A key feature of thermodynamics is the fact that the internal energy,
\(E_{\text {int }}\) of a system and its entropy, \(S\), are state variables; that
is, they depend only on the thermodynamic state of the system and not on the
processes by which it reached that state (unlike, for example, the heat
content, \(Q\) ). This means that the differentials \(d E_{\text {int }}=T d S-p
d V\) and \(d S=\) \(T^{-1} d E_{\text {int }}+p T^{-1} d V,\) where \(T\) is
temperature (in kelvins), \(p\) is pressure, and \(V\) is volume, are exact
differentials as defined in calculus. What relationships follow from this
fact?