三乘積法則(triple product rule)是關於偏導數的一個恆等關係式,其表達式為:
- 注釋:每一個變量可視作另外兩個變量的函數。偏導數的下標表示在此變量為常數的條件下求導。
三乘積法則用於熱力學關係式的推導。例如溫度、壓力和體積之間的關係滿足:
![{\displaystyle \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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa375a5650ca1974a52df8aba4acbe90b0e3eaef)
利用三乘積法則,可以將不易測量的關係用容易測得的物理量代替,如:
。
下面給出一個非正式的推導。設有函數f(x, y, z) = 0。若將z表示為x和y的函數,則全微分dz等於
![{\displaystyle dz=\left({\frac {\partial z}{\partial x}}\right)_{y}dx+\left({\frac {\partial z}{\partial y}}\right)_{x}dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20946f9c56cde2f4c82adfc970890b311f1d4b7d)
在dz = 0的軌跡上,x和y之間滿足
![{\displaystyle dy=\left({\frac {\partial y}{\partial x}}\right)_{z}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ab2a25f55fdb3279b6d90d0ea1c5222d5ec360)
於是將dz = 0帶入上式,
![{\displaystyle 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3edad542b6c12dc9e72251310b696c12b3f55d8d)
重排得
![{\displaystyle \left({\frac {\partial z}{\partial x}}\right)_{y}=-\left({\frac {\partial z}{\partial y}}\right)_{x}\left({\frac {\partial y}{\partial x}}\right)_{z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f04b8d6120124641051c2f77d7e4bdb195d003fe)
將所有偏導數移到等式左邊,
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46374395888e37b111c83826ef8d05663a2ab7c7)
此證明假定了偏導數存在,以及全微分dz存在,偏導數不為零從而能取倒數。數學分析的正式證明能避免這些隱含假定。
- 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.