A parametrized surface is described by a map \(\mathbf{s}:U\subset\mathbb{R}^2\to\mathbb{R}^3\), where \((u,v)\in U\) are coordinates in a flat parameter domain. The coordinates themselves do not yet say how far apart two surface points are or which angle two surface directions form. The first fundamental form supplies exactly this local metric information.
Its central idea is simple: map tangent vectors from the parameter domain onto the surface, measure their ordinary Euclidean dot product there, and express the result back in the original coordinates.
From Coordinates to Tangent Vectors
Fix a parameter point \(\mathbf{t}=(u,v)\). The differential of the parametrization at that point is the linear map
\[ D\mathbf{s}_{\mathbf{t}}: \mathbb{R}^2\longrightarrow T_{\mathbf{s}(\mathbf{t})}S. \]In standard coordinates this linear map is represented by the Jacobian whose columns are the two coordinate tangent vectors:
\[ J_{\mathbf{s}}(\mathbf{t}) =\begin{pmatrix}\mathbf{s}_u & \mathbf{s}_v\end{pmatrix}, \qquad D\mathbf{s}_{\mathbf{t}}\mathbf{v}=J_{\mathbf{s}}(\mathbf{t})\mathbf{v}. \]Thus a coordinate velocity \(\mathbf{v}=(v^1,v^2)^T\) becomes the actual surface-tangent vector \(v^1\mathbf{s}_u+v^2\mathbf{s}_v\). The parametrization is regular at \(\mathbf{t}\) when \(\mathbf{s}_u\) and \(\mathbf{s}_v\) are linearly independent. Only then does every nonzero coordinate vector map to a nonzero tangent vector.
Pulling the Dot Product Back to the Parameter Domain
Let \(\mathbf{v},\mathbf{w}\in\mathbb{R}^2\) represent two tangent directions. Their images are measured with the ordinary Euclidean dot product in \(\mathbb{R}^3\):
\[ \begin{aligned} \left\langle D\mathbf{s}_{\mathbf{t}}\mathbf{v},D\mathbf{s}_{\mathbf{t}}\mathbf{w}\right\rangle &=\left(J_{\mathbf{s}}\mathbf{v}\right)^T\left(J_{\mathbf{s}}\mathbf{w}\right)\\ &=\mathbf{v}^T\underbrace{J_{\mathbf{s}}^TJ_{\mathbf{s}}}_{\mathbf{I}^{\mathbf{s}}_{\mathbf{t}}}\mathbf{w}. \end{aligned} \]The symmetric matrix \(\mathbf{I}^{\mathbf{s}}_{\mathbf{t}}\) is the first fundamental form, or metric tensor, in the coordinates \((u,v)\):
\[ \begin{aligned} \mathbf{I}^{\mathbf{s}}_{\mathbf{t}} &=J_{\mathbf{s}}^TJ_{\mathbf{s}}\\ &=\begin{pmatrix} E & F\\ F & G \end{pmatrix}\\ &=\begin{pmatrix} \langle\mathbf{s}_u,\mathbf{s}_u\rangle & \langle\mathbf{s}_u,\mathbf{s}_v\rangle\\ \langle\mathbf{s}_u,\mathbf{s}_v\rangle & \langle\mathbf{s}_v,\mathbf{s}_v\rangle \end{pmatrix}. \end{aligned} \]In other words, \(\mathbf{I}^{\mathbf{s}}_{\mathbf{t}}\) is the Gram matrix of the coordinate tangent vectors. The entries have immediate geometric meanings: \(E\) and \(G\) are their squared lengths, while \(F\) measures their failure to be orthogonal. Regularity makes this matrix positive definite because
\[ \mathbf{v}^T\mathbf{I}^{\mathbf{s}}_{\mathbf{t}}\mathbf{v} =\|J_{\mathbf{s}}\mathbf{v}\|^2>0 \qquad\text{for every }\mathbf{v}\ne\mathbf{0}. \]Lengths and Angles in Surface Coordinates
The pulled-back inner product and its induced norm are
\[ \langle\mathbf{v},\mathbf{w}\rangle_{\mathbf{I}^{\mathbf{s}}_{\mathbf{t}}} =\mathbf{v}^T\mathbf{I}^{\mathbf{s}}_{\mathbf{t}}\mathbf{w}, \qquad \|\mathbf{v}\|_{\mathbf{I}^{\mathbf{s}}_{\mathbf{t}}} =\sqrt{\mathbf{v}^T\mathbf{I}^{\mathbf{s}}_{\mathbf{t}}\mathbf{v}}. \]These are not new measurements imposed by convention. They are exactly the usual Euclidean measurements of the mapped vectors on the surface. In particular, the angle \(\theta\) between two nonzero tangent directions obeys
\[ \cos\theta= \frac{\mathbf{v}^T\mathbf{I}^{\mathbf{s}}_{\mathbf{t}}\mathbf{w}} {\sqrt{\mathbf{v}^T\mathbf{I}^{\mathbf{s}}_{\mathbf{t}}\mathbf{v}} \sqrt{\mathbf{w}^T\mathbf{I}^{\mathbf{s}}_{\mathbf{t}}\mathbf{w}}}. \]For an infinitesimal coordinate displacement \((du,dv)\), the squared surface distance is conventionally written
\[ d\ell^2=E\,du^2+2F\,du\,dv+G\,dv^2. \]Consequently, the coordinate curves are orthogonal exactly when \(F=0\). If additionally \(E=G\), the parametrization preserves angles locally up to one common scale factor.
