Theory

Introduction

The calculation showing that divergence measures flow out of a loop can be done for a box-shaped loop instead of a circle. When two boxes are glued side-by-side, flow out of one box and into the other contributes positively to the line integral around the first box, and negatively to the line integral around the second box, so these contributions cancel in the sum. By subdividing a region into infinitesimal boxes, writing $\nabla \cdot \mathbf{F}dA$ for the total flow out of each box, summing the flow across all the box walls, and observing the inner cancellation of adjacent boxes, one obtains the divergence theorem:

Divergence Theorem

When $\mathcal{D}$ is a region of the $xy$-plane enclosed by its boundary curve $\partial D$, then:

$${\iint }_{\mathcal{D}}\nabla \cdot \mathbf{F}dA={\oint }_{\partial \mathcal{D}}\mathbf{F}\cdot {\mathbf{e}}_{\mathbf{n}}ds.$$

The way to think about this theorem is that $\nabla \cdot \mathbf{F}$ measures local expansion of a fluid that is flowing with velocity vectors $\mathbf{F}$. Then ${\iint }_{\mathcal{D}}\nabla \cdot \mathbf{F}dA$ measures the total expansion of fluid in the region $\mathcal{D}$. The theorem says that the total expansion in $\mathcal{D}$ must exactly match the total egress of fluid from the region $\mathcal{D}$ by flowing across the boundary $\partial \mathcal{D}$. Net expansion exactly matches net outflow.

When $\nabla \cdot \mathbf{F}=0$ in a region, then $\mathbf{F}$ is said to be divergence free. An incompressible fluid flowing in a closed container is divergence free. This concept is the analogue of being conservative, with $\nabla {\times }_z\mathbf{F}$ replaced by $\nabla \cdot \mathbf{F}$. Here is one mathematical application. For a divergence-free field, the flux between points $\mathbf{p}$ and $\mathbf{q}$ can be calculated using any path from $\mathbf{p}$ to $\mathbf{q}$ and does not depend on the chosen path. The reason is that any two paths create a loop, and the total flux across the loop is zero, so the flux into the loop on one side (the first path) equals the flux out of the loop on the other side (the second path). This is an analogue of path independence but for “across” line integrals instead of “aligned” line integrals.