Divergence

For a 3D vector field $\mathbf{F}=⟨P,Q,R⟩$, the divergence is the scalar field

$\nabla \cdot \mathbf{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}.$

In 2D: $\nabla \cdot \mathbf{F}=P_x+Q_y$.

Where Curl is formally a cross product $\nabla \times \mathbf{F}$, divergence is the dot product $\nabla \cdot \mathbf{F}$.

What divergence measures

A local property: the net outflow rate per unit volume at a point.

• Positive divergence: the point acts as a source, more goes out than comes in.
• Negative divergence: sink, more comes in than goes out.
• Zero divergence: locally balanced. Whatever enters a tiny region also leaves.

Divergence as flux density

The precise meaning comes from the Divergence theorem applied to a tiny ball $B$ of volume $\Delta V$ around a point:

$\iint \mathbf{F}\cdot d\mathbf{S}\approx (\nabla \cdot \mathbf{F})\Delta V.$

So $\nabla \cdot \mathbf{F}$ is the flux per unit volume out of an infinitesimal ball, which is what “source density” means.

Examples

Radial field $\mathbf{F}=⟨x,y,0⟩$: $\nabla \cdot \mathbf{F}=2$. Uniform source everywhere. Every point acts as a source.

Rotational field $\mathbf{F}=⟨-y,x,0⟩$: $\nabla \cdot \mathbf{F}=0$. Pure rotation, no source or sink. The field circulates without flowing into or out of any region.

Constant field: $\nabla \cdot \mathbf{F}=0$. Whatever enters a region also leaves.

Inverse-square field $\mathbf{F}=\mathbf{r}/|\mathbf{r}{|}^3$: $\nabla \cdot \mathbf{F}=0$ everywhere except at the origin, where the divergence has a delta-function singularity (a point source). This is the field of a point charge in electrostatics; the singularity is the charge.

Properties

• Linearity: $\nabla \cdot (a\mathbf{F}+b\mathbf{G})=a(\nabla \cdot \mathbf{F})+b(\nabla \cdot \mathbf{G})$.
• Product rule: $\nabla \cdot (f\mathbf{F})=f(\nabla \cdot \mathbf{F})+\nabla f\cdot \mathbf{F}$.

Divergence of a curl is zero

$\nabla \cdot (\nabla \times \mathbf{F})=0$

for any $\mathbf{F}$ with continuous second partials. Every term in the expansion cancels with another by equality of mixed partials.

Geometric content: rotation moves stuff in circles, not in or out, so a pure curl has no net source.

Companion identity: $\nabla \times (\nabla \varphi )=0$. Together:

• Gradients have zero curl.
• Curls have zero divergence.

On suitable domains the converses hold (Helmholtz / Poincaré-style results).

Source-free / solenoidal fields

A field with $\nabla \cdot \mathbf{F}=0$ is called source-free or solenoidal. Equivalent conditions on suitable domains:

• $\nabla \cdot \mathbf{F}=0$ (local).
• Every closed-surface flux is zero (global).
• $\mathbf{F}=\nabla \times \mathbf{G}$ for some vector potential $\mathbf{G}$ (3D structure result).

In 2D, the analogue: $\nabla \cdot \mathbf{F}=0$ iff $\mathbf{F}$ has a stream function $\psi$ such that $\mathbf{F}=⟨{\psi }_y,-{\psi }_x⟩$. Level curves of $\psi$ are streamlines of the flow. The “flux story” parallels the “circulation story” of conservative fields. When both conditions hold at once you get the Cauchy-Riemann equations, which is the same as analyticity.

Connection to Green’s flux form and divergence theorem

In 2D, Green’s flux form:

${\oint }_C\mathbf{F}\cdot \hat{\mathbf{N}}ds={\iint }_D\nabla \cdot \mathbf{F}dA.$

In 3D, the Divergence theorem:

$\iint \mathbf{F}\cdot d\mathbf{S}={\iiint }_V\nabla \cdot \mathbf{F}dV.$

Both say total flux out of the boundary equals integrated divergence over the interior: what flows out equals what is being created inside.

In physics

• Gauss’s law (electrostatics): $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$. Electric-field divergence equals charge density (up to constants).
• Magnetism: $\nabla \cdot \mathbf{B}=0$. No magnetic monopoles.
• Continuity equation (fluid dynamics): ${\partial }_t\rho +\nabla \cdot (\rho \mathbf{v})=0$. Conservation of mass: anything entering a region (positive divergence of flux) must equal mass increase.

Laplacian

The composition $\nabla \cdot \nabla$ is the Laplacian ${\nabla }^2=\Delta$:

${\nabla }^2\varphi =\nabla \cdot (\nabla \varphi )=\frac{{\partial }^2\varphi }{\partial x^2}+\frac{{\partial }^2\varphi }{\partial y^2}+\frac{{\partial }^2\varphi }{\partial z^2}.$

Solutions of ${\nabla }^2\varphi =0$ are harmonic functions: steady-state electrostatic potentials, irrotational incompressible flows, equilibrium heat distributions. They’re also the real and imaginary parts of analytic functions, the link between complex analysis and steady-state physics.