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How can we change variables from $(x,y)$ to $(r,\theta)$ for the metric on the open disc $r<\delta$ defined by $(dx^2+dy^2)\over g(\sqrt{x^2+y^2})^2$ where $g(\sqrt{x^2+y^2})>0$ $\forall r<\delta$?

I am tempted to say the transformed metric is $dr^2\over g(r)^2$, but There might be some monkey business with Jacobians or such?

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    @anon: okay, between you and Mar$s$, I understand the question now. :-)2012-03-11

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To change from cartesion to polar coordinates, write $x = r\cos\theta$ $y = r\sin\theta$ and then calculate $dx$ and $dy$ in terms of $dr$ and $d\theta$ the way we usually do in calculus, i.e. $dx =dr\cos\theta - r\sin\theta d\theta$ Then you may calculate $dx^2 + dy^2 = dr^2 + r^2 d\theta^2$ and you then have your expression of the metric tensor in polar coordinates, i.e.$ \frac{1}{g(r)^2}(dr^2 + r^2d\theta^2)$