Surface Integrals

_S f dS and _S F· d S — area weighted by f, or net flux of F through S.

The idea

A surface integral adds up a function over a surface $S$ in space, as a line integral adds one up along a curve, with small pieces of area in place of pieces of arc.

Chop $S$ into small patches. Weighting each patch by its area and by the value of a scalar $f$ on it gives the scalar surface integral $\iint_S f\, dS.$ With $f = 1$ every patch is weighted by one, and the result is the area of $S$ itself. Weighting each patch instead by its area and the direction it faces, dotted into a vector field, gives $\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_S \mathbf{F} \cdot \mathbf{n}\, dS,$ where $\mathbf{n}$ is a unit vector perpendicular to $S$ at the patch. This integral is the flux of $\mathbf{F}$ through $S$.

To compute either one, describe $S$ by a parameterization $\mathbf{r}(u, v)$, which sweeps out the surface as $(u, v)$ ranges over a flat region. Its two tangent vectors $\mathbf{r}_u$ and $\mathbf{r}_v$ carry both the size and the facing of each patch: their cross product $\mathbf{r}_u \times \mathbf{r}_v$ stands normal to the surface, and its length is the area of the small parallelogram they span — everything $dS$ and $d\mathbf{S}$ require.

A surface has two sides, so $\mathbf{n}$ always admits two choices, and every flux integral must declare which one is meant.

Ways to work on it

• Walkthrough. The two kinds of surface integral — scalar and flux — and the parameterization that computes them.
• Practice. Compute the flux of a radial field through a sphere.
• Hardest. Parameterize a curved surface and compute a flux integral from scratch.

Not sure where to start? Take the ten-question placement test.