Espai tangent: diferència entre les revisions
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Línia 24:
Suppose ''M'' is a C<sup>''k''</sup> manifold (''k'' ≥ 1) and ''x'' is a point in ''M''. Pick a [[chart (topology)|chart]] φ : ''U'' → '''R'''<sup>''n''</sup> where ''U'' is an [[open set|open subset]] of ''M'' containing ''x''. Suppose two curves γ<sub>1</sub> : (-1,1) → ''M'' and γ<sub>2</sub> : (-1,1) → ''M'' with γ<sub>1</sub>(0) = γ<sub>2</sub>(0) = ''x'' are given such that φ ∘ γ<sub>1</sub> and φ ∘ γ<sub>2</sub> are both differentiable at 0. Then γ<sub>1</sub> and γ<sub>2</sub> are called ''tangent at 0'' if the ordinary derivatives of φ ∘ γ<sub>1</sub> and φ ∘ γ<sub>2</sub> at 0 coincide. This defines an [[equivalence relation]] on such curves, and the [[equivalence class]]es are known as the tangent vectors of ''M'' at ''x''. The equivalence class of the curve γ is written as γ'(0). The tangent space of ''M'' at ''x'', denoted by T<sub>''x''</sub>''M'', is defined as the set of all tangent vectors; it does not depend on the choice of chart φ.
Suposi ''M'' és un C<sup>''k'' varietat de </sup> ''( k'' ≥; 1) i ''x'' és un punt en ''M'' . Triï una [[carta]] φ; : ''U'' → '''R'''<sup>''n'' </sup> on ''U'' és un [[conjunt obert|subconjunt obert]] de ''M'' contenint ''x'' . Suposi dues corbes γ<sub>1</sub> : (-1,1) → ''M'' i γ<sub>2</sub> : (-1,1) → ''M'' amb γ<sub>1</sub>(0) = γ<sub>2</sub>(0) = ''x'' són donats tal aquell φ; ∘ γ<sub>1</sub> i φ; ∘ γ<sub>2</sub> són els dos diferenciable a les 0. Llavors el γ<sub>1</sub> i γ<sub>2</sub> s'anomenen ''tangent a les 0'' si els derivats corrents de φ; ∘ γ<sub>1</sub> i φ; ∘ γ<sub>2</sub> a les 0 coincideixen. Això defineix una [[relació d'equivalència]] en tals corbes, i les [[classes d'equivalència]] es coneixen com els vectors
Línia 30:
[[Image:Tangentialvektor.svg|thumb|left|200px|The tangent space <math>\scriptstyle T_xM</math> and a tangent vector <math>\scriptstyle v\in T_xM</math>, along a curve traveling through <math>\scriptstyle x\in M</math>]]
[[Fitxer:Tangentialvektor.svg|thumb|deixat|200px|L'espai
Línia 58:
modeled on the [[product rule]] of calculus. These derivations form a real vector space in a natural manner; this is the tangent space T<sub>''x''</sub>''M''.
modeled sobre la [[regla del producte|regla de producte]] de càlcul. Aquestes derivacions formen un espai vectorial real en una conducta natural; això és l'espai
Línia 64:
The relation between the tangent vectors defined earlier and derivations is as follows: if γ is a curve with tangent vector γ'(0), then the corresponding derivation is ''D''(ƒ) = (ƒ ∘ γ)'(0) (where the derivative is taken in the ordinary sense, since ƒ ∘ γ is a function from (-1,1) to '''R''').
La relació entre els vectors
Línia 70:
Generalizations of this definition are possible, for instance to [[complex manifold]]s and [[algebraic variety|algebraic varieties]]. However, instead of examining derivations ''D'' from the full algebra of functions, one must instead work at the level of [[germ (mathematics)|germs]] of functions. The reason is that the [[structure sheaf]] may not be [[injective sheaf|fine]] for such structures. For instance, let ''X'' be an algebraic variety with [[structure sheaf]] ''F''. Then the [[Zariski tangent space]] at a point ''p''∈''X'' is the collection of ''K''-derivations ''D'':''F''<sub>p</sub>→''K'', where ''K'' is the [[groundfield]] and ''F''<sub>p</sub> is the stalk of ''F'' at ''p''.
