The Divergence Theorem

The divergence theorem (Gauss's theorem) links a volume integral of divergence to a surface integral of flux:

$$\oiint_S \vec{F}\cdot d\vec{S} = \iiint_V (\nabla\cdot\vec{F})\,dV$$

Here, $S$ is the closed surface bounding volume $V$ with an outward-pointing normal.

Physical meaning: Total flux leaving a closed surface equals total source strength inside.

Example: For $\vec{F} = x\hat{i} + y\hat{j} + z\hat{k}$ on the unit sphere, $\nabla\cdot\vec{F} = 3$. The volume integral is $\iiint 3\,dV = 3(\frac{4\pi}{3}) = 4\pi$, matching the flux integral.

Physics & Pitfalls

Applications in physics:

• Gauss's law: $\nabla\cdot\vec{E} = \rho/\varepsilon_0$
• Continuity equation: $\nabla\cdot\vec{J} + \frac{\partial\rho}{\partial t} = 0$

Key insight: It reduces 2D surface integrals to 3D volume integrals, or vice versa. Always pick the easier side.

Common pitfall: The theorem requires a closed surface and a field that is smooth throughout the volume.

An open surface, or a singularity inside like $1/r^2$ at the origin, invalidates the equality entirely.