[Phys-L] Re: geometric interpretation of partial derivatives

[jsd at AV8N.COM (John Denker)]

Referring to
   http://www.av8n.com/physics/partial-derivative.htm#eq-P/T

Bob Sciamanda wrote:

> Your Eq (14) is derived in standard texts ( Callen's, "Thermodynamics" or
> Margenau & Murphy's "The Mathematics of Physics & Chemistry")

Yup.

> The
> derivation is identical to yours, but uses standard partial derivative
> notation.

How identical?

The key step in "my" derivation is to divide

   (partial E / partial V @ const N,S)
  -------------------------------------     [*]
   (partial E / partial S @ const N,V)

and cancel the obvious common factor.  In "my" notation, it's obvious, but as
written here, using "standard partial derivative notation", I don't see any
obvious common factor.

I never said that the conventional calculation wasn't doable;  I just said the
method was hard to remember and provided little insight.

But maybe I'm wrong ... so I ask the phys-l readers:  Given the expression [*]
above, how to you simplify it?  What's the next step?  Can you prove the
correctness of that step, or are you just doing it from rote memory?  What's
the physical meaning of that step?