Inverse-square field

The inverse-square field is

$\mathbf{F}(\mathbf{r})=\frac{\mathbf{r}}{|\mathbf{r}{|}^3}=\frac{\hat{\mathbf{r}}}{r^2},$

defined on ${\mathbb{R}}^3\setminus \{0\}$. Magnitude falls as $1/r^2$; direction is radial.

This is the geometric prototype: a unit “source” at the origin producing a purely radial $1/r^2$ field. In SI units, a point charge $q$ produces the electric field $\mathbf{E}=(q/(4\pi {ϵ}_0))\hat{\mathbf{r}}/r^2$ — same shape as $\mathbf{F}$, with an overall factor of $q/(4\pi {ϵ}_0)$. A point mass $m$ produces a gravitational field $\mathbf{g}=-(Gm)\hat{\mathbf{r}}/r^2$ — same shape, factor of $-Gm$. Throughout this note we work with the bare geometric field $\hat{\mathbf{r}}/r^2$ (no charge or mass factor) so the flux comes out to $4\pi$ rather than $q/{ϵ}_0$ or $-4\pi Gm$. The physics is just the geometric result scaled by the source strength and the unit-system constant.

Two remarkable properties make this field the most-studied example in vector calculus.

Property 1: divergence is zero away from origin

$\nabla \cdot \mathbf{F}=0\text{for }\mathbf{r}\ne 0.$

Direct calculation: ${\partial }_x[x(x^2+y^2+z^2{)}^{-3/2}]=(x^2+y^2+z^2{)}^{-3/2}-3x^2(x^2+y^2+z^2{)}^{-5/2}=(r^2-3x^2)/r^5$. Summing the three analogous terms: $(3r^2-3(x^2+y^2+z^2))/r^5=0$.

So everywhere except at the origin, the field is source-free. The “source” lives entirely at $\mathbf{r}=0$, where the field blows up.

Property 2: flux out of any closed surface enclosing the origin is $4\pi$

Sphere of radius $a$. On $|\mathbf{r}|=a$, $\mathbf{F}=\hat{\mathbf{r}}/a^2$. The outward normal is $\hat{\mathbf{r}}$. Dot product: $1/a^2$. Surface area: $4\pi a^2$. Flux: $4\pi$ — independent of $a$.

Any other closed surface $\Sigma$ enclosing the origin. Between $\Sigma$ and a tiny sphere $\sigma$ around the origin, the field has zero divergence. By the Divergence theorem applied to the region between $\Sigma$ and $\sigma$ (with appropriate orientations),

$\iint \mathbf{F}\cdot d\mathbf{S}=\iint \mathbf{F}\cdot d\mathbf{S}=4\pi .$

So every closed surface enclosing the origin gives flux $4\pi$. Any closed surface not enclosing the origin gives flux $0$ (the divergence theorem applies to the interior, where divergence is zero).

Gauss’s law

This is the geometric content of Gauss’s law in electrostatics: the total electric flux through a closed surface is proportional to the enclosed charge, independent of the shape and size of the surface. A point charge contributes a fixed amount ($q/{ϵ}_0$) regardless of how you draw the surface around it.

The non-trivial step is the independence from the shape — that’s what the divergence theorem (and the inverse-square law) deliver.

The singularity as a “source”

Even though $\nabla \cdot \mathbf{F}=0$ everywhere $\mathbf{F}$ is defined (away from origin), there’s a “source” at the origin in a distributional sense:

$\nabla \cdot \mathbf{F}=4\pi {\delta }^3(\mathbf{r}),$

where ${\delta }^3$ is the 3D Dirac delta. Integrating both sides over a volume containing the origin gives $4\pi$ on each side, matching the flux result. The flux integral $\iint \mathbf{F}\cdot d\mathbf{S}=4\pi$ is detecting the delta-function source.

The 2D analog and the bridge to complex analysis

A direct 2D analog: the field $\mathbf{F}=\mathbf{r}/|\mathbf{r}{|}^2=\hat{\mathbf{r}}/r$ has zero divergence away from origin, and flux $2\pi$ around any closed curve enclosing the origin. The 2D rotational field $\mathbf{F}=⟨-y,x⟩/(x^2+y^2)$ has zero curl away from origin, and circulation $2\pi$ around any closed loop enclosing the origin. Both are “source / circulation localized at a singular point.”

This is the prototype of what becomes the Residue theorem in complex analysis. The function $1/z$ has a singularity at $z=0$; contour integrals enclosing the singularity give $2\pi i$, independent of the contour. The residue theorem packages the “singularity contributes a fixed quantity” idea systematically.

Vector Calculus and Complex Analysis explicitly frames the residue theorem as “the divergence theorem for closed contours encircling singular points” (Part III, wrap-up). The structural unity is real.

Connection to other contexts

• Coulomb’s law: force between two point charges falls as $1/r^2$. Comes from the inverse-square electric field of a point charge.
• Newtonian gravity: $\mathbf{F}=-GMm\hat{\mathbf{r}}/r^2$ for a point mass. Same field shape.
• Inverse-square law for light intensity: photon flux through a sphere is $L/(4\pi r^2)$ — same geometry, different physical quantity.

The recurring $4\pi$ is the surface area of the unit sphere — geometry, not physics.