Espai tangent: diferència entre les revisions

In [[mathematics]]MATHEMATICS , the '''((((( tangent space''' ))))) of a [[manifold]]MANIFOLD is a concept which facilitates the generalization of vectors from [[affineAFFINE space]]sSPACES to general manifolds , since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ..

(==) Informal description (==) ..

[[Image:Tangent-plane.svg| .3. thumb| .3. A pictorial representation of the tangent space of a single point , ''(( x'' )) , on a sphere. A vector in this tangent space can represent a possible velocity at ''(( x'' )) . After moving in that direction to another nearby point , one's velocity would then be given by a vector in the tangent space of that nearby point—a different tangent space , not shown.]] ..

In [[differentialDIFFERENTIAL GEOMETRY geometry]], one can attach to every point ''(( x'' )) of a differentiable [[manifold]]MANIFOLD a '''((((( tangent space''' ))))) , a real [[vectorVECTOR space]]SPACE which intuitively contains the possible "directions" in which one can tangentially pass through ''(( x'' )) . The elements of the tangent space are called '''((((( tangent vectors''' ))))) at ''(( x'' )) . This is a generalization of the notion of a [[boundBOUND vector]]VECTOR in a Euclidean space. All the tangent spaces have the same [[dimension ofDIMENSION a vector space|dimension]], equal to the dimension of the manifold. ..

For example , if the given manifold is a 2-[[sphere]]SPHERE , one can picture the tangent space at a point as the plane which touches the sphere at that point and is [[perpendicular]]PERPENDICULAR to the sphere's radius through the point. More generally , if a given manifold is thought of as an [[embedding|embedded]]EMBEDDED submanifold of [[EuclideanEUCLIDEAN space]]SPACE one can picture the tangent space in this literal fashion. ..

In [[algebraicALGEBRAIC GEOMETRY geometry]], in contrast , there is an intrinsic definition of '''((((( tangent space at a point P''' ))))) of a [[algebraicVARIETY variety|variety]](( ''V'' )) , that gives a vector space of dimension at least that of ''(( V'' )) . The points P at which the dimension is exactly that of ''(( V'' )) are called the '''((((( non-singular''' ))))) points ; the others are '''((((( singular''' ))))) points. For example , a curve that crosses itself doesn't have a unique tangent line at that point. The singular points of ''(( V'' )) are those where the 'test to be a manifold' fails. See [[ZariskiZARISKI tangentTANGENT space]]SPACE. ..

Once tangent spaces have been introduced , one can define [[vectorVECTOR FIELDS field]]s, which are abstractions of the velocity field of particles moving on a manifold. A vector field attaches to every point of the manifold a vector from the tangent space at that point , in a smooth manner. Such a vector field serves to define a generalized [[ordinaryORDINARY differentialDIFFERENTIAL equation]]EQUATION on a manifold: a solution to such a differential equation is a differentiable [[curve]]CURVE on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field. ..

All the tangent spaces can be "glued together" to form a new differentiable manifold of twice the dimension , the [[tangentTANGENT bundle]]BUNDLE of the manifold. ..

(==) Formal definitions (==) ..

There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via directions of curves is quite straightforward given the above intuition , it is also the most cumbersome to work with. More elegant and abstract approaches are described below. ..

(===) Definition as directions of curves (===) ..

Suppose ''(( M'' )) is a C^{''(( k'' ))} manifold (''(( k'' ≥)) ≥ 1) and ''(( x'' )) is a point in ''(( M'' )) . Pick a [[chartCHART (topology)|chart]] φφ : ''(( U'' →)) → ((((( '''R''' )))))^{''(( n'' ))} where ''(( U'' )) is an [[openOPEN set|open subset]]SUBSET of ''(( M'' )) containing ''(( x'' )) . Suppose two curves γγ₁ : (-1 ,1) →→ (( ''M'' )) and γγ₂ : (-1 ,1) →→ (( ''M'' )) with γγ₁(0) = γγ₂(0) = ''(( x'' )) are given such that φφ ∘∘ γγ₁ and φφ ∘∘ γγ₂ are both differentiable at 0. Then γγ₁ and γγ₂ are called ''(( tangent at 0'' )) if the ordinary derivatives of φφ ∘∘ γγ₁ and φφ ∘∘ γγ₂ at 0 coincide. This defines an [[equivalenceEQUIVALENCE relation]]RELATION on such curves , and the [[equivalenceEQUIVALENCE class]]esCLASSES are known as the tangent vectors of ''(( M'' )) at ''(( x'' )) . The equivalence class of the curve γγ is written as γγ'(0). The tangent space of ''(( M'' )) at ''(( x'' )) , denoted by T_{''(( x'' ))}''(( M'' )) , is defined as the set of all tangent vectors ; it does not depend on the choice of chart φφ. ..

