the thermodynamic square
The four common potentials and their Maxwell relations are easy to muddle: which sign goes where, which variable is conjugate to which. The thermodynamic square is a mnemonic — a labelled square you can sketch in a corner of the page — that lets you read off every differential and every Maxwell relation without memorizing any of them. It is pure bookkeeping, but wonderfully efficient bookkeeping.
Draw a square. Label its four sides with the potentials U, F, G, H and its four corners with the natural variables S, V, T, p. Each potential sits between its two natural variables, so reading its neighbours gives its differential (e.g. F lies between T and V, giving dF = -S dT - p dV, with signs supplied by diagonal arrows). The corners also generate the Maxwell relations directly. A classic mnemonic for the letters is 'Good Physicists Have Studied Under Very Fine Teachers'.
It is a teaching and recall aid, not new physics — everything on it follows from the definitions and the equality of mixed partials. Different textbooks orient the square differently and place the sign-carrying arrows differently, so always re-derive the sign convention from one relation you are sure of before trusting the rest.
Reading the Gibbs corner: G sits between T and p, so dG = -S dT + V dp. Following the two arrows out of the T-p edge to the opposite corners gives the Maxwell relation (dS/dp)_T = -(dV/dT)_p in one glance — no derivation needed.
One square, sketched from memory, regenerates all four differentials and all four Maxwell relations.
The thermodynamic square encodes no physics beyond the potentials' differentials and the Maxwell relations; its only pitfall is the sign convention, which varies between books — anchor it to one relation you know by heart.