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

Graphical or geometric meaning

Every small volume cell sends flux across its faces. Shared internal faces carry equal and opposite orientations, so they cancel. Only faces on the outside boundary survive, which is the three-dimensional source-to-flux balance.

Common mistakes and why they fail

Verification and reasonableness checks

• Confirm divergence is scalar and volume units convert it to flux units.
• Use symmetry or a direct simple-surface computation.
• Verify every boundary component is included.

Close the surface and orient it outward

For a bounded solid $E$ with suitably regular boundary, $\iint_{\partial E}\mathbf F\cdot\mathbf n\,dS=\iiint_E\nabla\cdot\mathbf F\,dV$, with the boundary normal outward. An open surface cannot be used directly; add convenient caps, apply the theorem to the closed surface, then subtract their fluxes with correct orientations. Field smoothness is required on a neighborhood of the solid, not merely on the boundary. A singularity inside can invalidate the theorem even when the surface integral exists. Check units: divergence integrated over volume has the same units as flux. Symmetry may simplify either side, but it does not change outward orientation or remove the need to account for every boundary piece.

Practice

1. Find outward flux of $\langle x,y,z\rangle$ through the unit sphere.
2. What orientation does the theorem use?
3. Can it be applied if the field is singular at an interior point?

Connections and next steps

• Surface Integrals
• Curl and Divergence
• Stokes' Theorem