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]

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 :)

 

[Ray Vickson]

Thank you :)

Also widely available on-line; see, eg, http://tutorial.math.lamar.edu/Classes/CalcIII/ChangeOfVariables.aspx

 

[mathman]

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