Look at a vector field — arrows everywhere. Now imagine placing a tiny box at some point. Divergence measures: is more stuff flowing out of the box than into it?

• Positive divergence = a source. More flows out than in. Like air escaping from a punctured balloon.
• Negative divergence = a sink. More flows in than out. Like water draining down a plughole.
• Zero divergence = balanced. What flows in, flows out. Like an incompressible fluid.

The intuition

Divergence at a point is the “net outflow per unit volume” from an infinitesimally small region around that point.

Think of dropping a small cork into a river: - Near a spring (source): water pushes the cork away — positive divergence. - Near a drain (sink): water pulls the cork in — negative divergence. - In a steady, uniform current: water carries the cork along but doesn’t spread or compress — zero divergence.

The formula

In two dimensions, the divergence of $\mathbf{F} = (F_x, F_y)$ is:

$$\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y}$$

Each term asks: “how much does the $x$-component change as you move in $x$?” and likewise for $y$. If both components are spreading out, divergence is positive.

For the source field $\mathbf{F} = (x, y)$: $\nabla \cdot \mathbf{F} = 1 + 1 = 2$. Positive everywhere — it’s a source at every point.

For the vortex $\mathbf{F} = (-y, x)$: $\nabla \cdot \mathbf{F} = 0 + 0 = 0$. The field swirls, but nothing is created or destroyed.

Why divergence matters

Divergence appears in two of Maxwell’s four equations:

• Gauss’s law: $\nabla \cdot \mathbf{E} = \rho / \varepsilon_0$ — electric charge is a source of electric field
• Gauss’s law for magnetism: $\nabla \cdot \mathbf{B} = 0$ — there are no magnetic charges (monopoles)

The divergence tells you where fields originate. Electric fields originate at charges. Magnetic fields originate... nowhere. They always form closed loops.