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Hetware
Posts:
148
Registered:
4/13/13


Re: What is a vector field?
Posted:
Apr 19, 2013 11:34 PM


On 4/19/2013 2:38 PM, Tom Roberts wrote: > On 4/17/13 4/17/13 10:14 PM, Hetware wrote: >> http://books.google.com/books?id=ztI6ezRvPXYC&lpg=PP1&pg=PA20#v=onepage&q&f=false >> >> Observe that on page 20 Weyl denotes the collection of position >> vectors in >> Euclidean nspace to be a vector field. > > No. He says that starting from point O and adding all elements belonging > to a hdimensional vector field M, one can reach all points in an > hdimensional submanifold. This is backwards from what you said.
I read halfway through the first sentence and realized what I had missed. Weyl is hard to read, and subtle, but that book is chockedfull of gems.
> Mathematicians often define things in a backwards manner from > what one might naively expect. > > As is usually discussed in intermediate physics courses, points in space > are not vectors at all, and the vector from one point to another is only > a vector in a limited sense;
Actually, the displacement vector is Weyl's urvector, if you will. See the first paragraph of Chapter 1, Section 2.
What got my head spun around is the (legitimate) view that some authors present of a vector space being represented by a collection of displacements from the origin of a mathematical coordinatized space.
> in general they form a vector space only as > long as they have an infinitesimal length (speaking loosely as a > physicist).
I'm not sure what you mean by that. Vectors in a tangent space are finite. The domain over which they are defined is infinitesimal. At least that's how I understand it.
>> I believe that means a field in the >> algebraic sense. > > No. An algebraic field is a commutative ring with a multiplicative > inverse (except for 0). There is no multiplication here, addition only.
Ah! Now I realize what's going on. Weyl avoided deriving multiplication of a vector by a scalar, so he gave a separate definition for multiplication. I believe the multiplication in an algebraic field would be an inner product of some sort.
> Mathematics has two completely different meanings of "field". One is the > algebraic field I mentioned above. The other came from physics and is a > function on a manifold  i.e. a map from each point in the manifold to > a value. This is the meaning used in differential geometry, relativity, > and in all of physics AFAIK. > > Weyl used the word "field" in a third sense no longer common AFAICT.
Perhaps part of my confusion.
> Note he does not consider nonEuclidean geometry, without saying so.
Correct.
> A scalar field maps every point in the manifold to a real number. For > instance, mass density is a scalar field. Within certain limits, > temperature is also a scalar field. > > A vector field maps a vector value to every point of the manifold. For > instance, the 3velocity of a fluid is a vector field, as are electric > and magnetic fields (considered as independent 3vectors). > > More appropriately, the electric and magnetic fields are combined > into a twoform, which is an antisymmetric rank2 tensor field (its > components in an inertial frame are the E and Bfields of that > frame).
Either that, or simply a rank2 tensor representable as either a 2form, rank2 mixed tensor or a rank2 contravariant tensor. See Weyl's discussion beginning on page 38 item 2.:
http://books.google.com/books?id=2VGc2QXyECAC&lpg=PP1&pg=PA38#v=onepage&q&f=false
> That is, the twoform maps every point in the manifold to an > antisymmetric rank2 tensor.
Best I can tell, differential forms are simply multilinear forms spoken with a French accent. Well, and a good deal of insight into how they function in the process of integration.
>> It seems to me Weyl was talking about something distinct from the >> currently >> prevailing (in physics, at least) notion of a vector field. > > Yes. Page 20 does not develop it fully. It is a precursor to today's usage. >
I'm not sure how it correlates to current usage.
> In modern mathematics (i.e. much later than Weyl's paper), the manifolds > of interest in physics have additional structure, including a tangent > space at every point. The tangent space at a given point can be thought > of as the collection of all vectors at that point; a vector field is > thus a specification of which of these vectors is used at each point of > the manifold (c.f. fiber bundle: a vector field is a section of the > manifold's tangent bundle, which is the fiber bundle over the manifold > with fiber being the tangent space at each point).
As in Schutz and Arnold.
> Actually, the tangent space at a point consists of all possible > directional derivatives at that point; using d/dx^i as a basis, > this is clearly an abstract vector space at that point.
In C++ we call that polymorphism. I find it confusing. AAMOF, that's why I picked up Weyl. I knew differential forms are closely related to contravariant (representations of) tensors.
> Physicists > ignore the subtleties....
Actually, I was reading MTW, who abuse the reader with the allegedly more insightful differential geometer's arcana. Of course I do understand the connection between old school and new school. I asked Wheeler about it one day, and he told me I was correct.
I also asked Wheeler what he thought of Weyl's SpaceTimeMatter. He took the book from my hand and paused for about a minute. Then he said, "I read that book about once every decade, and I learn something new from it every time." Weyl was John Archibald Wheeler's Wheeler.
> >> [...] > > I have no comment on the rest. The above should help. > > > Tom Roberts >
You were a great help. Thanks.



