2.3 Cylindrical and Spherical Coordinates

Rectangular coordinates measure three straight distances. Cylindrical coordinates $(r, \theta, z)$ keep the vertical axis but describe the horizontal plane by polar coordinates; spherical coordinates $(\rho, \phi, \theta)$ go further and measure one distance and two angles: $\rho$ is distance from the origin, $\phi$ is the angle down from the positive $z$ -axis, and $\theta$ is the azimuth around that axis, the same angle cylindrical coordinates use. The conversion to rectangular is

$x = \rho \sin\phi \cos\theta, \qquad y = \rho \sin\phi \sin\theta, \qquad z = \rho \cos\phi.$

The character of a coordinate system lives in its coordinate surfaces: the sets you get by freezing one coordinate and letting the other two roam. For rectangular coordinates all three families are flat planes. For spherical coordinates they are far more interesting, and the figure below shows all three at once.

Three intersecting wireframe surfaces: a transparent sphere of radius 2.5 centered at the origin, an upward-opening 45-degree cone with its tip at the origin, and a vertical half-plane with a pink mesh rising from the z-axis at 60 degrees from the x-axis. A small bright curved patch on the sphere, edged by a yellow mesh, marks the outer face of a spherical box. A slider named f, starting at pi over 4, sets the cone's angle down from the z-axis; raising it to pi over 2 flattens the cone into the horizontal plane. A slider named c, starting at pi over 6, slides the patch's angle interval down toward the equator; the patch grows as it moves down, although its angular widths stay the same. — Three intersecting coordinate surfaces: the sphere $\rho = 2.5$ , the cone $\phi = \pi/4$ , and the half-plane $\theta = \pi/3$ , drawn with a pink mesh. A fourth object, the bright patch edged in yellow on the sphere, is one face of a spherical box. The slider $f$ sets the cone's angle $\phi$ and the slider $c$ sets where the patch's $\phi$ -interval starts, both in radians.

Explore

Match each surface to its frozen coordinate by its shape alone. How does the shape give each one away? Why is the $\theta$ surface only half a plane?
Among the mesh lines on the sphere are horizontal circles and pole-to-pole arcs (the slanted lines only split each cell into triangles). On each circle or arc, one of the other two coordinates is constant. Which family is constant $\phi$ , and which is constant $\theta$ ?
Predict what happens to the cone as $\phi$ grows from $\pi/4$ toward $\pi/2$ , then test it: the cone's components are $\sin f \cdot u\cos v$ , $\sin f \cdot u\sin v$ and $\cos f \cdot u$ , so drag the slider $f$ from $\pi/4 \approx 0.785$ up to $\pi/2 \approx 1.571$ . What surface does the cone flatten into at exactly $\phi = \pi/2$ ?
Find the bright patch: the outer face of the spherical box $1.5 \le \rho \le 2.5$ , $\pi/6 \le \phi \le \pi/3$ , $\pi/6 \le \theta \le \pi/3$ . Orbit close and examine its four edges. Are any of them straight?
Drag the slider $c$ from $\pi/6 \approx 0.524$ up to $\pi/3 \approx 1.047$ . That slides the patch's $\phi$ -interval $c \le \phi \le c + \pi/6$ down toward the equator, keeping its width the same. The face grows. The angular intervals did not change — so what did?

The box with almost no flat sides

A "box" in any coordinate system is a region where each coordinate runs between two constants, so its faces are patches of coordinate surfaces. In spherical coordinates that means: two sphere patches (inner and outer), two cone patches, and two half-plane patches. Only the half-plane faces are flat. Of the twelve edges, only the four radial ones are straight segments — all the others are arcs of circles. The spherical box is a bite taken out of an orange: curved skin, curved core, flat cuts only where the knife followed a meridian.

The exploration's last step is the one to remember. A face with the same spread of angles is small near the pole and large near the equator, because the meridians crowd together as $\sin\phi$ shrinks. A patch at radius $\rho$ spanning $\Delta\phi$ by $\Delta\theta$ has area approximately $\rho^2 \sin\phi \,\Delta\phi\, \Delta\theta$ — and that $\sin\phi$ is exactly the factor in the spherical volume element $dV = \rho^2 \sin\phi \, d\rho\, d\phi\, d\theta$ used for triple integrals in spherical coordinates. Its two-dimensional cousin, the polar $r$ in $dA = r\,dr\,d\theta$ , is the one you will count cell by cell in the final chapter. You have now seen the reason before ever meeting the formula.

p^{2} \cdot \sin\left(f\right)

The Jacobian determinant of the spherical map ✓ Computed · mojocas 0.1.0 ✓ Agrees with the text

The volume factor you just watched crowd the meridians, computed from the spherical conversion formulas themselves: a $3 \times 3$ determinant of partial derivatives, collapsed by the computer algebra system to exactly $\rho^2 \sin\phi$ . ( $p$ , $f$ , $t$ stand for $\rho$ , $\phi$ , $\theta$ .) The orange-bite picture and the algebra agree — and now that agreement is checked on every rebuild.

Example 1 Naming surfaces from equations

Identify each set in spherical coordinates: (a) $\rho = 4$ ; (b) $\phi = \pi/2$ ; (c) $\theta = 0$ ; (d) $\rho\cos\phi = 1$ .

Show solution

(a) All points at distance 4 from the origin: a sphere of radius 4. (b) The angle down from the $z$ -axis is a right angle: the entire $xy$ -plane — the degenerate cone the slider in the exploration flattened into. (c) The half-plane hinged on the $z$ -axis containing the positive $x$ -axis. (d) Since $z = \rho\cos\phi$ , this is the horizontal plane $z = 1$ — a plane that spherical coordinates describe awkwardly, which is the point: each coordinate system makes its own family of surfaces simple, and choosing coordinates is choosing which surfaces you want to be easy.

An original work of XYZ Homework, built around interactive XYZ 3D figures. Aligned to OpenStax Calculus Volume 3 (Strang & Herman), © OpenStax (Rice University), licensed CC BY-NC-SA 4.0; no OpenStax content is reproduced, and this work is not affiliated with or endorsed by OpenStax or Rice University. License: CC-BY-NC-SA-4.0.