NOTE 1 When the space orientation is defined by a right-handed trihedron, the direction of the vector surface element defines the orientation of the surface at that point as being in the anti-clockwise direction for an observer looking in the direction opposite to that of the vector.

NOTE 2 For a surface defined by r = f(u, v) where $(u,v)\in \text{U}\subseteq R^{\text{2}}$, the vector surface element is given by

${\displaystyle \frac{\partial f}{\partial u}\times \frac{\partial f}{\partial v}\cdot \mathrm{d}u\mathrm{d}v}$.

NOTE 3 A vector surface element is preferably designated by $e_{\text{n}}\mathrm{d}A$ or sometimes by $n\mathrm{d}A$, where $e_{\text{n}}=n$ is a unit vector normal to the surface and dA is a scalar surface element.