Spherical coordinates

Spherical coordinates specify a point in 3D space by $(r,\theta ,\varphi )$, using the engineering/physics convention:

Image: Spherical coordinate system, CC BY-SA 2.0 IT

• $r\ge 0$: distance from the origin,
• $\theta$: polar angle from the $+z$-axis ($0\le \theta \le \pi$),
• $\varphi$: azimuthal angle in the $xy$-plane from the $+x$-axis ($0\le \varphi <2\pi$).

Spherical coordinates are the natural choice when a problem has rotational symmetry about a point (like a point charge or a sphere).

A notation heads-up

The labels $\theta$ and $\varphi$ get swapped in some sources — mathematicians often use $\varphi$ for polar and $\theta$ for azimuthal. Vector Calculus and Complex Analysis (following Ulaby) uses $\theta$ polar, $\varphi$ azimuthal for most of the material, but a few sphere-parameterization figures borrow the mathematicians’ convention. Always check the labels on a given figure before substituting into a formula; the underlying geometry is the same.

Conversion to Cartesian

$x=r\sin \theta \cos \varphi ,y=r\sin \theta \sin \varphi ,z=r\cos \theta .$

Visualize: $r$ sets the radius; $\theta$ sets how far from the north pole ($z$-axis); $\varphi$ rotates around the $z$-axis.

Going the other way: $r=\sqrt{x^2+y^2+z^2}$, $\theta =\arccos (z/r)$, $\varphi =atan2(y,x)$.

Unit vectors

$\hat{\mathbf{r}},\hat{\mathbf{\theta }},\hat{\mathbf{\varphi }},$

with

• $\hat{\mathbf{r}}$ radially outward from the origin,
• $\hat{\mathbf{\theta }}$ tangent to a meridian, in the direction of increasing $\theta$ (toward the south pole),
• $\hat{\mathbf{\varphi }}$ tangent to a parallel, eastward (same $\hat{\mathbf{\varphi }}$ as in Cylindrical coordinates).

All three are position-dependent. The mutual orthogonality holds at each point, but the absolute directions change as you move.

Differential elements

Differential length:

$d\mathbf{l}=dr\hat{\mathbf{r}}+rd\theta \hat{\mathbf{\theta }}+r\sin \theta d\varphi \hat{\mathbf{\varphi }}.$

The $r$ factor on $d\theta$ accounts for arc length on a meridian; the $r\sin \theta$ on $d\varphi$ accounts for the smaller circumference of parallels away from the equator.

Differential volume:

$dv=r^2\sin \theta drd\theta d\varphi .$

The Jacobian $r^2\sin \theta$ is the volume element factor.

Surface element on a sphere of radius $r_0$:

$d\mathbf{s}=r_0^2\sin \theta d\theta d\varphi \hat{\mathbf{r}}.$

Outward normal is $\hat{\mathbf{r}}$. Total surface area: $\iint r_0^2\sin \theta d\theta d\varphi =r_0^2\cdot 2\pi \cdot 2=4\pi r_0^2$.

When spherical is right

• A point charge or point mass (Coulomb, Newtonian gravity).
• A spherical conductor, ball, dielectric sphere.
• Anywhere the symmetry is around an origin point.

Classic example: the electric field of a point charge is $\mathbf{E}=(q/(4\pi {ϵ}_0{r}^2))\hat{\mathbf{r}}$. Pure radial. The divergence theorem applied to a sphere of radius $R$ gives the enclosed flux $q/{ϵ}_0$ in a few lines, recovering Gauss’s law. See Inverse-square field.

Distance between two points

For two points $P_1=(r_1,{\theta }_1,{\varphi }_1)$ and $P_2=(r_2,{\theta }_2,{\varphi }_2)$, the Euclidean distance is

$d=\sqrt{r_1^2+r_2^2-2r_1{r}_2[\sin {\theta }_1\sin {\theta }_2\cos ({\varphi }_2-{\varphi }_1)+\cos {\theta }_1\cos {\theta }_2]}.$

The bracketed expression is the dot product ${\hat{\mathbf{r}}}_1\cdot {\hat{\mathbf{r}}}_2$ — the cosine of the angle between the two radial directions, computed via the spherical angular addition formula. So equivalently $d^2=r_1^2+r_2^2-2r_1{r}_2\cos \Theta$ where $\Theta$ is the central angle between $P_1$ and $P_2$ as seen from the origin. This is the spherical law of cosines.

Special cases:

• $r_1=r_2=r$, ${\theta }_1={\theta }_2=\theta$, same parallel: substitute into the main formula and use $1-\cos x=2{\sin }^2(x/2)$ to get $d=2r\sin \theta |\sin (({\varphi }_2-{\varphi }_1)/2)|$. Geometrically: it’s a chord across a circle of latitude whose radius is $r\sin \theta$ (the parallel shrinks toward the poles). Along the equator $\theta =\pi /2$ this collapses to $d=2r|\sin (({\varphi }_2-{\varphi }_1)/2)|$, the chord across the equatorial circle.
• ${\varphi }_1={\varphi }_2$, same meridian: $d=\sqrt{r_1^2+r_2^2-2r_1{r}_2\cos ({\theta }_2-{\theta }_1)}$ — planar 2D distance.
• Origin to a point: $d=r$ (set $r_1=0$).

This formula appears in source/field integrations whenever the source charge or current distribution and the observation point are best located in spherical coordinates — e.g., radiation pattern integrals over a spherical antenna aperture.

Bridge

• $\theta =\pi /2$ (equator): reduces to cylindrical with $r\leftrightarrow \rho$.
• Cylindrical and spherical share the azimuthal angle $\varphi$.
• Cartesian: $z=r\cos \theta$, $\rho =r\sin \theta$ (where $\rho$ is the cylindrical radial coordinate).