= Vector area
{title2=$\mathbf S=\int_\Sigma\mathbf n\,dS$}
The vector area of an oriented surface is the integral of its unit <normal vector> times its area element. If its oriented boundary is a closed curve $C$, <Stokes theorem> gives
$$
\mathbf S=\frac12\oint_C\mathbf r\times d\mathbf r.
$$
Consequently it depends only on the oriented boundary, not on the spanning surface. For a planar surface it is signed area times a unit <normal vector>. A current loop has <magnetic dipole moment> $\mathbf m=I\mathbf S$; the scalar area of a nonplanar spanning surface is not interchangeable with this vector.
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