I am working with the Gaussian integral at the moment, and all the proofs I have seen involve a step where it is said that $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dxdy=\int_{0}^{2\pi}\int_{0}^{\infty}re^{-r^2}drdϕ$. I can see why that $might$ be the case, but I don't really understand it. Could anyone explain how exactly one arrives at this result?
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1This is just integration by change of variables to polar coordinates. – Oct 29 '16 at 18:28
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I know that but what I don't understand is how one does the switch to polar coordinates – El Duderino Oct 29 '16 at 18:31
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Set \begin{align} & x=r \cos \theta \\ & y=r \sin \theta \\ \end{align} Then $$x^2+y^2=r^2$$ and $$dxdy=\left| \frac{\partial (x,y)}{\partial (r ,\theta )} \right|dr\,d\theta={{r}\, dr\,d\theta }$$
Behrouz Maleki
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