(a) Show that, for any magnetizable material, the heat capacities at constant field \(H\) and at constant magnetization \(M\) are connected by the relation

\[
C_{H}-C_{M}=-T\left(\frac{\partial H}{\partial T}ight)_{M}\left(\frac{\partial M}{\partial T}ight)_{H}
\]

(b) Show that for a paramagnetic material obeying Curie's law

\[
C_{H}-C_{M}=C H^{2} / T^{2},
\]

where \(C\) on the right side of this equation denotes the Curie constant of the given sample.