Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 5.4-2

The fundamental equation of a system tells you how entropy depends on the system's natural variables—it's the master equation from which all thermodynamic properties flow. For electromagnetic radiation, you'll use the fact that pressure, energy density, and entropy density are all related through the Stefan–Boltzmann law and the radiation equation of state ($P = U/3V$). The key insight is recognizing which variables are "natural" for each representation: part (a) asks for entropy as a function of internal energy and volume (the $S[U,V]$ form), while part (b) asks you to perform a Legendre transformation to swap one variable for its conjugate, replacing $U$ with $P/T$ as the independent variable.

The fundamental equation of a system tells you how entropy depends on the system's natural variables—it's the master equation from which all thermodynamic properties flow. For electromagnetic radiation, you'll use the fact that pressure, energy density, and entropy density are all related through the Stefan–Boltzmann law and the radiation equation of state ($P = U/3V$). The key insight is recognizing which variables are "natural" for each representation: part (a) asks for entropy as a function of internal energy and volume (the $S[U,V]$ form), while part (b) asks you to perform a Legendre transformation to swap one variable for its conjugate, replacing $U$ with $P/T$ as the independent variable.