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Re: TdS is not dQ or d(anything)



This is helpful, Jack. But please clarify for me:

A 0-form on 3-space (x,y,z) is a smooth map f(x,y,z) of
3-space onto the real numbers.

How is this different from a scalar field - a scalar function of space?

The exterior derivative of the 0-form f is f_{x}dx+f_{y}dy+f_{z}dz
(f_{x} denotes the partial of f with respect to x.

How is this different from GRAD f (dot ) dr ?

What is this new language adding to good old fashioned vector analysis?

Bob Sciamanda (W3NLV)
Physics, Edinboro Univ of PA (em)
trebor@velocity.net
http://www.velocity.net/~trebor
----- Original Message -----
From: "Jack Uretsky" <jlu@HEP.ANL.GOV>
To: <PHYS-L@lists.nau.edu>
Sent: Tuesday, May 13, 2003 4:23 PM
Subject: Re: TdS is not dQ or d(anything)


| I think you guys are talking past each other, although I can't
for
| the life of me see why John is resorting to the calculus of manifolds to
| discuss elementary thermo. I also don't understand John's basis for
| distinguishing whether dV is, or is not, an "exact 1-form", so let's go
| back to basics. (I happen to have Nakahara, <Geometry, Topology and
| Physics> sitting next to me).
|
| A. A 1-form is, by definition, "exact" if it is the exterior derivative
of
| a 0-form. A 0-form on 3-space (x,y,z) is a smooth map f(x,y,z) of
| 3-space onto the real numbers.
| B. The exterior derivative of the 0-form f is f_{x}dx+f_{y}dy+f_{z}dz
| (f_{x} denotes the partial of f with respect to x. To quote Nahahara,
| "The simplest example of a 1-form is the differential df of a function f
| (where the function is defined over a suitable manifold -a point I don't
| think we have to get into).
|
| If you mean something else by "exact", then say so. I don't think that
| the notion of volume, as used in thermo, need get us into these esoteric
| mathematical discussions.
| Regards,
| Jack
|
|
|
|
| On Tue, 13 May 2003, Bob Sciamanda wrote:
|
| > Let us distinguish two different issues here:
| >
| > 1) Given a function of state, V, John wants to also call the exact
| > differential dV a function of state.
| > As far as I can see, this is just harmless terminology - though
confusing
| > to me. However, I have two notational problems with John's usage:
| >
| > | Anything you could legitimately do previously (considering
| > | d(V) as a small variation in V) you can do still (considering
| > | d(V) as the exterior derivative of V).
| >
| > (A) Why do you insist on d(V) and not just dV - what is the
distinction?
| > (B) Have you blurred the distinction between a differential and a
| > derivative? Is not dV a differential and not a derivative (consider
its
| > units!)? A derivative is a RATE of change.
| >
| > 2) The second issue shows in John's wish to obliterate inexact
| > differentials:
| >
| > |Bridgman _knew_ there
| > | was something fishy about d(W) but he couldn't explain it.
| >
| > You insult Bridgman! He well understood and explicitly explained the
| > difference between exact and inexact differentials, even acknowledging
the
| > need for different notations ( hence: dE vs dbarQ). dbarQ simply
| > represents an infinitesimal AMOUNT of energy in the form of "heat".
| > No-one pretends that it is the differential of a function. Are we
| > forbidden to use a symbol for such a quantity?
| >
| > Bob Sciamanda (W3NLV)
| > Physics, Edinboro Univ of PA (em)
| > trebor@velocity.net
| > http://www.velocity.net/~trebor
| >
|
| --
| "What did Barrow's lectures contain? Bourbaki writes with some
| scorn that in his book in a hundred pages of the text there are about
180
| drawings. (Concerning Bourbaki's books it can be said that in a
thousand
| pages there is not one drawing, and it is not at all clear which is
| worse.)"
| V. I. Arnol'd in
| Huygens & Barrow, Newton & Hooke