Divergence theorem

In vector calculus, the divergence theorem (sometimes named after Gauss or Ostrogradsky) states that the net flux of a vector field $\vec{F}$ through a closed surface $S$ is equal to the divergence of the field integrated over the volume enclosed by the surface $S$.

$$\oint {\oint }_S\vec{F}\cdot {\vec{n}}_{S}dS={\iiint }_R\nabla \cdot \vec{F}dV$$

where $S$ is oriented outwards, and for a closed surface we can use the closed integral symbol. The bounding surface of $R$ is sometimes denoted $\partial R$. The 2D case is given by Green’s theorem.

Thinking geometrically, what we are saying is that $\iint \nabla \cdot \vec{F}$ gives us the total vector field generated inside $R$ (if we evaluate at a point, how much of the field is generated at the point), and that ${\int }_{\partial R}\vec{F}\cdot \vec{n}$ gives how much $\vec{F}$ leaves the boundary. So the total field generated within the region is equal to the field that is leaving the boundary.

This lets us skip some steps in computations. Normally when computing flux we’d have to parameterise, compute $\hat{n}dS$, but by divergence theorem we have an alternate way to tackle the problem.

For regions with multiple bounding surfaces, it’s easier to take the triple integral.

The right-hand size dot product result is $\rho$, a scalar density. The left-hand side surface is commonly called the Gaussian surface.

Notes and tips

• When computing the left side, remember to take the flux through every surface! Not just the main one.

See also

• Cauchy-Goursat theorem, the complex equivalent
• Stokes’ theorem, the cross product (curl) equivalent
• Gauss’ law, an application of the theorem for finding the charge enclosed by a closed surface