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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