# Taylor expansion, thermodynamics

1. May 25, 2013

### fluidistic

1. The problem statement, all variables and given/known data
I'm asked to show that the relation $S(U+\Delta U, V+ \Delta V , n )+S(U-\Delta U, V- \Delta V , n ) \leq 2 S(U,V,n)$ implies that $\frac{\partial ^2 S}{\partial U ^2 } \frac{\partial ^2 S}{\partial V ^2}- \left ( \frac{\partial ^2 S }{\partial U \partial V} \right ) \geq 0$.

2. Relevant equations
Taylor expansion of a function of several variables: https://en.wikipedia.org/wiki/Taylor_series#Taylor_series_in_several_variables

3. The attempt at a solution
The book gives some help, it says to first make a Taylor expansion of the left side to the second order in $\Delta U$ and $\Delta V$. So I should reach the equation $S_{UU}(\Delta U)^2+2S_{UV}\Delta U \Delta V + S_{VV} (\Delta V)^2 \leq 0$, then I'd have to rewrite this expression and I should reach the answer.
However I'm having a problem at understanding what the book means by "in $\Delta U$ and $\Delta V$". Does it means I should expand S and U around the point $(\Delta U , \Delta V, n)$?
Or...

Edit: I've expanded S around the point (U,V,n) which seemed to make sense to me, but then I don't reach what I should. Instead I reach $-S+\Delta U S_U + \Delta V S_V + \frac{1}{2} \left [ (\Delta U)^2 S_{UU} + 2 \Delta U \Delta V S_{UV} + (\Delta V )^2 S_{VV} \right ] \leq 0$ where the functions are evaluated in $(U,V,n)$.

Edit2: NEVERMIND!!!! I forgot a term to make the Taylor expansion. I didn't do it yet but at first glance this is the right thing to do, i.e. making the expansion around (U,V,n)!
Edit3: problem solved, I've reached the final result!!!

Last edited: May 25, 2013