change of variables

Substituting new coordinates into a double or triple integral, trading a hard region or integrand for an easy one at the cost of the Jacobian.

The change of variables formula substitutes $x=x(u,v),\ y=y(u,v)$ (or the three-variable analogue) into a double or triple integral, at the cost of exactly one extra factor: the absolute value of the Jacobian. It is the multivariable analogue of single-variable $u$ -substitution, with the Jacobian playing the role $g'(u)$ plays there. The practical use case is almost always a badly-shaped region — a tilted parallelogram, or a region bounded by hyperbolas — becoming a plain rectangle or box after the right substitution. The one-to-one hypothesis matters concretely: a substitution that folds two different points onto the same image point would double-count that region's contribution.

Examples

With $u=x-y,\ v=x+y$ (Jacobian $\tfrac12$ ), the tilted region bounded by $x-y=0,2$ and $x+y=0,3$ maps to the rectangle $0\le u\le2$ , $0\le v\le3$ , giving area $\tfrac12(2)(3)=3$ .
With $x=2u,\ y=3v,\ z=4w$ (Jacobian $24$ ), the ellipsoid $\tfrac{x^2}{4}+\tfrac{y^2}{9}+\tfrac{z^2}{16}\le1$ maps to the unit ball, giving volume $24\cdot\tfrac43\pi=32\pi$ .