I understand the need for a factor to account for change of units of length/area/volume in multiple integration up to triple integration - & understand why the Jacobian is the appropriate factor, but cannot find any simple intuitive explanation or inductive proof that shows that the Jacobian gives the appropriate factor for change of variable in multiple integration in 4 variables or higher. Why is the volume of an element in the new variables J times the volume of the old? Would be grateful for an outline proof without any measure theory - happy to assume that the change of variables is 1:1 in the region of interest. David Kault, retired maths lecturer, JCU, Townsville, Australia

`$2\times2$`

determinant (multilinear, alternating,`$\det I=1$`

). Once the students get it, it is not a long stretch to extend that to three dimensions. And so it may not be altogether unreasonable to wave your arms about a bit and say it goes the same way in higher dimensions too. (But when it's time to get rigorous, look at the other comments.) – Harald Hanche-Olsen Feb 28 '10 at 13:45