digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

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 R3. 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,

$\!\mathrm{I}(x,y)= \langle x,y \rangle.$

Let X(uv) be a parametric surface. Then the inner product of two tangent vectors is

\begin{align} & {} \quad \mathrm{I}(aX_u+bX_v,cX_u+dX_v) \\ & = ac \langle X_u,X_u \rangle + (ad+bc) \langle X_u,X_v \rangle + bd \langle X_v,X_v \rangle \\ & = 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.

$\!\mathrm{I}(x,y) = x^T \begin{pmatrix} E & F \\ F & G \end{pmatrix}y$

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 X1 and X2:

$g_{ij} = X_i \cdot X_j$

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

$ds^2 = Edu^2+2Fdudv+Gdv^2 \,$.

The classical area element given by $dA = |X_u \times X_v| \ du\, dv$ can be expressed in terms of the first fundamental form with the assistance of Lagrange's identity,

$dA = |X_u \times X_v| \ du\, dv= \sqrt{ \langle X_u,X_u \rangle \langle X_v,X_v \rangle - \langle X_u,X_v \rangle^2 } \ du\, dv = \sqrt{EG-F^2} \, du\, dv.$

Example

The unit sphere in R3 may be parametrized as

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

Differentiating $X(u,v)$ with respect to u and v yields

$X_u = \begin{pmatrix} -\sin u \sin v \\ \cos u \sin v \\ 0 \end{pmatrix},\ X_v = \begin{pmatrix} \cos u \cos v \\ \sin u \cos v \\ -\sin v \end{pmatrix}.$

The coefficients of the first fundamental form may be found by taking the dot product of the partial derivatives.

$E = X_u \cdot X_u = \sin^2 v$
$F = X_u \cdot X_v = 0$
$G = X_v \cdot X_v = 1$

Length of a curve on the sphere

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

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

Area of a region on the sphere

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

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

Gaussian curvature

The Gaussian curvature of a surface is given by

$K = \frac{\det \mathit{II}}{\det I} = \frac{ LN-M^2}{EG-F^2 },$

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.

 157351 videos foundNext >
 Lec 19 | MIT 18.01 Single Variable Calculus, Fall 2007Lecture 19: First fundamental theorem of calculus View the complete course at: http://ocw.mit.edu/18-01F06 License: Creative Commons BY-NC-SA More informatio... The First Form of the Fundamental Theorem of CalculusLet f be a continuous function, then the first form of the Fundamental Theorem of Calculus says that the derivative of A(x) = ∫f(t) dt from a to x equals. Calculus - First Fundamental Theorem (2013)This lesson demonstrates examples of how to apply the First Fundamental Theorem of Calculus. SPM : base fundamental formsom bo jin, Paul Whitrod direction, during the study Sound : S.Yokoda. Easy Tai Chi Form: The 12 O'Clock Fundamental FormTo get the maximum benefits of practicing this form: •Watch this form fully first before following it. •Think of facing 12 O'Clock (center) when you begin th... Fundamental Theorem of Calculus Part 1Thanks for watching and please subscribe! Visit PatrickJMT.com and 'like' it! :) Fundamental Theorem of Calculus Part 1 - Derivatives of Integrals. In this v... The Fundamental Theorem of Calculus. Part 2The Fundamental Theorem of Calculus. Part 2. The FTC part 2 simply tells that to evaluate a definite integral, we find an antiderivative, plug in the limits ... Fundamental Form live @ Faktoriet - Klubb MementoFundamental Form live @ Faktoriet - Klubb Memento 090925. Ki Cho Hyung - Fundamental FormKi Cho Hyung is the Kuk Sool Fundamental (White Belt) form. For more info on Kuk Sool traditional Korean martial arts visit our site: http://www.martialartfi... Linda Vu's First Form of Wing ChunI've always been fascinated with martial arts and intrigued by the concept. I'm currently attending Las Vegas Wing Chun Kung Fu School training under Sifu Br...
 157351 videos foundNext >

We're sorry, but there's no news about "First fundamental form" right now.

 Limit to books that you can completely read online Include partial books (book previews) .gsc-branding { display:block; }

Oops, we seem to be having trouble contacting Twitter