Kas vektorruum on sama mis vektorväli? Kui ei, siis mis vahe neil on?


Vastus 1:

Ei

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.)


Vastus 2:

Vektorruum on objektide kogum, mis käitub nagu vektorid. Sarnaselt sündmusruumiga - sündmuste kogum, mis võib juhtuda.

Vektorväli on pigem funktsioon vektorruumist teise vektoriruumi.

Tavaliselt on vektorväli ka eristatav või pidev. Mis nõuaks täiendava struktuuri kehtestamist, et määratleda, mis on tuletis ja mida tähendab pidev.