> Traditionally:
> 1) V is a function of state; i.e. its value is uniquely
> specified for a given system state
Undoubtedly.
> 2) dV is an infinitesimal (often within the context of some
> limiting process) increment in the value of this state function V;
I don't have a problem with that.
> 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".
Oops, lost me there. If we were talking about DELTA V,
then I would agree that two states were involved. But
d(V) is defined, as we agreed above, as the limit where
the states become arbitrarily close together. So to
know d(V) we need only consider the properties of the
system in one arbitrarily small region. This makes
d(V) a function of state.
> Is the usurpation of the "d" notation by GA for its "one
> form" scheme now forbidding this language?
I see no usurpation here. I see nothing forbidden here.
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). The same goes for
d(E) and d(P) and d(S) and so forth.
Things that were illegitimate before are illegitimate still.
All that is changed is that we have a clearer understanding
of why they were/are illegitimate. Bridgman _knew_ there
was something fishy about d(W) but he couldn't explain it.
Now we can explain it. We can draw pictures of it. Go
ahead and say that d(V) is a small variation in V. But d(W)
is not a small variation in W, because there is no W function
to undergo small variations.
To say the same thing another way: Bob more-or-less defines
d(V) to be the directional derivative, but doesn't explicitly
state what direction. The direction can always be specified
later. That's fine. I don't have a problem with that.
But I consider that viewpoint formally indistinguishable
from a gradient!! Given a gradient, I can always form a
directional derivative later by contracting the gradient
with a vector that specifies the chosen direction.
I don't consider gradients as a usurpation of directional
derivatives. I don't consider gradients as forbidding
directional derivatives. I consider gradients as formally
indistinguishable from directional derivatives when the
direction hasn't been specified.
Secondarily, I prefer to visualize gradients as a field
of one-forms. That's safer, especially in thermodynamics
where there is no metric that to tell us the length of
V or the dot-product between V and E. But if you wish
to visualize the gradient as a field of pointy vectors,
go ahead. This is a minor point, and I hope it is not
too much of a distraction.
>> 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