Coordinate transformation and multiplying with size of J

[greisen]

Hi,

I am using the book "Advanced Engineering Mathematics" by Erwin Kreyszig where I am reading on the transformation of coordinates - when changing from \int f(x,y) to \int f(v(x,y),v(x,y) it is necessary to multiply with the size of the jacobian, |J| - I cannot find the proof in the book and I don't quite understand why one should multiply with this?
Any help or advise where to locate the proof in order to better understand the multiplication with the Jacobian.

Thanks in advance

Best

 

[malawi_glenn]

if you google "Coordinate transformation multiple integrals", you get lots of hits :)

 

[greisen]

Hi,

So if I transform and the volume of the transform is preserved the size of |J| is one?

 

[HallsofIvy]

Yes. If you have a "parallelopiped" (3 d figure like a "tilted" retangle) formed from 3 vectors $\vec{u}$, $\vec{v}$, and $\vec{w}$, the volume is given by the length of the "triple" product $\vec{u}\cdot(\vec{v}\times\vec{w})$ which, in turn, can be calculated by the determinant having those vectors as rows. That's basically what the Jacobian is.