Surface Integrals

Surface integrals often arise in Physics. For instance, the rate of flow of a liquid of velocity ${\bf v}$ through an infinitesimal surface of vector area $d{\bf S}$ is ${\bf v} \cdot d{\bf S}$ (i.e., the product of the normal component of the velocity, $v\,\cos\theta$, and the magnitude of the area, $dS$, where $\theta$ is the angle subtended between ${\bf v}$ and $d{\bf S}$). The net rate of flow through a surface $S$ made up of very many infinitesimal surfaces is

$$\int_{S} {\bf v}\cdot d{\bf S} = \lim_{d{\bf S}\rightarrow 0}\left[ \sum v\,\cos\theta \,dS\right],$$ (48)

where $\theta$ is the angle subtended between a surface element $d{\bf S}$ and the local flow velocity ${\bf v}({\bf r})$. If the surface is closed, and the surface elements all point outward, then the integral is conventionally written

$$\oint_{S} {\bf v}\cdot d{\bf S}.$$ (49)

In this case, the integral is often termed the flux of the velocity field ${\bf v}$ out of the closed surface $S$.