# Coordinate transformation and multiplying with size of J

1. Feb 14, 2008

### 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.

Best

2. Feb 14, 2008

### malawi_glenn

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

3. Feb 14, 2008

### greisen

Hi,

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

4. Feb 14, 2008

### HallsofIvy

Staff Emeritus
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.