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 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.