186.50.131.98: /* Example in C */

2015-01-06T21:21:57Z

Example in C

en>Qwertyus: /* In Ruby */ remove, this is not about pointers

2014-01-14T12:39:26Z

In Ruby: remove, this is not about pointers

← Older revision		Revision as of 14:39, 14 January 2014
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	~~Christine~~ is ~~what~~'~~s written on my birth certificate but my partner doesn~~'~~t may damage at every single~~. ~~One~~ of ~~his favorite hobbies is caving~~ and ~~he's been doing it for~~ a ~~long time. My house turn out to be~~ in ~~Arizona~~. The ~~job I've been occupying for years~~ is ~~an order clerk and~~ I~~'ll be promoted speedily. Go to my a website to find out more~~: ~~https://www.youtube.com/watch?v=pns2W-mSPU8~~<br><br>		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 [[canonical form\|canonically]] from the [[dot product]] of '''R'''<sup>''3''</sup>. 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,
			:<math>\!\mathrm{I}(x,y)= \langle x,y \rangle.</math>

	~~Feel free~~ to ~~visit my web page~~ ... [~~https~~://~~www~~.~~youtube~~.~~com~~/~~watch?~~v=~~pns2W~~-~~mSPU8 Scalar Energy Pendant Fake~~]		Let ''X''(''u'', ''v'') be a [[parametric surface]]. Then the inner product of two [[tangent vector]]s is

			:<math>
			\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}
			</math>

			where ''E'', ''F'', and ''G'' are the '''coefficients of the first fundamental form'''.

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

			:<math>\!\mathrm{I}(x,y) = x^T
			\begin{pmatrix}
			E & F \\
			F & G
			\end{pmatrix}y
			</math>

			==Further notation==
			When the first fundamental form is written with only one argument, it denotes the inner product of that vector with itself.
			:<math>\!\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 <math>g_{ij}</math>:
			:<math> \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 ''X''<sub>1</sub> and ''X''<sub>2</sub>:

			:<math>g_{ij} = X_i \cdot X_j</math>

			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
			:<math>ds^2 = Edu^2+2Fdudv+Gdv^2 \,</math>. <!--- "\," improves the display of this formula in Wikipedia. Do not delete --->

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

			:<math>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.</math>

			===Example===

			The unit [[sphere]] in '''R'''<sup>''3''</sup> may be parametrized as

			:<math>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).</math>

			Differentiating <math>X(u,v)</math> with respect to u and v yields

			:<math>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}.</math>

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

			:<math>E = X_u \cdot X_u = \sin^2 v</math>
			:<math>F = X_u \cdot X_v = 0</math>
			:<math>G = X_v \cdot X_v = 1</math>

			====Length of a curve on the sphere====

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

			:<math>\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</math>

			====Area of a region on the sphere====

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

			:<math>\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</math>

			==Gaussian curvature==

			The [[Gaussian curvature]] of a surface is given by

			:<math> K = \frac{\det \mathit{II}}{\det I} = \frac{ LN-M^2}{EG-F^2 }, </math>

			where ''L'', ''M'', and ''N'' are the coefficients of the [[second fundamental form]].

			[[Theorema egregium]] of [[Carl Friedrich Gauss\|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 [[Gaussian curvature#Alternative_formulas\|Brioschi formula]].

			==See also==
			*[[Metric tensor]]
			*[[Second fundamental form]]

			==External links==
			*[http://mathworld.wolfram.com/FirstFundamentalForm.html First Fundamental Form — from Wolfram MathWorld]
			*[http://planetmath.org/encyclopedia/FirstFundamentalForm.html PlanetMath: first fundamental form]

			{{curvature}}

			[[Category:Differential geometry of surfaces]]
			[[Category:Differential geometry]]
			[[Category:Surfaces]]

en>MastiBot: r2.7.2) (Robot: Modifying de:Zeiger (Informatik)#Funktionszeiger (Methodenzeiger)

2012-08-30T05:25:22Z

r2.7.2) (Robot: Modifying de:Zeiger (Informatik)#Funktionszeiger (Methodenzeiger)

New page

Christine is what's written on my birth certificate but my partner doesn't may damage at every single. One of his favorite hobbies is caving and he's been doing it for a long time. My house turn out to be in Arizona. The job I've been occupying for years is an order clerk and I'll be promoted speedily. Go to my a website to find out more: https://www.youtube.com/watch?v=pns2W-mSPU8<br><br>

Feel free to visit my web page ... [https://www.youtube.com/watch?v=pns2W-mSPU8 Scalar Energy Pendant Fake]

Function pointer - Revision history

186.50.131.98: /* Example in C */

en>Qwertyus: /* In Ruby */ remove, this is not about pointers

en>MastiBot: r2.7.2) (Robot: Modifying de:Zeiger (Informatik)#Funktionszeiger (Methodenzeiger)