TheoremProved

Divergence theorem (Gauss–Ostrogradsky theorem)

Statement

Let $V \subset \mathbb{R}^3$ be a bounded region with smooth boundary $\partial V$ oriented by the outward normal $\mathbf{n}$ , and let $\mathbf{F}$ be a continuously differentiable vector field on a neighborhood of $\overline{V}$ . Then $\iiint_V (\nabla \cdot \mathbf{F})\, dV = \iint_{\partial V} \mathbf{F} \cdot \mathbf{n}\, dS$ .

Why is it true?

The divergence $\nabla \cdot \mathbf{F}$ measures how much a vector field spreads out from each point, like the net rate fluid is created or destroyed there. Adding up all that local creation inside a region must equal the net flow escaping through its boundary — nothing is lost in between, it just has to leave through the surface.

Proof sketch

Prove the theorem first for a rectangular box by applying the fundamental theorem of calculus to each coordinate direction, so that each pair of opposite faces contributes the flux difference matching $\partial F_i/\partial x_i$ . Extend to a general elementary region (bounded above and below by graphs over each axis) using the same one-variable argument component by component, then cover a general region with a union of such elementary pieces and sum, noting that fluxes through shared interior faces cancel, leaving exactly the flux through the outer boundary.

Stated by

stated 1762Joseph-Louis Lagrange

Topics that use this theorem

Vector calculus

Related theorems

Stokes' theorem Fundamental theorem of calculus

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Jerrold E. Marsden, Anthony J. Tromba (2003). Vector Calculus