This equation shows how the internal energy, , depends on independent variations of the entropy, , and the volume, . If and are considered to be the two independent parameters that specify the system then

(6.83) |

It immediately follows that

Now, Equations (6.82) and (6.84) must be equivalent for all possible values of and . Hence, we deduce that

We also know that

(6.87) |

or

(6.88) |

In fact, this is a necessary condition for in Equation (6.84) to be an exact differential. (See Section 4.5.) It follows from Equations (6.85) and (6.86) that