r cos(t), y=r sin(t), the new coordinates are:

    [tex]\left(\begin{array}{cc}x'\\y'\end{array}\right)=det(Jacobian(x,y)) \left(\begin{array}{cc}x\\y\end{array}\right)[/tex]

    Where the Jacobian is merely:

    [tex]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)[/tex]
     
  4. Dec 1, 2012 #3
    Thanks for answer.

    Is parameter t in equation x = r cos(t) angle phi in the picture?
     
  5. Dec 3, 2012 #4
    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.

    [tex]r=\sqrt(x^2+y^2) , t=atan(y/x)[/tex]

    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.
     
  6. Dec 3, 2012 #5
    Ok. Thank you
     
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