Definition:Surface Integral/Physical Interpretation

Physical Interpretation of Surface Integral
Suppose $\mathbf v$ is interpreted as the velocity of some fluid in motion through a surface $S$.

Let $P$ be a point $P$ on $S$ at which a positive unit normal $\mathbf {\hat n}$ is constructed.

The expression $\mathbf v \cdot \mathbf {\hat n} \rd S$ denotes the amount of fluid passing through $\d S$ perpendicular to $S$ in unit time.

It is sufficient to consider this normal component, as the tangential component contributes nothing to the flow through $\d S$.

Hence the surface integral $I = \ds \iint_S \mathbf v \cdot \mathbf {\hat n} \rd S = \iint_S v \cos \theta \rd S$ expresses the total amount of fluid passing through $S$ in unit time.

If $I$ is positive, then this means there is a net outflow of fluid through $S$ from some source.

If $I$ is negative, then this means there is a net inflow of fluid through $S$ to some sink.

If $I$ is zero, the inflow equals the outflow, and either there are no sources or sinks within $S$, or that if there are some, their net inflow and outflow are equal.