Maxwell Relations
Definition
Four elegant equations linking quantities you can't measure (like entropy change with volume) to ones you can (like pressure change with temperature).
Technical definition
Derived from exact differentials of U, H, F, and G — e.g., (∂T/∂V)_S = −(∂P/∂S)_V. They convert hard derivatives into measurable ones.
Example
To find how entropy changes when a gas expands, use (∂S/∂V)_T = (∂P/∂T)_V — measure pressure versus temperature instead of the impossible entropy directly.
Common confusion
Maxwell relations are not new physics — they are mathematical identities following from state functions having exact differentials. The physics went in when U, H, F, G were defined.