Specific fundamental relation and valid response charts

Let $U\subset (0,\mathrm{\infty })\times \mathbb{R}$ be open, with coordinates $(v,s)$, and let $e:U\to \mathbb{R}$ have continuous total derivatives through order two (write $e\in C^2(U)$). The pair $(U,e)$ is a simple-fluid state relation for the purposes of this mathematical library. At $(v,s)\in U$, $v$ is the specific-volume coordinate, $s$ the specific-entropy coordinate, and $e(v,s)$ the specific-energy value. Define $T(v,s):=\frac{\mathrm{\partial }e}{\mathrm{\partial }s}(v,s),p(v,s):=-\frac{\mathrm{\partial }e}{\mathrm{\partial }v}(v,s),\rho :=\frac{1}{v}\mathrm{.}$ In these partial derivatives the other state coordinate is held fixed. Write $e_{vv},e_{vs},e_{sv},e_{ss}$ for the corresponding second partial derivatives, set $H_e:=\left(\begin{array}{cc}{e}_{vv}& e_{vs}\\ e_{sv}& e_{ss}\end{array}\right),$ and define the coordinate-Hessian determinant $J_e:=e_{vv}{e}_{ss}-e_{vs}{e}_{sv}\mathrm{.}$ The local thermal-chart hypothesis at a state is $T>0$ and $e_{ss}>0$; it is a pointwise hypothesis and does not assert a global inverse. The conditions $e_{vv}>0$ and $J_e>0$, when required, are separate pointwise response restrictions. The definition itself imposes no sign condition on $e_{vv}$ or $J_e$ and no invertibility or convexity condition on all of $U$.

Under the physical SI interpretation only, $v$ has units ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}$, $s$ has units $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$, $e$ has units $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$, $T$ has units $\mathrm{K}$, $p$ has units $\mathrm{P}\mathrm{a}$, and $\rho$ has units $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$. This is a mathematical definition of a state relation and makes no claim that an actual material obeys it.

Rohan Abeyaratne, Continuum Mechanics, Volume II, Chapter 12, §§12.1–12.2, printed pp. 347–360