[[Image:Tangentialvektor.svg| .3. thumb| .3. left| .3. 200px| .3. 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>]] ..

To define the vector space operations on T_{''(( x'' ))}''(( M'' )) , we use a chart φ&#966; : ''(( U'' →)) &#8594; ((((( '''R''' )))))^{''(( n'' ))} and define the [[MapMAP (mathematics)|map]] (dφd&#966;)_{''(( x'' ))} : T_{''(( x'' ))}''(( M'' →)) &#8594; ((((( '''R''' )))))^{''(( n'' ))} by (dφd&#966;)_{''(( x'' ))}(γ&#947;'(0)) = <math>\scriptstyle\frac{d}{dt}</math>(φ&#966; ∘&#8728; γ&#947;)(0). It turns out that this map is [[bijective]]BIJECTIVE and can thus be used to transfer the vector space operations from '''((((( R''' )))))^{''(( n'' ))} over to T_{''(( x'' ))}''(( M'' )) , turning the latter into an ''(( n'' )) -dimensional real vector space. Again , one needs to check that this construction does not depend on the particular chart φ&#966; chosen , and in fact it does not. ..

(===) Definition via derivations (===) ..

Suppose ''(( M'' )) is a C^∞&#8734; manifold. A real-valued function ƒ&#402;: ''(( M'' →)) &#8594; ((((( '''R''' ))))) belongs to C^∞&#8734;(''((( M'' )))) if ƒ&#402; ∘&#8728; φ&#966;^−1&#8722;1 is infinitely often differentiable for every chart φ&#966; : ''(( U'' →)) &#8594; ((((( '''R''' )))))^{''(( n'' ))}. C^∞&#8734;(''((( M'' )))) is a real [[associativeASSOCIATIVE algebra]]ALGEBRA for the [[pointwisePOINTWISE product]]PRODUCT and sum of functions and scalar multiplication. ..

Pick a point ''(( x'' )) in ''(( M'' )) . A ''[[Derivation (abstract( DERIVATION algebra)|derivation]]'') at ''(( x'' )) is a [[linearLINEAR map]]MAP (( ''D'' )) : C^∞&#8734;(''((( M'' )))) → '''&#8594; ((((( R''' ))))) which has the property that for all ƒ&#402; , ''(( g'' )) in C^∞&#8734;(''((( M'' )))) : ..

: <math>D(fg) = D(f)\cdot g(x) + f(x)\cdot D(g)</math> ..

modeled on the [[productPRODUCT rule]]RULE of calculus. These derivations form a real vector space in a natural manner ; this is the tangent space T_{''(( x'' ))}''(( M'' )) . ..

The relation between the tangent vectors defined earlier and derivations is as follows: if γ&#947; is a curve with tangent vector γ&#947;'(0) , then the corresponding derivation is ''(( D'' )) (ƒ&#402;) = (ƒ&#402; ∘&#8728; γ&#947;)'(0) (where the derivative is taken in the ordinary sense , since ƒ&#402; ∘&#8728; γ&#947; is a function from (-1 ,1) to '''((((( R''' )))))). ..

