Added Hopkins 2013 formalism definitions in text, as well as fixed Eqn. 63 (f_{ij} for Pressure-Entropy) as it was unclear.
There is some conversation to be had here about the formalism that is used in this documentation. I think sticking to defining x, \tilde{x}, y, and \tilde{y} and then just following the equations through would be clearer. That is something for another time, though.