>> I don't know what "dbar" means. I'll take it
>> to mean the same as "d" until I hear otherwise.
On 05/12/2003 08:32 AM, Vern Lindberg wrote:
>
> As seen in any standard classical thermodynamics text, d-bar, sometimes
> written d' is the symbol for an inexact or imperfect differential.
> Refer for example to Sears and Salinger (Thermodynamics, Kinetic
> Theory, and Statistical Thermodynamics), Zemansky (Heat and
> Thermodynamics), or Callen, (Thermodynamics and an introduction to
> Thermostatistics).
I still don't like it. I know it is rude and dangerous
to speculate about other people's thought processes, but
I suspect something like the following may be going on:
1) Fact: any exact one-form can be written as d(something).
2) Fact: a non-exact one-form cannot be written as d(anything).
3) Non-experts think it would be nice to write all
one-forms as d(something), perhaps thinking that
the "d" shows it is a one-form.
4) They've been told they can't write d(W) so they
write dbar(W).
IMHO this violates several principles of good notation
and good pedagogy. One principle is "lead them not into
temptation". Alas dbar(W) practically invites people
to confuse it with d(W).
It's like starting a restaurant called MacDonalds. The
people at McDonald's are not going to be amused. There's
too much potential for confusion.
In any case it is unnecessary. There is no rule that
says every one-form must have a name starting with "d".
Indeed fact (2) tells us that non-exact one-forms must
not have names starting with "d". Choose another name.
Speculating again, I suspect some people have put
themselves into a small box, where all their thoughts
about one-forms come from their experience with exact
one-forms, with little or no experience with non-exact
one-forms. Writing a "d" in front of all one-forms is
presumably a symptom of this. People need to break out
of this box!
> As I recall, the 1/T in math is called an
> integrating factor to turn an inexact (path dependant) differential
> into a perfect (path independant) differential.
Well, it rather depends on what non-exact one-form
we're talking about. Clearly:
-- if you start with T dS, then 1/T is an integrating
factor, since dS is manifestly exact.
-- if you start with P dV, then 1/P is an integrating
factor (and 1/T almost certainly is not).
========================
By the way, not quite "any" thermo book uses dbar. The
really dumb ones just write
dW "=" P dV
(without the dbar) and the really smart ones use neither
d(W) nor dbar(W). To my ears, dbar sounds like a
sophomoric worst-case situation, such as comes from
having been told that there's a problem but not knowing
how to fix it.
My recommended fixes are:
*) Preferably, T dS can just be called T dS.
P dV can just be called P dV.
*) If you really need to give these things nicknames,
you can write:
w = P dV
q = T dS
where w and q are perfectly good non-exact one-forms.
-) Do not write P dV = d(anything) or dbar(anything).
Do not write T dS = d(anything) or dbar(anything).
*) If you integrate a non-exact one-form along some
path, say exactly what the path is.
==================
I was just flipping through some of my books. I would
like to say some good things about
_Thermal Physics_ by Kittel and Kroemer.
I didn't read it closely just now, but I skimmed many
pages without seeing anything objectionable, which is
a lot more than I can say for most textbooks, especially
thermo textbooks. And indeed I saw several things that
I really liked:
-- It did _not_ start with ideal gasses. This is good.
Waaay too many students get the impression that
thermo only applies to ideal gasses, and/or they
develop misconceptions from spending too much time
on ideal gasses (e.g. the notion that all thermal
energy is kinetic energy).
-- It takes a statistical approach from the get-go,
including a statistical definition of entropy.
-- It states the first law of thermodynamics the way
I like to see it: Energy is conserved. Period.
-- Nowhere do I see d(W) or d(Q) or anything like that.
W is calculated by integrating P dV along a
specified path.