First fundamental form

In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of $R^{3}$. It permits the calculation of curvature and metric properties of a surface such as length and area in a manner consistent with the ambient space. The first fundamental form is denoted by the Roman numeral $I$, $$\mathrm{I}(x,y)= \langle x,y \rangle.$$

Definition
Let $X(u, v)$ be a parametric surface. Then the inner product of two tangent vectors is $$ \begin{align} & \mathrm{I}(aX_u+bX_v,cX_u+dX_v) \\[5pt] = {} & ac \langle X_u,X_u \rangle + (ad+bc) \langle X_u,X_v \rangle + bd \langle X_v,X_v \rangle \\[5pt] = {} & Eac + F(ad+bc) + Gbd, \end{align} $$ where $E$, $F$, and $G$ are the coefficients of the first fundamental form.

The first fundamental form may be represented as a symmetric matrix. $$\mathrm{I}(x,y) = x^\mathsf{T} \begin{bmatrix} E & F \\ F & G \end{bmatrix}y $$

Further notation
When the first fundamental form is written with only one argument, it denotes the inner product of that vector with itself. $$\mathrm{I}(v)= \langle v,v \rangle = |v|^2$$

The first fundamental form is often written in the modern notation of the metric tensor. The coefficients may then be written as $g_{ij}$: $$ \left(g_{ij}\right) = \begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix} =\begin{pmatrix} E & F \\ F & G \end{pmatrix}$$

The components of this tensor are calculated as the scalar product of tangent vectors $X_{1}$ and $X_{2}$: $$g_{ij} = \langle X_i, X_j \rangle $$ for $i, j = 1, 2$. See example below.

Calculating lengths and areas
The first fundamental form completely describes the metric properties of a surface. Thus, it enables one to calculate the lengths of curves on the surface and the areas of regions on the surface. The line element $ds$ may be expressed in terms of the coefficients of the first fundamental form as $$ds^2 = E\,du^2+2F\,du\,dv+G\,dv^2 \,.$$

The classical area element given by $dA = |X_{u} × X_{v}| du dv$ can be expressed in terms of the first fundamental form with the assistance of Lagrange's identity, $$dA = |X_u \times X_v| \ du\, dv= \sqrt{ \langle X_u,X_u \rangle \langle X_v,X_v \rangle - \left\langle X_u,X_v \right\rangle^2 } \, du\, dv = \sqrt{EG-F^2} \, du\, dv.$$

Example: curve on a sphere
A spherical curve on the unit sphere in $R^{3}$ may be parametrized as $$X(u,v) = \begin{bmatrix} \cos u \sin v \\ \sin u \sin v \\ \cos v \end{bmatrix},\ (u,v) \in [0,2\pi) \times [0,\pi].$$ Differentiating $X(u,v)$ with respect to $u$ and $v$ yields $$\begin{align} X_u &= \begin{bmatrix} -\sin u \sin v \\ \cos u \sin v \\ 0 \end{bmatrix},\\[5pt] X_v &= \begin{bmatrix} \cos u \cos v \\ \sin u \cos v \\ -\sin v \end{bmatrix}. \end{align}$$ The coefficients of the first fundamental form may be found by taking the dot product of the partial derivatives.

$$\begin{align} E &= X_u \cdot X_u = \sin^2 v \\ F &= X_u \cdot X_v = 0 \\ G &= X_v \cdot X_v = 1 \end{align}$$ so: $$ \begin{bmatrix}E & F \\F & G\end{bmatrix} =\begin{bmatrix} \sin^2 v & 0 \\0 & 1\end{bmatrix}.$$

Length of a curve on the sphere
The equator of the unit sphere is a parametrized curve given by $$(u(t),v(t))=(t,\tfrac{\pi}{2})$$ with $t$ ranging from 0 to 2$\pi$. The line element may be used to calculate the length of this curve.

$$\int_0^{2\pi} \sqrt{ E\left(\frac{du}{dt}\right)^2 + 2F \frac{du}{dt} \frac{dv}{dt} + G\left(\frac{dv}{dt}\right)^2 } \,dt = \int_0^{2\pi} \left|\sin v\right| \, dt = 2\pi \sin \tfrac{\pi}{2} = 2\pi$$

Area of a region on the sphere
The area element may be used to calculate the area of the unit sphere.

$$\int_0^\pi \int_0^{2\pi} \sqrt{ EG-F^2 } \ du\, dv = \int_0^\pi \int_0^{2\pi} \sin v \, du\, dv = 2\pi \Big[ {-\cos v} \Big]_0^{\pi} = 4\pi$$

Gaussian curvature
The Gaussian curvature of a surface is given by $$ K = \frac{\det \mathrm{I\!I}_p}{\det \mathrm{I}_p} = \frac{ LN-M^2}{EG-F^2 }, $$ where $L$, $M$, and $N$ are the coefficients of the second fundamental form.

Theorema egregium of Gauss states that the Gaussian curvature of a surface can be expressed solely in terms of the first fundamental form and its derivatives, so that $K$ is in fact an intrinsic invariant of the surface. An explicit expression for the Gaussian curvature in terms of the first fundamental form is provided by the Brioschi formula.