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 n-space to be a vector field. > > No. He says that starting from point O and adding all elements belonging > to a h-dimensional vector field M, one can reach all points in an > h-dimensional 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 chocked-full 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 ur-vector, 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 non-Euclidean geometry, without saying so.
> 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 3-velocity of a fluid is a vector field, as are electric > and magnetic fields (considered as independent 3-vectors). > > More appropriately, the electric and magnetic fields are combined > into a two-form, which is an antisymmetric rank-2 tensor field (its > components in an inertial frame are the E- and B-fields of that > frame).
Either that, or simply a rank-2 tensor representable as either a 2-form, rank-2 mixed tensor or a rank-2 contravariant tensor. See Weyl's discussion beginning on page 38 item 2.:
> That is, the two-form maps every point in the manifold to an > anti-symmetric rank-2 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 Space-Time-Matter. 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 >