# I Gaussian Integral Coordinate Change

#### looseleaf

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

Homework Helper
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
$$\frac{\partial(x,y)}{\partial(u,v)} \equiv \left| \begin{array}{cc} \partial x/ \partial u & \partial x /\partial v\\ \partial y /\partial u & \partial y / \partial v \end{array} \right|$$

#### looseleaf

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

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

"Gaussian Integral Coordinate Change"

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