Wednesday, 11 January 2017

calculus - Limits and Jacobian for Gaussian Integrals



Ahoy everyone!
I am new to Gaussian Integrals and my teachers cannot help me out (because they don't get it). So I turn to the Internet for answers. I have very basic doubts and would really appreciate a clear explanation.
1) How do we explain $dxdy = r\cdot dr d\theta$? I am looking for either a simple and intuitive geometric interpretation and/or an algebraic proof which starts from the very basics.
2)While converting cartesian coordinates to polar, I believe the limits for '$r$' are the same as those of $x$ (and $y$) but how do we exactly decide the limits of theta?

I would appreciate any help.
Thanks.


Answer



The limits of theta ($\theta$) are $0$ and $\infty$, i.e. one full circle. And you should look at the following: https://www.math24.net/double-integrals-polar-coordinates/



I hope this helps you.


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