First fundamental form

A "classical" surface is a two-dimensional surface embedded in an three-dimensional Euclidean space, i.e., a surface in the most normal sense.
We need two parameters ${\theta^1,\theta^2}$ to point an arbitrary point of a surface. These are called coordinates.
It would be natural to represent the shape of a surface by three functions that take coordinates as variables, i.e., ${x(\theta^1,\theta^2),y(\theta^1,\theta^2),z(\theta^1,\theta^2)}$.
A complete set of "ordered" three real values is written as $\mathbb{R}^3$, and we express

\begin{align} \boldsymbol{x}(\theta^1,\theta^2)\in\mathbb{R}^{3}\mid \boldsymbol{x}\equiv\left[\begin{array}{c}y(\theta^1,\theta^2)\\x(\theta^1,\theta^2)\\z(\theta^1,\theta^2)\end{array}\right]. \end{align}

When ${\theta^1,\theta^2}$ moves continuously, $\boldsymbol{x}(\theta^1,\theta^2)$ renders a surface.
We refer $\boldsymbol{x}$ to as position vector. We usually introduce standard inner product, $a^ib^i$, to $\mathbb{R}^3$.
There reason why the indices in $(\theta^1,\theta^2)$ are superscripts is explained later.
A tangent vector at a point specified by coordinates $( \theta^1,\theta^2)$ can be represented by $\boldsymbol{b}=b^1\boldsymbol{g}_1+b^2\boldsymbol{g}_2$, which is a linear combination of $\boldsymbol{g}_1,\boldsymbol{g}_2$. These are calculated by

\begin{align} \boldsymbol{g}_1(\theta^1,\theta^2)=\left[\begin{array}{c}\frac{\partial x}{\partial \theta^1}\\\frac{\partial y}{\partial \theta^1}\\\frac{\partial z}{\partial \theta^1}\end{array}\right],\ \ \ \ \boldsymbol{g}_2(\theta^1,\theta^2)=\left[\begin{array}{c}\frac{\partial x}{\partial \theta^2}\\\frac{\partial y}{\partial \theta^2}\\\frac{\partial z}{\partial \theta^2}\end{array}\right] \end{align}

and called as natural basis or simply as basis. Sometimes, they are called covariant basis, when dual basis are defined.
On a point of a surface, a complete set of tangent vectors at that point is called tangent plane. When it is generalized to an arbitrary dimensional space, it is called tangent space.
Suppose we are given inner products between basis. For example, if we can use standard inner product, the inner products between basis can be calculated as

\begin{align} \boldsymbol{g}_i\cdot\boldsymbol{g}_j=\frac{\partial x}{\partial \theta^i}\frac{\partial x}{\partial \theta^j}+\frac{\partial y}{\partial \theta^i}\frac{\partial y}{\partial \theta^j}+\frac{\partial z}{\partial \theta^i}\frac{\partial z}{\partial \theta^j}. \end{align}

Such inner products between basis are conventionally written as $g_{ij}=\boldsymbol{g}_i\cdot\boldsymbol{g}_j$, and are called Riemannian metric (or simply metric).
By using Riemannian metric, the inner product of an arbitrary combination of tangent vectors is given by

\begin{align} \boldsymbol{a}\cdot\boldsymbol{b}=\left(a^1\boldsymbol{g}_1+a^2\boldsymbol{g}_2\right)\cdot\left(b^1\boldsymbol{g}_1+b^2\boldsymbol{g}_2\right)=a^1b^1g_{11}+a^1b^2g_{12}+a^2b^1g_{21}+a^2b^2g_{22}. \end{align}

Using Einstein's summation convention, this can be re-expressed as

\begin{align} \left<\boldsymbol{a},\boldsymbol{b}\right>\equiv a^ib^jg_{ij}, \end{align}

and is called generalized inner product. By using Einstein's summation convention, a generalization of a quantity defined on an orthonormal frame to a curvilinear coordinate can be done by simply raising and lowering indices.
Incidentally, a standard inner product can be written as

\begin{align} \boldsymbol{a}\cdot\boldsymbol{b}\equiv a_ib_j\delta_{ij}, \end{align}

hence, it is known that standard inner product is a special case of generalized inner product.

A form like (5) is called a bilinear form, because it is linear with respect to both $a^i$ and $b^i$. In general, a form obtained by substituting an identical pairs of vectors to a bilinear form is called a quadratic form.
Particularly, a quadratic form obtained by substituting $d\boldsymbol{x}=d\theta^i\boldsymbol{g}_i$ to (5), i.e.,

\begin{align} \mathrm{I}\equiv dx^idx^jg_{ij} \end{align}

is called the first fundamental form. Historically, the first fundamental form is written in the following form:

\begin{align} \mathrm{I}\equiv Edu^2+2Fdudv+Gdv^2. \end{align}

Note that the former can be used for general dimensional space whereas the latter is for surface only.
The first fundamental form is a square of infinitesimal tangent vector $d\boldsymbol{x}$, i.e.,

\begin{align} d\boldsymbol{x}\cdot d\boldsymbol{x}=\mathrm{I}. \end{align}

Hence, is is a fundamental "measure" to measure a length or an angle on a given surface.