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I have a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given in polar coordinates by $f(r,\theta) = (r,\theta + k)$ and i want to compute if it is volume-preserving (i.e the determinant of the Jacobian is $+1$ or $-1$).

Can I just compute the partial derivatives of this function as if it were given in euclidean coordinates? Or do I have to compose it with coordinate transformations and then compute the determinant of the Jacobian of the composed function?

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up vote 1 down vote accepted

The transformation is simply a rotation by angle $k$ about the origin. A rotation is an isometry, so preserves area.

More precisely, this is the transformation given by $$\left[\begin{array}{c}x\\y\end{array}\right]\mapsto\left[\begin{array}{cc}\cos k & \sin k\\- \sin k & \cos k\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right].$$

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The answer to your question is "Yes." When you compute the Jacobian matrix, you just have the identity matrix, which has determinant 1.

"Preserving volume" is a property of the transformation, and not of any coordinate system you write it in, so you should expect the same method to work across admissible coordinate systems.

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Thanks a lot! That's what i was somehow expecting/hoping. – eddard Oct 25 '12 at 21:45

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