first fundamental form

Let {{math|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 {{mvar|E}}, {{mvar|F}}, and {{mvar|G}} are the coefficients of the first fundamental form.

The first fundamental form may be represented as a symmetric matrix.

$$\backslash mathrm\{I\}(x,y)\; =\; x^\backslash mathsf\{T\}$$

\begin{bmatrix}

E & F \\

F & G

\end{bmatrix}y

When the first fundamental form is written with only one argument, it denotes the inner product of that vector with itself.

$$\backslash mathrm\{I\}(v)=\; \backslash langle\; v,v\; \backslash rangle\; =\; |v|^2$$

The first fundamental form is often written in the modern notation of the metric tensor. The coefficients may then be written as {{mvar|g_ij}}:

$$\backslash left(g\_\{ij\}\backslash right)\; =\; \backslash begin\{pmatrix\}$$

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 {{math|X₁}} and {{math|X₂}}:

$$g\_\{ij\}\; =\; \backslash langle\; X\_i,\; X\_j\; \backslash rangle$$

for {{math|1=i, j = 1, 2}}. See example below.

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 {{math|ds}} may be expressed in terms of the coefficients of the first fundamental form as

$$ds^2\; =\; E\backslash ,du^2+2F\backslash ,du\backslash ,dv+G\backslash ,dv^2\; \backslash ,.$$

The classical area element given by {{math|1=dA = {{abs|X_u × X_v}} du dv}} can be expressed in terms of the first fundamental form with the assistance of Lagrange's identity,

$$dA\; =\; |X\_u\; \backslash times\; X\_v|\; \backslash \; du\backslash ,\; dv=\; \backslash sqrt\{\; \backslash langle\; X\_u,X\_u\; \backslash rangle\; \backslash langle\; X\_v,X\_v\; \backslash rangle\; -\; \backslash left\backslash langle\; X\_u,X\_v\; \backslash right\backslash rangle^2\; \}\; \backslash ,\; du\backslash ,\; dv\; =\; \backslash sqrt\{EG-F^2\}\; \backslash ,\; du\backslash ,\; dv.$$

A spherical curve on the unit sphere in {{math|R³}} may be parametrized as

$$X(u,v)\; =\; \backslash begin\{bmatrix\}\; \backslash cos\; u\; \backslash sin\; v\; \backslash \backslash \; \backslash sin\; u\; \backslash sin\; v\; \backslash \backslash \; \backslash cos\; v\; \backslash end\{bmatrix\},\backslash \; (u,v)\; \backslash in\; [0,2\backslash pi)\; \backslash times\; [0,\backslash pi].$$

Differentiating {{math|X(u,v)}} with respect to {{mvar|u}} and {{mvar|v}} yields

$$\backslash begin\{align\}$$

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.

$$\backslash begin\{align\}$$

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:

The equator of the unit sphere is a parametrized curve given by

$$(u(t),v(t))=(t,\backslash tfrac\{\backslash pi\}\{2\})$$

with {{mvar|t}} ranging from 0 to 2{{pi}}. The line element may be used to calculate the length of this curve.

$$\backslash int\_0^\{2\backslash pi\}\; \backslash sqrt\{\; E\backslash left(\backslash frac\{du\}\{dt\}\backslash right)^2\; +\; 2F\; \backslash frac\{du\}\{dt\}\; \backslash frac\{dv\}\{dt\}\; +\; G\backslash left(\backslash frac\{dv\}\{dt\}\backslash right)^2\; \}\; \backslash ,dt\; =\; \backslash int\_0^\{2\backslash pi\}\; \backslash left|\backslash sin\; v\backslash right|\; \backslash ,\; dt\; =\; 2\backslash pi\; \backslash sin\; \backslash tfrac\{\backslash pi\}\{2\}\; =\; 2\backslash pi$$

The area element may be used to calculate the area of the unit sphere.

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

The Gaussian curvature of a surface is given by

$$K\; =\; \backslash frac\{\backslash det\; \backslash mathrm\{I\backslash !I\}\_p\}\{\backslash det\; \backslash mathrm\{I\}\_p\}\; =\; \backslash frac\{\; LN-M^2\}\{EG-F^2\; \},$$

where {{mvar|L}}, {{mvar|M}}, and {{mvar|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 {{mvar|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.

• Metric tensor
• Second fundamental form
• Third fundamental form
• Tautological one-form
• Gram matrix

• [http://mathworld.wolfram.com/FirstFundamentalForm.html First Fundamental Form — from Wolfram MathWorld]

{{curvature}}

Category:Differential geometry of surfaces

Category:Differential geometry

Category:Surfaces