Re: TdS is not dQ or d(anything)

[Jack Uretsky <jlu@HEP.ANL.GOV>]

        See below, and also the Arnold quote at the end.

On Tue, 13 May 2003, Bob Sciamanda wrote:

> 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?
In short, it's not.
> > 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 ?
        For all practical purposes it's identical.  The difference comes
into play when you hit the 1-form with the exterior derivative (not
defined in "vector analysis") again, to get a 2-form which is
antisymmetric in the dx's.  However the exterior derivative of an exact
form always vanishes (equivalent of curl(grad f)=0.).

> What is this new language adding to good old fashioned vector analys.
        Nothing, in my opinion.  It doesn't really come into play until
you are dealing with higher rank tensors, as in MTW's <Gravitation> book,
where it is used to simplify the calculation of the elements of a
curvature tensor.
                                        Regards,
                                                Jack


> 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

--
        "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