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Re: [Phys-l] thermodynamic dot products, or not

On 01/12/2010 06:44 AM, Rauber, Joel asked:

If one can identify curves of constant E (or whatever coordinate),
doesn't the idea of a tangent to the curve imply some sort of notion
of directionality?

That's a good, incisive question.

Here's the deal:

Remember that except in trivial cases, thermodynamics
is highly multi-dimensional ... and the set of
interesting variables is overcomplete. As a simple
example, suppose we have a three dimensional state-
space and five variables (C, D, E, F, G). Then the
contour of constant E is a two-dimensional shell,
not a "curve" strictly speaking.

Now suppose we pick two variables, E and F. Then dE
represents the shells of constant E and dF represents
the shells of constant F. The intersection dE /\ dF
suffices to define a good "notion of directionality".
It is the direction of constant E and F.

Note that the direction of constant E and F cannot be
considered the "E axis" or the "E direction", since
usually we think of the E axis as "the" direction in
which E is changing ... and by construction E is not
changing at all along the dE /\ dF direction. Similarly
the dE /\ dF direction cannot be considered the C, D,
F, or G direction.

If we had only three variables, we could *maybe*
think in terms of a direction where "everything" is
constant except E. But that is a bad idea in general,
and fails miserably when the set of variables is
overcomplete. Some days you want to let E change
along the dC /\ dD direction, sometimes along the
dC /\ dF direction, et cetera; there is no good
way to decide what is "the" direction in which E

I suppose I could paraphrase by asking if tangent spaces exist for
the Thermodynamic State Space?

Sure, a tangent space exists at each point.

But there's no dot product in the tangent space.