Area element of a surface ($dA$)

In a positively oriented surface chart,The coordinate-change Jacobian and the metric determinant transform inversely, so this defines a global area form.

The surface area of a measurable region $U\subseteq S$ is ${\int }_{U}dA$. For a parametrization $X(u,v)$ it is $\int |X_u\times X_v|dudv$.

For an oriented parametrized surface $\mathbf{r}(u,v)$,

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 givesConsequently 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.