Category:Examples of Surface Integrals

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This category contains examples of Surface Integral.

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 $\mathbf {\hat n}$.



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


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

The surface integral of $\mathbf v$ over $S$ is therefore defined as:

$\ds \iint_S \mathbf v \cdot \mathbf {\hat n} \rd S = \iint_S v \cos \theta \rd S$