Surface Integrals

Surface Integrals & Parametrisation

A surface integral extends integration from curves to 2D surfaces in 3D space.

A parametric surface $S$ is defined by parameters $(u,v)$:

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

The differential area vector is built from partial derivatives:

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

A scalar surface integral integrates a function $f$ over $S$:

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

Flux Integrals & Orientation

A flux integral measures the net flow of a vector field $\vec{F}$ through a surface $S$:

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

Physics link: Electric flux through a closed surface equals the enclosed charge via Gauss's law:

$\Phi_E = \oiint_S \vec{E}\cdot d\vec{S} = \frac{Q_{\text{enc}}}{\varepsilon_0}$

Orientation is critical: the choice of unit normal $\hat{n}$ (outward vs. inward for closed surfaces) determines the sign of the flux.

Common pitfall: Flux integrals are orientation-sensitive. Flipping the surface normal flips the sign of your final answer. Always fix the normal direction before you begin integrating.