Surface Integrals

A surface integral extends integration from curves to two-dimensional surfaces in $\mathbb{R}^{3}$.

Parametric surface — A surface $S$ parametrised by $(u,v)$:

$$\vec{r}(u,v) = x(u,v)\,\hat{i} + y(u,v)\,\hat{j} + z(u,v)\,\hat{k}$$

Normal vector and area element

$$d\vec{S} = \left(\frac{\partial\vec{r}}{\partial u}\times\frac{\partial\vec{r}}{\partial v}\right)du\,dv$$

Scalar surface integral

$$\iint_S f\,dS = \iint_D f(\vec{r}(u,v))\,\left\|\frac{\partial\vec{r}}{\partial u}\times\frac{\partial\vec{r}}{\partial v}\right\|\,du\,dv$$

Flux integral — The flow of $\vec{F}$ through $S$:

$$\Phi = \iint_S \vec{F}\cdot d\vec{S} = \iint_S \vec{F}\cdot\hat{n}\,dS$$

Special case — If $S$ is a graph $z = g(x,y)$:

$$d\vec{S} = (-g_x\,\hat{i} - g_y\,\hat{j} + \hat{k})\,dx\,dy$$

Orientation matters: the choice of $\hat{n}$ (outward vs. inward for a closed surface) determines the sign of the flux.

Physics link: Electric flux through a closed surface gives the enclosed charge: $\Phi_E = \oiint_S \vec{E}\cdot d\vec{S} = Q_{\text{enc}}/\varepsilon_0$ (Gauss's law in integral form).

Common pitfall: Flux integrals are orientation-sensitive: flipping the surface normal flips the sign. Fix the normal’s direction (outward? upward?) before integrating, or the "right" magnitude arrives with the wrong sign.