Surface Integral/Examples/Magnetic Flux

Example of Surface Integral
Let $\mathbf M$ be an magnetic field acting over a region of space $R$.

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

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

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

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