Divergence

Prerequisites: Vector Calculus. Full path (3 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

Divergence (div) is “flux density”—the amount of flux entering or leaving a point. Think of it as the rate of flux expansion (positive divergence) or flux contraction (negative divergence). If you measure flux in bananas (and c’mon, who doesn’t?), a positive divergence means your location is a source of bananas. You’ve hit the Donkey Kong jackpot.

Vector Calculus: Understanding Divergence by Kalid Azad

Concrete

For a vector field $\vec F = (F_x, F_y, F_z)$ in Cartesian coordinates, the divergence is the scalar field

$$\nabla\cdot\vec F = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} .$$

The coordinate-free definition is the outward flux per unit volume through the surface $\partial V$ of a small region $V$ that shrinks to the point $p$:

$$(\nabla\cdot\vec F)(p) = \lim_{V \to p} \frac{1}{|V|} \oint_{\partial V} \vec F\cdot d\vec A .$$

For a small box with sides $dx, dy, dz$, the flux through the two faces perpendicular to $x$ is $\left[F_x(x+dx) - F_x(x)\right] dy\, dz \approx \frac{\partial F_x}{\partial x}\, dx\, dy\, dz$. Adding the three pairs of faces and dividing by the volume gives the Cartesian formula.

Divergence theorem

Adding up the flux of many small boxes, the contributions of interior faces cancel and only the outer surface remains:

$$\int_V \nabla\cdot\vec F \; dV = \oint_{\partial V} \vec F\cdot d\vec A .$$

Examples

• $\vec F = (x, y, z)$ has $\nabla\cdot\vec F = 3$: the field points away from the origin everywhere and grows with distance.
• $\vec F = (-y, x, 0)$ has $\nabla\cdot\vec F = 0$: a pure rotation neither creates nor destroys flow.
• The Coulomb field $\vec E = \frac{q}{4\pi\varepsilon_0}\frac{\hat r}{r^2}$ has $\nabla\cdot\vec E = 0$ for $r \neq 0$, while its flux through any sphere around the origin is $q/\varepsilon_0$. All the divergence sits at the origin: $\nabla\cdot\vec E = \frac{q}{\varepsilon_0}\delta^3(\vec x)$.

In physics

• Gauss's law: $\nabla\cdot\vec E = \rho/\varepsilon_0$, and $\nabla\cdot\vec B = 0$ (no magnetic charges). See the Maxwell equations.
• Conservation laws: a density $\rho$ with current density $\vec j$ obeys the continuity equation $\partial_t \rho + \nabla\cdot\vec j = 0$. Integrated over a volume and combined with the divergence theorem, it says that the amount inside changes only by what flows through the surface.
• Incompressible flow: the velocity field of an incompressible fluid satisfies $\nabla\cdot\vec v = 0$.

Curvilinear coordinates

In spherical coordinates $(r, \theta, \varphi)$ with orthonormal components,

$$\nabla\cdot\vec F = \frac{1}{r^2}\frac{\partial}{\partial r}\left( r^2 F_r \right) + \frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}\left( \sin\theta\, F_\theta \right) + \frac{1}{r\sin\theta}\frac{\partial F_\varphi}{\partial\varphi} .$$

The extra factors account for the change of the area of the faces of a volume element with position. For $F_r = 1/r^2$ the first term vanishes, which is the Coulomb result above.

• Vector Calculus: Understanding Divergence by Kalid Azad
• Another great discussion of the divergence in vector calculus can be found in Section 3 of Vol. 2 of Feynman's Lectures on Physics, which is available here
• No-Nonsense Electrodynamics by Jakob Schwichtenberg, section "Divergence"

Abstract

Defining the divergence requires only a volume form; no metric is needed. Let $M$ be an oriented $n$-dimensional manifold with volume form $\mu$ and $X$ a vector field. The divergence of $X$ is the function defined by

$$\mathcal{L}_X \mu = (\operatorname{div}_\mu X)\, \mu .$$

By Cartan's formula $\mathcal{L}_X = d\,\iota_X + \iota_X d$ and $d\mu = 0$, this is $d(\iota_X \mu) = (\operatorname{div}_\mu X)\, \mu$. The Lie derivative describes how the flow $\varphi_t$ of $X$ changes volumes, so

$$\frac{d}{dt}\int_{\varphi_t(U)} \mu = \int_{\varphi_t(U)} (\operatorname{div}_\mu X)\, \mu ,$$

and a flow preserves volume if and only if $X$ is divergence-free. Liouville's theorem is the special case of a Hamiltonian vector field on phase space with the Liouville volume form $\omega^n/n!$.

Riemannian case

On a (pseudo-)Riemannian manifold one takes $\mu$ to be the metric volume form, $\mu = \sqrt{|g|}\, dx^1\wedge\dots\wedge dx^n$ in a positively oriented chart. Then

$$\operatorname{div} X = \frac{1}{\sqrt{|g|}}\, \partial_\mu\left( \sqrt{|g|}\, X^\mu \right) = \nabla_\mu X^\mu ,$$

with $\nabla$ the Levi-Civita connection. The spherical-coordinate formula of the Concrete layer is this expression with $\sqrt{g} = r^2\sin\theta$, after converting the coordinate components $X^\mu$ to orthonormal components. In the language of forms, $\operatorname{div} X = \star\, d \star X^\flat$ (up to a sign that depends on dimension and signature), so the divergence is minus the codifferential acting on 1-forms.

Divergence theorem

For compact $M$ with boundary, Stokes' theorem applied to the $(n-1)$-form $\iota_X\mu$ gives

$$\int_M (\operatorname{div}_\mu X)\, \mu = \int_M d(\iota_X\mu) = \int_{\partial M} \iota_X \mu ,$$

and in the Riemannian case $\iota_X\mu$ restricted to $\partial M$ equals $g(X, N)\, \mu_{\partial M}$ with $N$ the outward unit normal. Together with the product rule $\operatorname{div}(fX) = f\operatorname{div} X + X(f)$, this shows that $-\operatorname{div}$ is the formal adjoint of the gradient with respect to the $L^2$ inner products on functions and vector fields, and that $\operatorname{div}\operatorname{grad}$ is the Laplace-Beltrami operator.

Why is it interesting?

The divergence is a crucial tool, for example, in electrodynamics. This can be seen immediately because the fundamental equations of electrodynamics (called Maxwell equations) contain the divergence of the electric and magnetic fields.

Path to this page

1. Calculus
2. Linear Algebra
3. Vector Calculus