Divergence Theorem

The central idea

Let $E\subset\mathbb R^3$ be a bounded solid with piecewise smooth closed boundary $S=\partial E$, oriented outward, and let $\mathbf F$ be $C^1$ on an open set containing $E$. Then $\iint_S\mathbf F\cdot\mathbf n\,dS=\iiint_E\nabla\cdot\mathbf F\,dV$. The surface must be closed.

Definitions, hypotheses, and notation

For a region with an internal cavity, the boundary includes the outer surface and the cavity surface. 'Outward from the solid' points into the cavity on that inner component. This orientation ensures internal cancellation and is easy to reverse accidentally. Piecewise smooth boundaries allow corners and edges while retaining well-defined face normals almost everywhere.

A singularity can sometimes be handled by excising a small surrounding surface and applying the theorem on the punctured region, but the added flux must be analyzed. Simply integrating divergence across a point where the field is undefined violates the theorem's hypotheses.

Conceptual meaning

Total outward flux through a closed boundary equals total source strength inside. Local divergence contributions accumulate over volume; internal exchanges cancel, leaving only boundary flow. Orientation determines the flux sign.

A dependable method and decision rule

1. Verify the surface is closed and choose outward orientation.
2. Check the field is continuously differentiable throughout the enclosed solid.
3. Compute scalar divergence.
4. Describe the solid in convenient coordinates and integrate divergence.
5. If the original surface is open, add a cap and subtract its flux afterward.

Fully worked example

Common mistakes and why they fail