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已知：气体的状态方程（即已知 p 、 T 、 V 之间任意的偏微分关系 \left (\frac {\partial p} {\partial T}\right)_V 等等）和 C_p （或 C_V，二者知道一个即可），也知道体系的 热力学函数 p 、 T 、 V 、 S。 推求：热力学函数 p 、 V 、 T 、 U 、 H 、 S 、 A （有的地方写作 F）、 G 之间的偏微分关系 \left (\frac {\partial x} {\partial y}\right)_z ，共336个偏微分式。 如 \left (\frac {\partial U} {\partial H}\right)_G. 本文使用的推导思路大概是教材中的方法，比较适合写在试卷的证明题上。

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P.V.\int_{-\infty }^{+\infty}f(x)dx=\lim_{a \rightarrow +\infty}{\int_{-a}^{a}f(x)dx} 等号理解为若等式右边极限存在则左边取值为右边的极限值 类似地可以定义有界区间上的无界函数反常积分的柯西主值