Length of a Curve on the Surface
Let \(\mathbf{c}:[a,b]\to U\) be a curve in the parameter domain and \(\boldsymbol{\gamma}=\mathbf{s}\circ\mathbf{c}\) its image on the surface. The chain rule gives
\[ \boldsymbol{\gamma}'(t) =J_{\mathbf{s}}(\mathbf{c}(t))\mathbf{c}'(t). \]Substituting this velocity into the ordinary Euclidean arc-length integral yields
\[ \begin{aligned} L(\boldsymbol{\gamma}) &=\int_a^b\|\boldsymbol{\gamma}'(t)\|\,dt\\ &=\int_a^b \sqrt{\mathbf{c}'(t)^T \mathbf{I}^{\mathbf{s}}_{\mathbf{c}(t)} \mathbf{c}'(t)}\,dt\\ &=\int_a^b \|\mathbf{c}'(t)\|_{\mathbf{I}^{\mathbf{s}}_{\mathbf{c}(t)}}\,dt. \end{aligned} \]It is therefore the same geometric arc-length integral as in \(\mathbb{R}^3\), expressed entirely in parameter coordinates. The metric changes with \(\mathbf{c}(t)\), so each coordinate velocity is measured using the local stretching of the surface map at that point.
Sanity Check: A Flat Plane
For \(\mathbf{s}(u,v)=(u,v,0)\),
\[ J_{\mathbf{s}}= \begin{pmatrix} 1 & 0\\ 0 & 1\\ 0 & 0 \end{pmatrix}, \qquad \mathbf{I}^{\mathbf{s}}=J_{\mathbf{s}}^TJ_{\mathbf{s}}= \begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}. \]Nothing is stretched or skewed, so the first fundamental form is the identity matrix. Lengths and angles in the parameter domain already equal those on the plane.
Example: A Cylinder
Parametrize a cylinder of radius \(R\) by
\[ \mathbf{s}(u,v)=(R\cos u,R\sin u,v). \]Its coordinate tangent vectors and first fundamental form are
\[ \mathbf{s}_u=(-R\sin u,R\cos u,0), \qquad \mathbf{s}_v=(0,0,1), \] \[ E=R^2,\qquad F=0,\qquad G=1, \qquad \mathbf{I}^{\mathbf{s}}= \begin{pmatrix}R^2 & 0\\0 & 1\end{pmatrix}. \]A change \(du\) around the cylinder has physical length \(R|du|\), while a change \(dv\) along the axis keeps length \(|dv|\). The coordinates remain orthogonal because \(F=0\), but they have different scale factors unless \(R=1\).
As a concrete curve, let \(\mathbf{c}(t)=(t,at)\) for \(0\le t\le T\). Its surface image is a helix, and the metric gives its length directly:
\[ L=\int_0^T \sqrt{\begin{pmatrix}1 & a\end{pmatrix} \begin{pmatrix}R^2 & 0\\0 & 1\end{pmatrix} \begin{pmatrix}1\\a\end{pmatrix}}\,dt =T\sqrt{R^2+a^2}. \]Area and the Determinant
The same matrix also records local area scaling. The parallelogram spanned by \(\mathbf{s}_u\) and \(\mathbf{s}_v\) has area
\[ \|\mathbf{s}_u\times\mathbf{s}_v\| =\sqrt{EG-F^2} =\sqrt{\det\mathbf{I}^{\mathbf{s}}}. \]Hence the surface-area element is
\[ dA=\sqrt{EG-F^2}\,du\,dv. \]Lengths, angles, and areas therefore come from one object: the Gram matrix of the differential. The first fundamental form contains all metric information visible from within the surface, without requiring a preferred coordinate system.