Surface Integral/Examples/Fluid in Motion

Example of Surface Integral
Let $\mathbf v$ be the velocity within a body of fluid $B$ as a point-function.

Let $S$ be a surface through which $B$ is in motion.

Let $\d S$ be a small element of $S$ whose center is at a point $P$.

Then the flow rate of $B$ through $S$ is given by the surface integral:
 * $\ds \iint_S \mathbf v \cdot \mathbf {\hat n} \rd S$

where $\mathbf {\hat n}$ denotes the unit normal to $S$ at $\d S$ in the direction of flow of $B$.