I Gaussian Integral Coordinate Change

  • Thread starter looseleaf
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Hi everyone, sorry for the basic question. But I was just wondering how one does the explicit coordinate change from dxdy to dr in the polar-coordinates method for solving the gaussian. I can appreciate that using the polar element and integrating from 0 to inf covers the same area, but how do we do this in a rigorous way? I know it's a basic multivariable calculus question, but I couldn't find what I was looking for by googling.

Thanks!
 

Ray Vickson

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Hi everyone, sorry for the basic question. But I was just wondering how one does the explicit coordinate change from dxdy to dr in S3:=sum(1/n^3,n=2..infinity);evalf(S3);the polar-coordinates method for solving the gaussian. I can appreciate that using the polar element and integrating from 0 to inf covers the same area, but how do we do this in a rigorous way? I know it's a basic multivariable calculus question, but I couldn't find what I was looking for by googling.

Thanks!
Standard textbook result:
$$ dx \, dy = \frac{\partial(x,y)}{\partial(u,v)} \: du \, dv,$$
Here
$$ \frac{\partial(x,y)}{\partial(u,v)} \equiv \left| \begin{array}{cc} \partial x/ \partial u & \partial x /\partial v\\
\partial y /\partial u & \partial y / \partial v \end{array} \right| $$
 
Standard textbook result:
$$ dx \, dy = \frac{\partial(x,y)}{\partial(u,v)} \: du \, dv,$$
Here
$$ \frac{\partial(x,y)}{\partial(u,v)} \equiv \left| \begin{array}{cc} \partial x/ \partial u & \partial x /\partial v\\
\partial y /\partial u & \partial y / \partial v \end{array} \right| $$
Thank you :)
 

mathman

Science Advisor
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##dxdy=rdrd\theta##. You need to describe the integration limits on ##x,y##.
 

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