First fundamental form
Template:Short description 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 Template:Math. 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 Template:Math, <math display="block">\mathrm{I}(x,y)= \langle x,y \rangle.</math>
DefinitionEdit
Let Template:Math be a parametric surface. Then the inner product of two tangent vectors is <math display="block"> \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} </math> where Template:Mvar, Template:Mvar, and Template:Mvar are the coefficients of the first fundamental form.
The first fundamental form may be represented as a symmetric matrix. <math display="block">\mathrm{I}(x,y) = x^\mathsf{T} \begin{bmatrix} E & F \\ F & G \end{bmatrix}y </math>
Further notationEdit
When the first fundamental form is written with only one argument, it denotes the inner product of that vector with itself. <math display="block">\mathrm{I}(v)= \langle v,v \rangle = |v|^2</math>
The first fundamental form is often written in the modern notation of the metric tensor. The coefficients may then be written as Template:Mvar: <math display="block"> \left(g_{ij}\right) = \begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix} =\begin{pmatrix} E & F \\ F & G \end{pmatrix}</math>
The components of this tensor are calculated as the scalar product of tangent vectors Template:Math and Template:Math: <math display="block">g_{ij} = \langle X_i, X_j \rangle </math> for Template:Math. See example below.
Calculating lengths and areasEdit
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 Template:Math may be expressed in terms of the coefficients of the first fundamental form as <math display="block">ds^2 = E\,du^2+2F\,du\,dv+G\,dv^2 \,.</math>
The classical area element given by Template:Math can be expressed in terms of the first fundamental form with the assistance of Lagrange's identity, <math display="block">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.</math>
Example: curve on a sphereEdit
A spherical curve on the unit sphere in Template:Math may be parametrized as <math display="block">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].</math> Differentiating Template:Math with respect to Template:Mvar and Template:Mvar yields <math display="block">\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}</math> The coefficients of the first fundamental form may be found by taking the dot product of the partial derivatives.
<math display="block">\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}</math> so: <math display="block"> \begin{bmatrix}E & F \\F & G\end{bmatrix} =\begin{bmatrix} \sin^2 v & 0 \\0 & 1\end{bmatrix}.</math>
Length of a curve on the sphereEdit
The equator of the unit sphere is a parametrized curve given by <math display="block">(u(t),v(t))=(t,\tfrac{\pi}{2})</math> with Template:Mvar ranging from 0 to 2Template:Pi. The line element may be used to calculate the length of this curve.
<math display="block">\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</math>
Area of a region on the sphereEdit
The area element may be used to calculate the area of the unit sphere.
<math display="block">\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</math>
Gaussian curvatureEdit
The Gaussian curvature of a surface is given by <math display="block"> K = \frac{\det \mathrm{I\!I}_p}{\det \mathrm{I}_p} = \frac{ LN-M^2}{EG-F^2 }, </math> where Template:Mvar, Template:Mvar, and Template:Mvar 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 Template:Mvar 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.