Un espace vectoriel est-il identique à un champ vectoriel? Sinon, quelle est la différence entre eux?


Réponse 1:

Non.

AvectorspaceoverafieldFisaset[math]V[/math],togetherwithtwooperations,commonlyknownasvectoraddition(whichtakestwoelementsof[math]V[/math]andoutputsanotherelementof[math]V[/math])andscalarmultiplication(whichtakesanelementof[math]F[/math]andanelementof[math]V[/math]andoutputsanotherelementof[math]V[/math]),suchthat,if[math]u,v,wV[/math]and[math]a,bF[/math]:A vector space over a field F is a set [math]V[/math], together with two operations, commonly known as vector addition (which takes two elements of [math]V[/math] and outputs another element of [math]V[/math]) and scalar multiplication (which takes an element of [math]F[/math] and an element of [math]V[/math] and outputs another element of [math]V[/math]), such that, if [math]\textbf{u},\textbf{v},\textbf{w}\in V[/math] and [math]a,b\in F[/math]:

  1. u+v=v+u[math]u+(v+w)=(u+v)+w[/math]Thereexistssomevector[math]0[/math]suchthat,forevery[math]v[/math],[math]v+0=v[/math]Foreveryvector[math]v[/math],thereexistssomevector[math]v[/math]suchthat[math]v+(v)=0[/math][math]a(bv)=(ab)v[/math]Foreveryvector[math]v[/math],[math]1Fv=v[/math],where[math]1F[/math]isthemultiplicativeidentityin[math]F[/math][math]a(u+v)=au+av[/math][math](a+b)v=av+bv[/math]\textbf{u}+\textbf{v} = \textbf{v}+\textbf{u}[math]\textbf{u}+(\textbf{v}+\textbf{w}) = (\textbf{u}+\textbf{v})+\textbf{w}[/math]There exists some vector [math]\textbf{0}[/math] such that, for every [math]\textbf{v}[/math], [math]\textbf{v}+\textbf{0} = \textbf{v}[/math]For every vector [math]\textbf{v}[/math], there exists some vector [math]-\textbf{v}[/math] such that [math]\textbf{v}+(-\textbf{v}) = \textbf{0}[/math][math]a(b\textbf{v}) = (ab)\textbf{v}[/math]For every vector [math]\textbf{v}[/math], [math]1_F\textbf{v} = \textbf{v}[/math], where [math]1_F[/math] is the multiplicative identity in [math]F[/math][math]a(\textbf{u}+\textbf{v}) = a\textbf{u}+a\textbf{v}[/math][math](a+b)\textbf{v} = a\textbf{v}+b\textbf{v}[/math]

Avectorfieldisafunctionthattakespointsinsomemanifoldasinputsandreturnstangentvectorstothemanifoldasoutputs.Alotofthetime,themanifoldwillbeRn,butitdoesnthavetobe.Itcanbeanyarbitrarymanifold.(Initially,Ihadsaidthatavectorfieldwasamapbetweenvectorspaces,but,aspointedoutbyRuskoRuskov,thisisnotcorrect.)A vector field is a function that takes points in some manifold as inputs and returns tangent vectors to the manifold as outputs. A lot of the time, the manifold will be \mathbb{R}^n, but it doesn’t have to be. It can be any arbitrary manifold. (Initially, I had said that a vector field was a map between vector spaces, but, as pointed out by Rusko Ruskov, this is not correct.)


Réponse 2:

L'espace vectoriel est un ensemble d'objets qui se comportent comme des vecteurs. Similaire à l'espace des événements - ensemble d'événements qui peuvent se produire.

Le champ vectoriel ressemble plus à une fonction d'un espace vectoriel à un autre espace vectoriel.

Habituellement, un champ vectoriel est également différenciable ou continu. Ce qui nécessiterait d'imposer une structure supplémentaire pour définir ce qui est dérivé et ce que l'on entend par continu.