Definition:Surface Integral

Definition
Let $S$ be a surface in a vector field $\mathbf F$.

Let $\d S$ be a small element of $S$.

Let $\mathbf v$ be the vector induced by $\mathbf F$ at the middle of $\d S$.

Let $\mathbf {\hat n}$ denote the positive unit normal to $S$ at $\d S$.

Let $\mathbf v$ make an angle $\theta$ with $\mathbf {\hat n}$.


 * Surface-integral.png

Hence:
 * $\mathbf v \cdot \mathbf {\hat n} = v \cos \theta \rd S$

where:
 * $\cdot$ denotes dot product
 * $v$ denotes the magnitude of $\mathbf v$.

The surface integral of $\mathbf v$ over $S$ is therefore defined as:
 * $\ds \iint_S \mathbf v \cdot \mathbf {\hat n} \rd S = \iint_S v \cos \theta \rd S$