Math Help - Change to polar coordinates

1. Change to polar coordinates

$\displaystyle{ \hdots=\int_0^\infty\int_0^\infty e^{-(u^2+v^2)}du dv }$

We write $u=r\cos(\theta)$ and $v=r\sin(\theta)$ which simplifies $\displaystyle{ e^{-(u^2+v^2)} }$ into $\displaystyle{ e^{-r^2} }$, but why is differential area now $\mathbf{r} dr d\theta$?

$\frac{du}{dr}=\cos(\theta)$ and $\frac{dv}{d\theta}=r\cos(\theta) \Rightarrow dudv = r dr d\theta\cos^2(\theta)$

Also on a sidenote, if I rewrite $r$ as $r=\frac{u}{\cos(\theta)}=\frac{v}{\sin(\theta)}\Ri ghtarrow v =u\tan(\theta) \Rightarrow \theta=\tan^{-1}(\frac{v}{u})\stackrel{u,v\to\infty}{=}\tan^{-1}(\frac{\infty}{\infty})\stackrel{?}{=}\frac{\pi} {2}$.
(I know that $\tan^{-1}(\infty)=\frac{\pi}{2}$.)

2. Originally Posted by courteous
$\displaystyle{ \hdots=\int_0^\infty\int_0^\infty e^{-(u^2+v^2)}du dv }$

We write $u=r\cos(\theta)$ and $v=r\sin(\theta)$ which simplifies $\displaystyle{ e^{-(u^2+v^2)} }$ into $\displaystyle{ e^{-r^2} }$, but why is differential area now $\mathbf{r} dr d\theta$?

$\frac{du}{dr}=\cos(\theta)$ and $\frac{dv}{d\theta}=r\cos(\theta) \Rightarrow dudv = r dr d\theta\cos^2(\theta)$

Also on a sidenote, if I rewrite $r$ as $r=\frac{u}{\cos(\theta)}=\frac{v}{\sin(\theta)}\Ri ghtarrow v =u\tan(\theta) \Rightarrow \theta=\tan^{-1}(\frac{v}{u})\stackrel{u,v\to\infty}{=}\tan^{-1}(\frac{\infty}{\infty})\stackrel{?}{=}\frac{\pi} {2}$.
(I know that $\tan^{-1}(\infty)=\frac{\pi}{2}$.)
Here is why
Polar coordinate system - Wikipedia, the free encyclopedia

You can also draw a picture of a polar rectangle and calculate is area directly this is a very informal justification

Finally you need to calculate the new limits of integration in Polar coordinates.

Sketch the first quadrant and now decided what values $r$ and $\theta$ need to vary between to sweep out the first quadrant.

I hope this helps

3. I am not questioning the correctness of $rdrd\theta$. So, you have to consider the Jacobian determinant (of the coordinate conversion formula)? And if you just differentiate as usual (the way I did it) it is just wrong?

4. Originally Posted by courteous
I am not questioning the correctness of $rdrd\theta$. So, you have to consider the Jacobian determinant (of the coordinate conversion formula)? And if you just differentiate as usual (the way I did it) it is just wrong?
You don't differentiate it, you integrate it.

5. I meant differentiate $\frac{du}{dr}$ and $\frac{dv}{d\theta}$. What is wrong with my algebraic "reasoning" above?

6. Originally Posted by courteous
I meant differentiate $\frac{du}{dr}$ and $\frac{dv}{d\theta}$. What is wrong with my algebraic "reasoning" above?