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

[trebor at WINBEAM.COM (Bob Sciamanda)]

For completeness, here is the usual (extremely simple and brief) derivation
of JD's Eq(14) from JD's Eq(8):


In the general case, arbitrary increments in the variables E,S,V and N are
related as:

dE=(dE/dS)v,n (dS) + (dE/dV)n,s (dV) + (dE/dN)s,v (dN)    =>   JD's Eq  (8)

Impose Constant E and Constant N :

0 =(dE/dS)v,n (dS)e,n +(dE/dV)n,s (dV)e,n + 0

Divide by (dV)e,n :

0 =(dE/dS)v,n (dS)e,n/(dV)e,n +(dE/dV)n,s

0 = (dE/dS)/v,n (dS/dV)e,n + (dE/dV)n,s

(dE/dV)n,s  /   (dE/dS)/v,n = - (dS/dV)e,n              => JD's Eq (13)

P/T =  (dS/dV)e,n                                          => JD's Eq (14)


Bob Sciamanda
Physics, Edinboro Univ of PA (Em)
http://www.winbeam.com/~trebor/
trebor@winbeam.com