Cylindrical and Spherical Coordinates

Definition Fix right-handed Cartesian axes. A point $P = (x, y, z)$ has cylindrical coordinates $(r, \theta, z)$ when

with $r \ge 0$. Here $(r, \theta)$ are polar coordinates of the projection $(x, y, 0)$ of $P$ onto the $xy$-plane, so $r = \sqrt{x^2 + y^2}$ is the distance from $P$ to the $z$-axis, and $z$ is the usual height.

Definition A point $P = (x, y, z)$ has spherical coordinates $(\rho, \theta, \phi)$ when

with $\rho \ge 0$ and $0 \le \phi \le \pi$. The number $\rho = \sqrt{x^2 + y^2 + z^2}$ is the distance from the origin to $P$, the angle $\phi$ is the angle between the positive $z$-axis and the segment from the origin to $P$, and $\theta$ is the same angle as in cylindrical coordinates. The two systems are linked by $r = \rho\sin\phi$ and $z = \rho\cos\phi$, where $\sin\phi \ge 0$ guarantees $r \ge 0$.

Definition The azimuth $\theta$ is measured in the $xy$-plane from the positive $x$-axis toward the positive $y$-axis, which is counterclockwise when you look down from the positive $z$-axis. The polar angle $\phi$ is measured from the positive $z$-axis, so $\phi = 0$ points straight up, $\phi = \frac{\pi}{2}$ lies in the $xy$-plane and $\phi = \pi$ points straight down. Angles are in radians. This course uses the letters $\theta$ for the azimuth and $\phi$ for the polar angle, and lists spherical coordinates in the order $(\rho, \theta, \phi)$. Many physics books swap the letters, so always check which angle a source means.

Definition The standard domains are

for cylindrical coordinates and

for spherical coordinates. Any half-open interval of length $2\pi$ works just as well for $\theta$, because it still covers each direction exactly once. Some books use $-\pi < \theta \le \pi$, and when you describe a particular curve or solid you may pick whichever such interval keeps the description simplest, as long as you say which one you use. The radial coordinates $r$ and $\rho$ are never negative here, and $\phi$ never leaves $[0, \pi]$, since $\phi$ outside that range would let $\sin\phi$ turn negative and break the link $r = \rho\sin\phi$.

Definition To go from Cartesian to cylindrical coordinates, use

valid when $r > 0$, and keep $z$. For spherical coordinates add

valid when $\rho > 0$, with the same $\theta$ as before, which again needs $r > 0$. The formula $\tan\theta = \frac{y}{x}$ alone does not fix $\theta$, since $\arctan\frac{y}{x}$ lies in $(-\frac{\pi}{2}, \frac{\pi}{2})$ and gives the right direction only when $x > 0$, and even then a negative value needs $2\pi$ added to land in $[0, 2\pi)$. Choose the angle in the quadrant given by the signs of $x$ and $y$, and when $x = 0$ take $\theta = \frac{\pi}{2}$ if $y > 0$ and $\theta = \frac{3\pi}{2}$ if $y < 0$. No such care is needed for $\phi$, because $\arccos$ already returns values in $[0, \pi]$.

Definition A coordinate surface is the set of points where one coordinate equals a constant. In the table $c > 0$, $k$ is any real number, $0 \le \beta < 2\pi$, and $0 < \alpha < \pi$ with $\alpha \ne \frac{\pi}{2}$.

The equation $\phi = \frac{\pi}{2}$ gives the $xy$-plane, and $\phi = 0$ and $\phi = \pi$ give the nonnegative and the nonpositive $z$-axis. The equation $r = 0$ gives the $z$-axis and $\rho = 0$ gives the origin. A half-plane $\theta = \beta$ includes its edge, the $z$-axis, since points with $r = 0$ may take any $\theta$.

Definition Even within the standard domains, some points have more than one coordinate triple. In cylindrical coordinates every point of the $z$-axis has $r = 0$, and $\theta$ can be anything. In spherical coordinates the origin has $\rho = 0$ with $\theta$ and $\phi$ arbitrary, and every other point of the $z$-axis has $\phi = 0$ or $\phi = \pi$ with $\theta$ arbitrary. Off the $z$-axis, the coordinates in the standard domains are unique. If the domain of $\theta$ is allowed to be a closed interval such as $[0, 2\pi]$, then $\theta = 0$ and $\theta = 2\pi$ also name the same half-plane.