[THEOREMS] Change of variables and polar coordinates

Change of variables

Let $\Omega$ be a region in $(x,y)$ -coordinates and let $\Phi$ be a region in $(u,v)$ -coordinates. Assume that $f$ is integrable on $\Omega$ , that $g$ is $C^1$ on $\Phi$ , that $\Phi$ can be split into finitely many pieces on which $g$ is one-to-one with non-overlapping interiors and that the Jacobian is nonzero away from boundary or other exceptional sets of area zero. The images of those pieces must cover $\Omega$ once, apart from boundaries of area zero; otherwise the transformed integral can omit or count part of the region more than once. If a map covers the target with multiplicity, split the parameter region into one-to-one pieces (or account for that multiplicity) before applying the formula. Then

\iint_\Omega f(x,y)\,dx\,dy = \iint_\Phi f\bigl(x(u,v),y(u,v)\bigr) \left|\det J_g(u,v)\right|\,du\,dv.

Applying the theorem involves two separate tasks: rewriting the integrand in terms of $u,v$ and describing the old region in the new coordinates. The second task is geometric and cannot be skipped.

[1]:Jacobian determinant
[2]:Euler's Basel sum
[3]:Choose and apply a coordinate change
[4]:Polar area element

Polar area element

For $r\ge0$ , the polar-coordinate change of variables is

x=r\cos\theta,\qquad y=r\sin\theta,

and its Jacobian satisfies

\left|\frac{\partial(x,y)}{\partial(r,\theta)}\right| =r.

Consequently,

dA=r\,dr\,d\theta,

and

\iint_\Omega f(x,y)\,dA =\iint_\Phi f(r\cos\theta,r\sin\theta)\,r\,dr\,d\theta.

The factor $r$ is also visible geometrically. A small polar rectangle has radial side approximately $dr$ and angular side approximately $r\,d\theta$ , so its area is approximately $r\,dr\,d\theta$ . The same $(r,\theta)$ pair parametrises complex numbers in polar form, with $x+iy=re^{i\theta}$ .

A usual polar description takes $r\ge0$ and an angular interval of length at most $2\pi$ . Each point then has one representation, except on a boundary ray and at the origin. Those exceptional sets have area zero, so the change-of-variables theorem still applies.

The radial limits are read along each ray from the boundary of the region; they need not be constant.

Theorems from Change of variables and polar coordinates

Change of variables

Polar area element

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