Surface Integral/Examples/Flow of Heat

Example of Surface Integral
Let $\mathbf h$ be the flow of heat within a body $B$ as a point-function..

Let $S$ be a surface through which $\mathbf h$ acts.

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

Then the heat flow through $S$ to which $\mathbf h$ gives rise is given by the surface integral:
 * $\ds \iint_S \mathbf h \cdot \mathbf {\hat n} \rd S$

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