热力学中两个常用偏导关系的证明

三个变量 $A$、$B$、$C$，任意一个变量都是其他两个变量的可微函数，证明如下关系：

1. $\left (\frac{\partial A}{\partial B} \right)_C\left (\frac{\partial B}{\partial C} \right)_A\left (\frac{\partial C}{\partial A} \right)_B=-1$

2. $\left (\frac{\partial A}{\partial C} \right)_B=1\bigg/\left (\frac{\partial C}{\partial A} \right)_B$

证明：将 $A$ 对 $B$ 和 $C$ 的依赖关系设为 $f(A,B,C)=0$，于是有

\begin{equation} \left (\frac{\partial f}{\partial A} \right)_{B,C}\mathrm dA+\left (\frac{\partial f}{\partial B} \right)_{A,C}\mathrm dB+\left (\frac{\partial f}{\partial C} \right)_{A,B}\mathrm dC=0 \label{d} \end{equation}

如果 $A$ 不变，由上式，可得

\begin{equation} \left (\frac{\partial f}{\partial B} \right)_{A,C}\left (\frac{\partial B}{\partial C} \right)_A=-\left (\frac{\partial f}{\partial C} \right)_{A,B} \label{pA} \end{equation}

即

\begin{equation} \left (\frac{\partial B}{\partial C} \right)_A=-\left (\frac{\partial f}{\partial C} \right)_{A,B}\Bigg/\left (\frac{\partial f}{\partial B} \right)_{A,C} \label{pAn} \end{equation}

同理，还有

\begin{equation} \left (\frac{\partial C}{\partial A} \right)_B=-\left (\frac{\partial f}{\partial A} \right)_{B,C}\Bigg/\left (\frac{\partial f}{\partial C} \right)_{A,B} \label{pB} \end{equation}

\begin{equation} \left (\frac{\partial A}{\partial B} \right)_C=-\left (\frac{\partial f}{\partial B} \right)_{A,C}\Bigg/\left (\frac{\partial f}{\partial A} \right)_{B,C} \label{pC} \end{equation}

将以上三式相乘，得

\begin{equation} \left (\frac{\partial A}{\partial B} \right)_C\left (\frac{\partial B}{\partial C} \right)_A\left (\frac{\partial C}{\partial A} \right)_B=-1 \label{1} \end{equation}

将 \eqref{pB} 式，交换 $A$ 和 $C$，得

\begin{equation} \left (\frac{\partial A}{\partial C} \right)_B=-\left (\frac{\partial f}{\partial C} \right)_{A,B}\Bigg/\left (\frac{\partial f}{\partial A} \right)_{B,C} \label{pBi} \end{equation}

与 \eqref{pB} 式对照，得

\begin{equation} \left (\frac{\partial A}{\partial C} \right)_B=1\bigg/\left (\frac{\partial C}{\partial A} \right)_B \label{2} \end{equation}