Divergence of a Vector Field
The big idea: divergence measures how much a vector field is pouring out of a point, per unit volume, right there. It turns the global idea of flux — how much flow crosses a whole closed surface — into a purely local number attached to a single point.
- State the limit definition of divergence honestly as flux-per-unit-volume as a surface shrinks.
- Derive the coordinate formula on a small box, face pair by face pair.
- Read $\nabla\cdot\mathbf F$ fluently and compute it across a gallery of fields.
- Translate the number into sources, sinks and incompressible flow.
- Use linearity, the product rule, and recognize the Laplacian as $\nabla\cdot\nabla f$.
- Preview the continuity equation and the divergence theorem as what comes next.
Flux answers "how much of $\mathbf F$ crosses this whole closed surface?" Divergence asks the same question at a single point: shrink the closed surface down around that point and ask how much flux escapes per unit volume enclosed. That ratio, in the limit, is the divergence:
$$\operatorname{div}\mathbf F(p) = \lim_{V\to\{p\}} \frac{1}{|V|}\oiint_{S=\partial V} \mathbf F\cdot d\mathbf S$$
This limit definition is the concept's honest face — everything else, including the coordinate formula below, is a computational shortcut for exactly this number.