三乘积法则

三乘积法则（triple product rule）是关于偏导数的一个恒等关系式，其表达式为：

\left({\frac {\partial x}{\partial y}}\right)_{z}\left({\frac {\partial y}{\partial z}}\right)_{x}\left({\frac {\partial z}{\partial x}}\right)_{y}=-1.

注释：每一个变量可视作另外两个变量的函数。偏导数的下标表示在此变量为常数的条件下求导。

三乘积法则用于热力学关系式的推导。例如温度、压力和体积之间的关系满足：

\left({\frac {\partial p}{\partial T}}\right)_{V}\left({\frac {\partial V}{\partial p}}\right)_{T}\left({\frac {\partial T}{\partial V}}\right)_{p}=-1.

利用三乘积法则，可以将不易测量的关系用容易测得的物理量代替，如：

\left({\frac {\partial x}{\partial y}}\right)_{z}=-{\frac {\left({\frac {\partial z}{\partial y}}\right)_{x}}{\left({\frac {\partial z}{\partial x}}\right)_{y}}}

。

推导

下面给出一个非正式的推导。设有函数f(x, y, z) = 0。若将z表示为x和y的函数，则全微分dz等于

dz=\left({\frac {\partial z}{\partial x}}\right)_{y}dx+\left({\frac {\partial z}{\partial y}}\right)_{x}dy

在dz = 0的轨迹上，x和y之间满足

dy=\left({\frac {\partial y}{\partial x}}\right)_{z}dx

于是将dz = 0带入上式，

0=\left({\frac {\partial z}{\partial x}}\right)_{y}\,dx+\left({\frac {\partial z}{\partial y}}\right)_{x}\left({\frac {\partial y}{\partial x}}\right)_{z}\,dx

重排得

\left({\frac {\partial z}{\partial x}}\right)_{y}=-\left({\frac {\partial z}{\partial y}}\right)_{x}\left({\frac {\partial y}{\partial x}}\right)_{z}

将所有偏导数移到等式左边，

\left({\frac {\partial x}{\partial y}}\right)_{z}\left({\frac {\partial y}{\partial z}}\right)_{x}\left({\frac {\partial z}{\partial x}}\right)_{y}=-1

此证明假定了偏导数存在，以及全微分dz存在，偏导数不为零从而能取倒数。数学分析的正式证明能避免这些隐含假定。

Elliott, JR, and Lira, CT. Introductory Chemical Engineering Thermodynamics, 1st Ed., Prentice Hall PTR, 1999. p. 184.
Carter, Ashley H. Classical and Statistical Thermodynamics, Prentice Hall, 2001, p. 392.

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