How change from one to another coordinate system


by soks
Tags: coordinate
soks
soks is offline
#1
Nov22-12, 04:26 PM
P: 5
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.

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meldraft
meldraft is offline
#2
Nov26-12, 11:25 AM
P: 280
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]
soks
soks is offline
#3
Dec1-12, 05:24 PM
P: 5
Thanks for answer.

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

meldraft
meldraft is offline
#4
Dec3-12, 04:02 AM
P: 280

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.

[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.
soks
soks is offline
#5
Dec3-12, 05:30 AM
P: 5
Ok. Thank you


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