Traditionally:
1) V is a function of state; i.e. its value is uniquely specified for a given
system state
2) dV is an infinitesimal (often within the context of some limiting process)
increment in the value of this state function V; i.e. its meaning and value are
defined for a PAIR of neighboring states. If you will, you might call it a
"two-state function", or a "function of two states", but in no sense is it a
"single state function".
Is the usurpation of the "d" notation by GA for its "one form" scheme now
forbidding this language?
Bob Sciamanda (W3NLV)
Physics, Edinboro Univ of PA (em)
trebor@velocity.net http://www.velocity.net/~trebor
----- Original Message -----
From: "John S. Denker" <jsd@MONMOUTH.COM>
To: <PHYS-L@lists.nau.edu>
Sent: Tuesday, May 13, 2003 12:27 AM
Subject: Re: TdS is not dQ or d(anything)
| I wrote:
| > |
| > | P is a function of state
| > | V is a function of state
| > | P dV is a function of state
| > | W is not a function of state
|
|
| On 05/12/2003 02:54 PM, Bob Sciamanda wrote:
| >
| > P dV is a function of state ???????
|
| Aha. Now we are getting somewhere. At
| least we are getting to the point where we
| can begin to have a discussion.
|
| We need to figure out what we mean by "d".
| There are at least three somewhat-related
| things it could mean:
| -- "d" could indicate some notion of smallness
| -- "d" could indicate differentiation
| -- "d" could indicate the measure as part
| of the notation for integrals
|
|
| I find that the technology of _differential forms_
| provides a unified viewpoint of all of the above.
| I find this viewpoint very very compelling. It
| is a powerful light that illuminates some previously
| dark corners.
|
| I realize that not everyone was born with this
| viewpoint. This is why I worked so hard to invent
| the fish-scale diagrams. I find one-forms easy to
| visualize ... but I need the diagrams if I'm going
| to explain them to other folks.
|
| One forms are, technically, vectors. A one-form
| that is a function of position is a vector field.
| There is a contrast:
| pointy vector one-form
| familiar less familiar
| arrow with tip and tail contour lines and/or fish scales
| column vector row vector
|
| So, yes, I am quite serious when I say:
|
| T is a function of state, a scalar field
| V is a function of state, a scalar field
| P is a function of state, a scalar field
| dT is a function of state, a one-form field, exact
| dV is a function of state, a one-form field, exact
| dP is a function of state, a one-form field, exact
| P dV is a function of state, a one-form field, non-exact
| W is not a function of state.
| P dV does not equal d(W) or d(anything)
|
| I recently added pictures of dT, dV, and dP to
| my writeup on this subject:
| http://www.monmouth.com/~jsd/physics/thermo-forms.htm
|
| I need to add a picture of P dV and/or other non-exact
| forms, but I haven't gotten around to it.
|
| To say the same thing in slightly different words:
| You don't need to think of dE as a "small change in E".
| You can quite advantageously think of it as the
| exterior derivative of E, i.e. the slope of E, i.e.
| the contour-lines of E. This is a huge win.
|
| To my ears, writing d(W) "=" P dV is equivalent to saying:
|
| "W is not a function, but if it were a function, and if
| we knew how to differentiate it, its derivative would
| equal P dV."
|
| What a load of doubletalk! Yuuuuuck!
|
| A lot of people, including some people on this list,
| have gotten into trouble by assuming, tacitly or
| otherwise, that d(W) is the derivative of some
| function W. It's high time to formulate things in
| ways that make that mistake less tempting.
|
| ================
|
| Bob also quoted a passage from Bridgman. I find
| that it supports my position quite nicely:
|
| -- Bridgman is happy that dE is exact.
| It can be integrated freely.
| -- He is aware the P dV and T dS are not exact.
| -- He is aware the writing them as "dW" and "dQ"
| is not kosher. But he doesn't understand the
| problem well enough to fix it.
| -- He can't make up his mind whether "d" means
| a differential, which can be integrated, or
| whether "d" merely means something small,
| even though it isn't a differential. So he
| winds up with a misch-mash, with one thing
| on one side of the equation and something
| else on the other side.