In thermodynamics, for a system such as an enclosed gas, the internal energy \(U(S, V)\) can be expressed in terms of the independent variables of entropy \(S\) and volume \(V\), such that \(d U=T d S-P d V\), where \(T\) is the temperature and \(P\) the pressure. Suppose we want to find a related function in which the volume is to be eliminated in favor of the pressure, using a Legendre transformation.

(a) Which is the passive variable, and which are the active variables?

(b) Find an expression for the new function in terms of \(U, P\), and \(V\). (The result is the enthalpy \(H\) or its negative, where the enthalpy \(H\) is unrelated to the Hamiltonian \(H\).)