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*From*: John Denker <jsd@av8n.com>*Date*: Tue, 12 Jan 2010 09:48:20 -0700

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

changes.

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.

**References**:**[Phys-l] T dS versus dQ***From:*John Denker <jsd@av8n.com>

**[Phys-l] thermodynamics of dissipation***From:*John Denker <jsd@av8n.com>

**Re: [Phys-l] thermodynamics of dissipation***From:*Stefan Jeglinski <jeglin@4pi.com>

**[Phys-l] thermodynamic dot products, or not***From:*John Denker <jsd@av8n.com>

**Re: [Phys-l] thermodynamic dot products, or not***From:*"Rauber, Joel" <Joel.Rauber@SDSTATE.EDU>

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