[BUG] partial derivative product rule

[Tevian Dray]

>>>>> Tevian Dray writes:
    TD> I see no physical context here ...

>>>>> Matthias Kawski writes:
    MK> ... some plausible examples from thermodynamics (a little more
    MK> sophisticated models than Boyle's law) where it might be hard
    MK> experimentally to measure one desired derivative (forgot which was
    MK> difficult to hold constant), but this nice relationship allowed one to
    MK> measure two other partials (supposedly easy to hold the corresponding
    MK> quantities fixed), and then obtain the desired one via simple
    MK> arithmetic (-- but the minus sign is critical!).

>>>>> Martin Jackson writes:
    MJ> A physicist looking at this result might think in terms of dimensions.
    MJ> If x, y, and z represent physical quantities (pressure, volume,
    MJ> temperature in the Boyle's law example Matthias mentions in his post)
    MJ> with specified physical dimensions, then this particular product of
    MJ> partial derivatives is dimensionless.

>>>>> Allen Wasserman writes:
    AW> This result is so important ... that I give it a silly name so that
    AW> students will remember it.  I have called it a "twisted" chain rule or
    AW> a cyclic chain rule, depending upon which came to mind at the moment.

I discussed this with Allen Wasserman, the thermodynamics expert in our
physics department.  Looks like I blew it -- this is clearly an important
result!  But what Allen calls the "twisted chain rule" is slightly different
from the problem Suda pointed out in Stewart.  I'll start by paraphrasing
Allen's derivation of this chain rule, relating it to Suda's question, then
discuss Allen's example.

In thermodynamics, there's an equation of state of the form
				f(p,V,T) = 0,
so that any one of these variables can be viewed as an implicit function of the
other two.  For instance,
		       dT = (dT/dp)_V dp + (dT/dV)_p dV
where I have tried to mimic standard thermodynamics notation as much as
possible: the derivatives are of course partial derivatives, and the subscript
(denoted by _) indicates which variable is being held fixed.  If T is
constant, the left-hand-side of this expression vanishes, resulting in
		     (dV/dp)_T = - (dT/dp)_V / (dT/dV)_p
This is the "twisted chain rule".  To get the "triple chain rule" in Stewart,
note first of all that the same argument shows that
			  (dp/dV)_T = 1 / (dV/dp)_T
and similarly for other combinations of the variables, so that
		    (dT/dV)_p * (dV/dp)_T * (dp/dT)_V = -1
which is Stewart's result after suitably relabeling the variables.

In Allen's example (attached as a PDF file), the goal is to rewrite the
isoentropic compressibility in terms of other, more easily measured
quantities, namely the isothermal compressibility and the heat capacities
(C_V and C_p), all of which are defined in the attachment.  To follow the
derivation, you also need to know that the entropy S can be viewed as a
function of any two of the basic thermodynamic variables p, V, T, so that in
particular there is also a "twisted chain rule" involving p, V, and S.

Tevian

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