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 — the precise meaning of “source density.”

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. At the origin 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. See Inverse-square field.

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 — 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 — see Cauchy-Riemann equations for the case when both conditions hold simultaneously, which turns out to be 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 = 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. Harmonic functions are also the real and imaginary parts of analytic functions — the link between complex analysis and steady-state physics.