Els Generalizations d'aquesta definició són possibles, per exemple a [[varietats complexes]] i [[varietat algebraica|varietats algebraiques]]. Tanmateix, en comptes de derivacions que examinen ''D'' de la plena àlgebra de funcions, un ha de treballar en canvi a l'altura de [[gèrmens]] de funcions. La raó és que el [[feix d'estructura]] pot no estar BÉ per a tals estructures. Per exemple, sia ''X'' una varietat algebraica amb [[feix d'estructura]] ''F'' . Llavors l'[[Espai
Línia 88:
If ''D'' is a derivation, then ''D''(ƒ) = 0 for every ƒ in ''I''<sup>2</sup>, and this means that ''D'' gives rise to a linear map ''I'' / ''I''<sup>2</sup> → '''R'''. Conversely, if ''r'' : ''I'' / ''I''<sup>2</sup> → '''R''' is a linear map, then ''D''(ƒ) = ''r''((ƒ - ƒ(''x'')) + ''I''<sup> 2</sup>) is a derivation. This yields the correspondence between the tangent space defined via derivations and the tangent space defined via the cotangent space.
Si ''D'' és una derivació, llavors ''D'' (ƒ;) = 0 per a tots els ƒ; en ''jo'' <sup>2</sup>, i aquest mitjà allò ''D'' dóna aconseguir un mapa lineal ''jo'' / ''jo'' <sup>2</sup> →; '''R'''. Al contrari, si ''r'' : ''jo'' / ''jo'' <sup>2</sup> →; '''R''' és un mapa lineal, llavors ''D'' (ƒ;) = ''r'' ''ƒ - ƒ(''x'') ) + (( jo'' <sup> 2</sup>) és una derivació. Això produeix la correspondència entre l'espai
== Propietats ==
If ''M'' is an open subset of '''R'''<sup>''n''</sup>, then ''M'' is a C<sup>∞</sup> manifold in a natural manner (take the charts to be the [[Identity function|identity maps]]), and the tangent spaces are all naturally identified with '''R'''<sup>''n''</sup>.
Si ''M'' és un subconjunt obert de '''R'''<sup>''n'' </sup>, llavors ''M'' és una varietat de C<sup>∞</sup> en una conducta natural (consideri que les cartes són els [[Funció identitat|mapes d'identitat)]], i els espais
=== Vectors tangent com a derivades direccionals ===
One way to think about tangent vectors is as [[directional derivative]]s. Given a vector ''v'' in '''R'''<sup>''n''</sup> one defines the directional derivative of a smooth map ƒ: '''R'''<sup>''n''</sup>→'''R''' at a point ''x'' by
Una manera de pensar en vectors
:<math> D_v f(x) = \frac{d}{dt}f(x+tv)\big|_{t=0}=\sum_{i=1}^{n}v^i\frac{\partial f}{\partial x^i}(x).</math>
This map is naturally a derivation. Moreover, it turns out that every derivation of C<sup>∞</sup>('''R'''<sup>''n''</sup>) is of this form. So there is a one-to-one map between vectors (thought of as tangent vectors at a point) and derivations.
Aquest mapa és naturalment una derivació. A més, es presenta que totes les derivacions de C<sup>∞</sup>'''( R'''<sup>''n'' </sup>) és d'aquesta forma. Així hi ha un mapa exacte entre vectors (pensat en ells com vectors
Línia 108:
Since tangent vectors to a general manifold can be defined as derivations it is natural to think of them as directional derivatives. Specifically, if ''v'' is a tangent vector of ''M'' at a point ''x'' (thought of as a derivation) then define the directional derivative in the direction ''v'' by
Des de vectors
:<math> D_v(f) = v(f)\,</math>
where ƒ: ''M'' → '''R''' is an element of C<sup>∞</sup>(''M'').
Línia 131:
Every smooth (or differentiable) map ''φ'' : ''M'' → ''N'' between smooth (or differentiable) manifolds induces natural [[linear map]]s between the corresponding tangent spaces:
Cada mapa llis (o diferenciable) ''φ;'' : ''M'' → ''N'' entre varietats llises (o diferenciable) indueix [[mapes lineals]] naturals entre els espais
:<math> \mathrm d\varphi_x\colon T_xM \to T_{\varphi(x)}N.</math>
If the tangent space is defined via curves, the map is defined as
Si l'espai
:<math> \mathrm d\varphi_x(\gamma'(0)) = (\varphi\circ\gamma)'(0).</math>
If instead the tangent space is defined via derivations, then
Si en canvi l'espai
:<math> \mathrm d\varphi_x(X)(f) = X(f\circ \varphi).</math>
Línia 177:
* [http://mathworld.wolfram.com/TangentPlane.html Tangent Planes] at MathWorld
* [Avions
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