The Divergence Theorem

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

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

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

Physical meaning — The total flux leaving a closed surface equals the total "source strength" inside the volume.

When to use it

Example — Verify for $\vec{F} = x\,\hat{i} + y\,\hat{j} + z\,\hat{k}$ over the unit sphere:

• $\nabla\cdot\vec{F} = 1+1+1 = 3$
• Volume integral: $\iiint_V 3\,dV = 3\cdot\dfrac{4\pi}{3} = 4\pi$
• Flux integral: $\oiint_S \vec{F}\cdot\hat{n}\,dS = 4\pi$ (confirms the theorem)

Applications in physics

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

Key insight: The divergence theorem reduces a 2D surface integral to a 3D volume integral (or vice versa). Choose whichever side is easier to compute.

Common pitfall: The divergence theorem needs a closed surface enclosing a volume where the field is smooth. An open surface, or a singularity inside (like $1/r^{2}$ at the origin), voids the equality.