Change of variable integral problem

[NewtonianAlch]

Homework Statement

http://img716.imageshack.us/img716/7453/28042782.jpg

The Attempt at a Solution

What I do not understand is how the Jacobian suddenly gets inverted when doing the integral, I have looked over my other tutorial problems for similar solutions and I do not recall doing something like this.

http://img140.imageshack.us/img140/7459/45340834.jpg
http://img220.imageshack.us/img220/7548/63185150.jpg

 

[vela]

The Jacobian calculated in the solution is
$$J = \begin{vmatrix}
\partial u/\partial x & \partial u/\partial y \\
\partial v/\partial x & \partial v/\partial y
\end{vmatrix}$$ so you have that $du\,dv = J\,dx\,dy$. Because you want to find $\int dx\,dy$, you need to integrate $\int (1/J)\,du\,dv$. If you used the inverse transformations and found instead
$$J' = \begin{vmatrix}
\partial x/\partial u & \partial x/\partial v \\
\partial y/\partial u & \partial y/\partial v
\end{vmatrix},$$ then you'd have $dx\,dy = J'\,du\,dv$.

 

[algebrat]

Yes, it looks like velas response is correct, but let me add a notation that might help you remember which way you are going. There is a visual believability about

$$dudv=|\frac{\partial(u,v)}{\partial(x,y)}|dxdy$$

and so also for

$$dxdy=|\frac{\partial(x,y)}{\partial(u,v)}|dudv$$

See? It sort of looks like things cancel correctly.

 

[SammyS]

Yes, it looks like velas response is correct, ...

vela has a history of giving correct responses!

 

[NewtonianAlch]

Thanks for the responses, yea I sort of see it now. I guess a bit more practice with it is required.