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#19
Mar3107, 11:13 AM

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If you wish to assign to each point a vector, somehow representing its position, then that vector is in the vector space R^{2} you have attached to each point... which can be added to other vectors in the vectorspacebasedatthatpoint. 


#20
Mar3107, 11:57 AM

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#21
Mar3107, 01:18 PM

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I can also try to find the definition of a vector from one of my relativity professors.[/QUOTE]Why? I wasn't looking toward my friend for supporting me. I was looking to him to give me an example which will clarify this issue. The rest I mentioned because the source of an example is sometimes important to (very few) people, but I put it there anyway. Bad move on my part! Let's put this on hold until I determined if I made an error or I can come back with a strong determining arguement (which I seem to lacking right now). Cheers Rob Pete 


#22
Mar3107, 01:23 PM

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I'm curious. How is it you came to believe that all geometric vectors are the same kind of vectors which are elements of a vector space? Pete 


#23
Mar3107, 01:50 PM

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Linearity is the key feature of a vector. 


#24
Mar3107, 02:13 PM

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The position vector is defined in places such as Ohanian's GR text. I'm curious as to your position. Any two elements in a vector space must be able to be added to yield another vector. How do you see this in working on a curved manifold when only vectorss defined at the same point can be added? I assume you're response will be related to parallel transporting vectors defined at other points. But this is seems to me to be an over qualification, i.e. you will need something besides a vector and a scalar to create the resulting vector. Pete 


#25
Apr107, 05:09 PM

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Maybe I need a clarification of your question.
Are you asking whether or not a "geometric vector" (for now, let's instead call it a gvector) [which you defined as a quantity that transforms like a displacement vector], must be an element of a vectorspace [using Schutz's GMoMP definition]? If so, what is your answer? To me, a vector is an element of a vector space... and I believe it can be shown to be compatible with the above "transformation"viewpoint [probably due to F.Klein]. In a vector space [like Euclidean space or Minkowski spacetime], I can define a displacement vector between two points... or, if I choose an origin, I can define a position vector of a distant point. In a general manifold, I can't as easily make such definitions that join two points on that manifold with a vector. Can one define 2D positionvectors on a sphere? The best I can do is to talk about vectors in the tangent vectorspaces attached to the different points... and, as you say, addition can only take place in the same vector space... possibly using a paralleltransported vector from another point's tangent vector space. When the manifold is a vectorspace, the distinction between the tangent vectorspaces and the vectorspace manifold can be blurred... parallel transport isn't much of an issue. 


#26
Apr107, 05:23 PM

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Let me take some time and think this over. But I'm curious as to why you call it my definition? Had it been me that originated such a definition and it found its way into all the physics texts then I could see your point. But I'm going by what I gather from the literature, e.g. from all the books I've read on the subject and what comes up in conversation with other relativists I know. If it was me that created ut ya think I'd get some royalties huh? Best wishes and I'll get back to this later. Pete 


#27
Apr1907, 08:07 PM

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Hi Rob
Okay. I found the passage that I read which is one of the resources I came acrossed. This is from Schutz's "Geometrican Methods of Mathematical Physics[/i], page 15 towards bottom of page Pete 


#28
Apr1907, 08:20 PM

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#29
Apr1907, 08:36 PM

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Then let me ask you this. A great deal of textual sources define vectors according to their transportation properties. That is neccesary condition to be a vector according to those sources. Now consider an element of a vector space. This property of transformation is not a condition to be a vector. Therefore an element of a vector space does not imply that the element is also a vector whose components transforms as a vector. Do you or do you not agree with this. If not then please elaborate. Thanks. That is what I've been trying to get across. I guess I failed. Pete 


#30
Apr1907, 08:41 PM

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#31
Apr1907, 10:14 PM

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In a setting that adopts the convention that the word "vector" is used exclusively to refer to tangent vectors of manifolds, and furthermore that tangent vectors are only to be thought of as coordinatechartdependenttuplesof'components', I would completely agree with what you have written in that post. But the tangent vectors at a point P are elements of a vector space: the tangent space at P. So this isn't an example of vectors that are not elements of a vector space. 


#32
Apr1907, 10:23 PM

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Pete 


#33
Apr2007, 06:34 AM

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[itex]e'_{i} = L^{j} {}_{i} e_{j}.[/tex] Any [itex]v[/itex] in [itex]V[/itex] can be expanded with respect to both bases: [tex]v = v^j e_j = v'^i e'_i = v'^i L^{j} {}_{i} e_{j}.[/tex] Thus, for example, [tex]v^j = v'^i L^{j} {}_{i}.[/tex] Some books blur the distinction between (tangent) vectors (at a point) and vector fields. The set of vector fields is not a vector space, but is a generalization (a module over the ring of scalar fierlds) of a vector space. Again, transformation properties are derived properties. Consider an ndimensional real differentiable manifold, and suppose further that the bases above are coordinate tangent vector fields that arise from 2 overlapping charts: [tex]e_{i} =\frac{\partial}{\partial x^{i}}[/tex] and [tex]e'_{i} =\frac{\partial}{\partial x'^{i}}.[/tex] Then, by the chain rule, the change of basis relation is [tex]e'_{i} = L^{j} {}_{i} e_{j} = \frac{\partial x^{j}}{\partial x'^{i}} e_{j},[/tex] and [tex]v^j = \frac{\partial x^{j}}{\partial x'^{i}} v'^i.[/tex] 


#34
Apr2007, 07:12 AM

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I'll get back after I do some more searching on this topic. The idea is to get a good example. 


#35
Apr2007, 08:42 AM

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Now, for the space of 10x10 matrices there are (at least) two types of transformations that can induced by a change of basis, depending on whether the change of basis is for the space of matrices (this treats the matrices as singleindex objects), or for the space on which the matrices act (this treats the matrices as twoindex objects). This discussion is getting somewhat surreal, so I'll think I'll bow out now. 


#36
Apr2007, 10:07 AM

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I can't, for the life of me, understand why you think that any vector can be transformed from one coordinate system to another when such a transformation may be entirely meaningless for a particuar kind of vector. E.g. I meant for the matrices to be ten dimensional, not 10x10
(In case you were thinking about 4D spacetime). As I said, I'm looking for such an example right now and will post it when I find it. But such a criteria never exist for a vector space where such a critera always exists for geometrical vectors. A good example which comes to mind now is the vector space whose elements belong to Fourier series, i.e. a sequance of sines and cosines in increasing frequency which can approximate any function of a given interval. How would you transform these elements to another coordinate system when, even when you can write down something that lokks like a transformation (God only knows what that transformation looks like or means) such a transformation is meaningless. Pete 


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