Generalizations of this definition are possible , for instance to [[complexCOMPLEX manifold]]sMANIFOLDS and [[algebraicALGEBRAIC variety|algebraic varieties]]VARIETIES. However , instead of examining derivations ''(( D'' )) from the full algebra of functions , one must instead work at the level of [[germ (mathematics)|germs]]GERMS of functions. The reason is that the [[structureSTRUCTURE sheaf]]SHEAF may not be [[injective sheaf|fine]]FINE for such structures. For instance , let ''(( X'' )) be an algebraic variety with [[structureSTRUCTURE sheaf]]SHEAF (( ''F'' )) . Then the [[ZariskiZARISKI tangentTANGENT space]]SPACE at a point ''(( p''∈'' )) &#8712;(( X'' )) is the collection of ''(( K'' )) -derivations ''(( D'' )) :''(( F'' )) _p→''&#8594;(( K'' )) , where ''(( K'' )) is the [[groundfield]]GROUNDFIELD and ''(( F'' )) _p is the stalk of ''(( F'' )) at ''(( p'' )) . ..

(===) Definition via the cotangent space (===) ..

Again we start with a C^∞&#8734; manifold , ''(( M'' )) , and a point , ''(( x'' )) , in ''(( M'' )) . Consider the [[idealIDEAL , (ring( I theory)|ideal]],) ''I'', in C^∞&#8734;(''((( M'' )))) consisting of all functions , ƒ&#402; , such that ƒ&#402;(''((( x'' )))) = 0. Then ''(( I'' )) and ''(( I'' )) ^{&nbsp ;2} are real vector spaces , and T_{''(( x'' ))}''(( M'' )) may be defined as the [[dualDUAL space]]SPACE of the [[quotientQUOTIENT spaceSPACE (linear( I algebra)|quotient) space]] ''I'' / ''(( I'' )) ^{&nbsp ;2}. This latter quotient space is also known as the [[cotangentCOTANGENT space]]SPACE of ''(( M'' )) at ''(( x'' )) . ..

While this definition is the most abstract , it is also the one most easily transferred to other settings , for instance to the [[algebraic variety|varieties]]VARIETIES considered in [[algebraicALGEBRAIC GEOMETRY. geometry]]..

If ''(( D'' )) is a derivation , then ''(( D'' )) (ƒ&#402;) = 0 for every ƒ&#402; in ''(( I'' )) ² , and this means that ''(( D'' )) gives rise to a linear map ''(( I'' )) / ''(( I'' )) ² →&#8594; ((((( '''R''' ))))). Conversely , if ''(( r'' )) : ''(( I'' )) / ''(( I'' )) ² →&#8594; ((((( '''R''' ))))) is a linear map , then ''(( D'' )) (ƒ&#402;) = ''(( r'' )) ((ƒ&#402; - ƒ&#402;(''((( x'' )))) ) + ''(( I'' )) ^{&nbsp ;2}) is a derivation. This yields the correspondence between the tangent space defined via derivations and the tangent space defined via the cotangent space. ..

(==) Properties (==) ..

If ''(( M'' )) is an open subset of '''((((( R''' )))))^{''(( n'' ))} , then ''(( M'' )) is a C^∞&#8734; manifold in a natural manner (take the charts to be the [[IdentityIDENTITY function|identity maps]]MAPS) , and the tangent spaces are all naturally identified with '''((((( R''' )))))^{''(( n'' ))}. ..

(===) Tangent vectors as directional derivatives (===) ..

One way to think about tangent vectors is as [[directionalDIRECTIONAL derivative]]sDERIVATIVES. Given a vector ''(( v'' )) in '''((((( R''' )))))^{''(( n'' ))} one defines the directional derivative of a smooth map ƒ&#402;: '''((((( R''' )))))^{''(( n'' ))}→'''&#8594;((((( R''' ))))) at a point ''(( x'' )) by ..

: <math> D_v f(x) = \frac{d}{dt}f(x+tv)\big| .3. _{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^∞&#8734;('''((((( R''' )))))^{''(( n'' ))}) is of this form. So there is a one-to-one map between vectors (thought of as tangent vectors at a point) and derivations. ..

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 ..

: <math> D_v(f) = v(f)\ ,</math> ..

where ƒ&#402;: ''(( M'' →)) &#8594; ((((( '''R''' ))))) is an element of C^∞&#8734;(''((( M'' )))) . ..

If we think of ''(( v'' )) as the direction of a curve , ''(( v'' )) = γ&#947;'(0) , then we write ..

: <math> D_v(f) = (f\circ\gamma)'(0).</math> ..

(===) The derivative of a map (===) ..

{{ main| .3. Pushforward (differential)}} ..

