Normal

To orient the tangent plane $T_{x_0}S$, it is required to choose a unitary normal $\vec{n}_0$. A positive base $(\vec{e}_1, \vec{e}_2)$ of $\overrightarrow{T_{x_0}S}$ is therefore the one for which $(\vec{e}_1, \vec{e}_2, \vec{n}_x)$ is a positive base of $\vec{E}$. A parameterization $(U, \xi)$ allows to orient the tangent planes of $\xi(U)$ thanks to the unitary normal:

$$\frac{\xi_u\times\xi_v}{\vert\vert\xi_u\times\xi_v\vert\vert}$$

An implicit equation provides also a priviledged unitary normal:

$$\frac{grad_{x_0}\Xi}{\vert\vert grad_{x_0}\Xi\vert\vert}$$

A surface is oriented when a chosen unitary normal $\vec{n}:S\mapsto\vec{E}$ depends continuously on $x\in S$, i.e. for all parameterization $(U, \xi)$, $\vec{n}(\xi(U))$ is continuous.

There are three types of fundamental forms. The most important are the first and second, since the third can be expressed in terms of these. The fundamental forms are useful in determining the metric properties of a surface, such as line element, area element, normal curvature, Gaussian curvature, and mean curvature. They are bilinear forms defined on the tangent space. Their output is a number ( $\overrightarrow{T_xS}^2\mapsto\mathbb{R}$).