هل مساحة المتجه هي نفس مجال المتجهات؟ إذا لم يكن كذلك ، ما هو الفرق بينهما؟


الاجابه 1:

لا.

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.)


الاجابه 2:

مساحة المتجه عبارة عن مجموعة من الكائنات التي تتصرف مثل المتجهات. على غرار مساحة الحدث - مجموعة من الأحداث التي يمكن أن تحدث.

يشبه حقل المتجه وظيفة من مساحة متجه إلى مساحة متجه أخرى.

عادةً ما يكون حقل المتجه مختلفًا أو مستمرًا. الأمر الذي يتطلب فرض هيكل إضافي لتحديد ما هو المشتق وما هو المقصود بالمستمر.