Every smooth (or differentiable) map ''φ''(( &#966; )) : ''(( M'' →)) &#8594; (( ''N'' )) between smooth (or differentiable) manifolds induces natural [[linearLINEAR map]]sMAPS between the corresponding tangent spaces: ..

: <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 ..

: <math> \mathrm d\varphi_x(\gamma'(0)) = (\varphi\circ\gamma)'(0).</math> ..

If instead the tangent space is defined via derivations , then ..

: <math> \mathrm d\varphi_x(X)(f) = X(f\circ \varphi).</math> ..

The linear map d''φ''(( &#966; )) _{''(( x'' ))} is called variously the ''(( derivative'' )) , ''(( total derivative'' )) , ''(( differential'' )) , or ''(( pushforward'' )) of ''φ''(( &#966; )) at ''(( x'' )) . It is frequently expressed using a variety of other notations: ..

: <math> D\varphi_x ,\quad (\varphi_*)_x ,\quad \varphi'(x).</math> ..

In a sense , the derivative is the best linear approximation to ''φ''(( &#966; )) near ''(( x'' )) . Note that when ''(( N'' )) = '''((((( R''' ))))) , the map d''φ''(( &#966; )) _{''(( x'' ))} : T_{''(( x'' ))}''(( M''→''' )) &#8594;((((( R''' ))))) coincides with the usual notion of the [[Differential (calculus)|differential]]DIFFERENTIAL of the function ''φ''(( &#966; )) . In [[localLOCAL coordinates]]COORDINATES the derivative of ƒ&#402; is given by the [[JacobianJACOBIAN. matrix and determinant|Jacobian]]..

An important result regarding the derivative map is the following: ..

:''' ((((( Theorem''' ))))). If ''φ''(( &#966; )) : ''(( M'' →)) &#8594; (( ''N'' )) is a [[localLOCAL diffeomorphism]]DIFFEOMORPHISM at ''(( x'' )) in ''(( M'' )) then d''φ''(( &#966; )) _{''(( x'' ))} : T_{''(( x'' ))}''(( M'' →)) &#8594; T_{''φ''(''( &#966; )) (((( x'' ))))}''(( N'' )) is a linear [[isomorphism]]ISOMORPHISM. Conversely , if d''φ''(( &#966; )) _{''(( x'' ))} is an isomorphism then there is an [[openOPEN set|openNEIGHBORHOOD neighborhood]](( ''U'' )) of ''(( x'' )) such that ''φ''(( &#966; )) maps ''(( U'' )) diffeomorphically onto its image. ..

This is a generalization of the [[inverseINVERSE functionFUNCTION theorem]]THEOREM to maps between manifolds. ..

(==) References (==) ..

* {{ citation| .3. first=Peter W.| .3. last=Michor| .3. title=Topics in Differential Geometry| .3. series=Graduate Studies in Mathematics| .3. volume=Vol. 93| .3. publisher=American Mathematical Society| .3. publication-place=Providence| .3. year=2008}} (''((( to appear'' )))) . ..

* {{ Citation | .3. last1=Spivak | .3. first1=Michael | .3. author1-link=Michael Spivak | .3. title=Calculus on Manifolds | .3. publisher=[[HarperCollins]]HARPERCOLLINS .3. | isbn=978-0-8053-9021-6 | .3. year=1965}} ..

(==) External links (==) ..

* [http://mathworld.wolfram.com/TangentPlane.html Tangent Planes] at MathWorld ..

{{DEFAULTSORT Defaultsort:Tangent Space}} ..

[[Category:Differential topology]] ..

[[Category:Differential geometry]] ..

[[da:Tangentrum]] ..

[[de:Tangentialraum]] ..

[[es:Espacio tangente]] ..

[[fr:Espace tangent]] ..

[[it:Spazio tangente]] ..

[[he:המרחב המשיק]] ..

[[nl:Raakruimte]] ..

[[ja:接ベクトル空間]] ..

[[pl:Przestrzeń styczna]] ..

[[pt:Espaço tangente]] ..

[[ru:Касательное пространство]] ..

[[fi:Tangenttiavaruus]] ..

[[sv:Tangentrum]] ..

[[uk:Дотичний простір]] ..

[[zh:切空间]] ..