The Change of Variables

This chapter explains why you must multiply by a Jacobian determinant when you change coordinate systems. The answer is already present in the double integral — the area element $r\,dr\,d\theta$ in polar coordinates is obviously not the same as $dx\,dy$ — and this chapter generalises it.

The geometric meaning of the Jacobian determinant is the ratio by which the volume is scaled by the transformation, and this is exactly the meaning of a determinant in linear algebra; the two subjects come together here. The intuition behind the general formula is this: the factor by which an infinitesimal region is stretched or compressed under the transformation is the value of the Jacobian determinant at that point.

The chapter also gives a practical strategy for choosing a transformation: first look at which part of the integrand and of the region is “asymmetric”, and choose the transformation that removes that part. Most examination problems can be dispatched quickly with this line of thinking.