Vector surface integrals

Surface integrals often occur during vector analysis. For instance, the rate of flow of a liquid of velocity ${\bf v}$ through an infinitesimal surface of vector area $d{\bf S}$ is ${\bf v} \cdot d{\bf S}$. The net rate of flow through a surface ${\bf S}$ made up of lots of infinitesimal surfaces is

$$\int\!\int_S {\bf v}\cdot d{\bf S} = \lim_{dS\rightarrow 0}\left[ \sum v \cos\theta dS\right],$$ (86)

where $\theta$ is the angle subtended between the normal to the surface and the flow velocity.

Analogously to line integrals, most surface integrals depend both on the surface and the rim. But some (very important) integrals depend only on the rim, and not on the nature of the surface which spans it. As an example of this, consider incompressible fluid flow between two surfaces $S_1$ and $S_2$ which end on the same rim. The volume between the surfaces is constant, so what goes in must come out, and

$$\int\!\int_{S_1} {\bf v}\cdot d{\bf S} = \int\int_{S_2} {\bf v}\cdot d{\bf S}.$$ (87)

It follows that

$$\int\!\int {\bf v}\cdot d{\bf S}$$ (88)

depends only on the rim, and not on the form of surfaces $S_1$ and $S_2$.