dE = dq + dw

Depending on how one was educated, the First Law can be written as

dE = TdS - pdV

which means that E is a function of S and V, i.e., E(S,V). Thus

$$\left ( \frac{\partial E}{\partial S} \right )_V = T$$

and

$$\left ( \frac{\partial E}{\partial V} \right )_S =-p$$

The question asked in this piece is, how does it happen that when one invokes the ``classical'' Legendre Transformation:

dE + d(pV) = dH = TdS - pdV + pdV + Vdp = Tds + V dp

H turns out to be a function of S and p! Why does this work? The answer is, of course, that the Legendre Transformation is more than meets the eve (above).