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Topic: What is a vector field?
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Posts: 148
Registered: 4/13/13
Re: What is a vector field?
Posted: Apr 19, 2013 11:34 PM
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On 4/19/2013 2:38 PM, Tom Roberts wrote:
> On 4/17/13 4/17/13 10:14 PM, Hetware wrote:
>> 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

You were a great help. Thanks.

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