Divergence measures local net outflow from each point, while flux measures total flow through a surface. The divergence theorem states that these are two descriptions of the same overall effect for a closed surface.

Let $V$ be a suitable volume with outward-oriented boundary $\partial V$. Then

$$\iint_{\partial V}\mathbf F\cdot\mathbf n,dS

\iiint_V\nabla\cdot\mathbf F,dV.$$

Local sources add up to boundary flow

Imagine dividing the volume into many tiny cells. Flux leaving one cell and entering its neighbor cancels at their shared internal face. After all internal cancellations, only flux through the outer boundary remains.

Meanwhile, the divergence inside each small cell measures its local excess outflow. Adding those contributions throughout the volume therefore produces the same total as the outward boundary flux.

Example

For

$$\mathbf F(x,y,z)=(x,y,z),$$

we have

$$\nabla\cdot\mathbf F=3.$$

Over a volume $V$,

$$\iiint_V\nabla\cdot\mathbf F,dV =3\operatorname{Vol}(V).$$

The theorem says that the outward flux through the boundary is exactly the same quantity.

Choosing the easier side

The theorem can replace a difficult closed-surface flux integral with a volume integral of divergence, or the other way around. The useful representation depends on the geometry of the region and the field.

The divergence theorem is a local-to-global principle: infinitesimal source strength throughout a volume becomes total outward flow through its boundary.