# 麦克斯韦关系式

$\frac{\partial }{\partial x_j}\left(\frac{\partial \Phi}{\partial x_i}\right)= \frac{\partial }{\partial x_i}\left(\frac{\partial \Phi}{\partial x_j}\right)$

## 四个最常见的麦克斯韦关系式

$\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial p}{\partial S}\right)_V\qquad= \frac{\partial^2 U }{\partial S \partial V}$
$\left(\frac{\partial T}{\partial p}\right)_S = +\left(\frac{\partial V}{\partial S}\right)_p\qquad= \frac{\partial^2 H }{\partial S \partial p}$
$\left(\frac{\partial S}{\partial V}\right)_T = +\left(\frac{\partial p}{\partial T}\right)_V\qquad= - \frac{\partial^2 A }{\partial T \partial V}$
$-\left(\frac{\partial S}{\partial p}\right)_T = \left(\frac{\partial V}{\partial T}\right)_p\qquad= \frac{\partial^2 G }{\partial T \partial p}$

$U(S,V)\,$内能
$H(S,p)\,$
$A(T,V)\,$亥姆霍兹自由能
$G(T,p)\,$吉布斯自由能

## 麦克斯韦关系式的推导

$dU = TdS-pdV \,$
$dH = TdS+Vdp \,$
$dA =-SdT-pdV \,$
$dG =-SdT+Vdp \,$

$dz = \left(\frac{\partial z}{\partial x}\right)_y\!dx + \left(\frac{\partial z}{\partial y}\right)_x\!dy$

$dz = Mdx + Ndy \,$

$M = \left(\frac{\partial z}{\partial x}\right)_y, \quad N = \left(\frac{\partial z}{\partial y}\right)_x$

$T = \left(\frac{\partial H}{\partial S}\right)_p, \quad V = \left(\frac{\partial H}{\partial p}\right)_S$

$\frac{\partial}{\partial y}\left(\frac{\partial z}{\partial x}\right)_y = \frac{\partial}{\partial x}\left(\frac{\partial z}{\partial y}\right)_x = \frac{\partial^2 z}{\partial y \partial x} = \frac{\partial^2 z}{\partial x \partial y}$

$\frac{\partial}{\partial p}\left(\frac{\partial H}{\partial S}\right)_p = \frac{\partial}{\partial S}\left(\frac{\partial H}{\partial p}\right)_S$

$\left(\frac{\partial T}{\partial p}\right)_S = \left(\frac{\partial V}{\partial S}\right)_p$

## 一般的麦克斯韦关系式

$\left(\frac{\partial \mu}{\partial p}\right)_{S, N} = \left(\frac{\partial V}{\partial N}\right)_{S, p}\qquad= \frac{\partial^2 H }{\partial p \partial N}$

$\left(\frac{\partial y}{\partial x}\right)_z = 1\left/\left(\frac{\partial x}{\partial y}\right)_z\right.$