For a change of variables one needs the mapping to be injective.
In the book I'm reading, we restrict the mapping of polar coordinates $g(r,\theta)$ to the domain $r>0$ and $0<\theta<2\pi$. However, why can't we also use $0\leq\theta<2\pi$ or $0<\theta\leq2\pi$ to define the domain?
The same situation happens when we're talking about cylindrical, and spherical coordinates. In these cases, can we extend the domain in a similar manner?
Any help would be appreciated.
Best Answer
$\newcommand{\Reals}{\mathbf{R}}$Briefly, domain restrictions depend on what one is trying to do. For example:
A "change of coordinates" must, by customary definition, be a diffeomorphism, a smooth bijection with smooth inverse. Allowing the radius $r$ or $\rho$ to be zero spoils bijectivity, as noted in the comments.
Allowing the angle $\theta$ to range over a half-open interval of length $2\pi$ gives a bijective map whose inverse is discontinuous at each point on the ray/half plane hit by the endpoint value of $\theta$. For example, the inverse of the bijection $\theta \in [0, 2\pi) \mapsto (\cos\theta, \sin\theta)$ is discontinuous at $(1, 0)$, the image of the endpoint $0$.
In complex analysis, a holomorphic function, such as the $n$th root or logarithm, $$ z = re^{i\theta} \mapsto r^{1/n} e^{i\theta/n},\qquad z = re^{i\theta} \mapsto \log(r) + i\theta, $$ is only guaranteed to be analytic on a region of the $z$-plane where $z \mapsto (r, \theta)$ is smooth. In fact, analytic branches of these functions can be defined on a slit plane (the plane with a ray removed from the origin to $\infty$, on which a "branch of angle" is smoothly defined), but do not exist on a punctured plane (where $r \neq 0$), even though the origin is the only "intrinsic trouble spot" for these functions.
For purposes of parametrizing a region of integration, bijectivity can be relaxed on the boundary. For example, the disk of radius $R > 0$ centered at the origin in the plane can be parametrized by polar coordinates taking $0 \leq r \leq R$ and $0 \leq \theta \leq 2\pi$.
The polar coordinates map sends the edge $R = 0$ to the origin, and identifies the edges $\theta = 0$ and $\theta = 2\pi$. For integrating, this is harmless because these edges have area zero, and their images have area zero, so each contributes nothing to a two-dimensional integral. (Similar comments hold for spherical coordinates and three-dimensional integrals.)
Though the polar coordinates mapping is smooth on all of $\Reals^{2}$, and the cylindrical and spherical coordinates mappings are smooth on $\Reals^{3}$, one seldom wants to consider domains in which $\theta$ (for a fixed $r$) ranges over an interval of length $2\pi$, or where $r$ (over an interval of $\theta$) ranges over an open interval containing $0$: The corresponding mapping fails to be bijective on some open set, making it neither a diffeomorphism nor a suitable region for integrating over its image.