The Chemical Potential is defined such that: 
.
The Enthalpy, H, is defined as H = U = pv and its identity is dH = τ dσ + V dp + μ dN.
An extensive variable of a system doubles upon duplication of the system. Important extensive variables are the volume V, the entropy σ, the number N and the energy U. Compare to intensive variables.
The Gibbs Free Energy, G, is defined as G = U - τσ + pV and its identity is dG = - σ dτ + V dp + μ dN.
The Helmholtz Free Energy, F, is defined as F = U - τσ and its identity is dF = - σ dτ - p dV + μ dN.
An intensive variable of a system remains constant upon duplication of the system. Important intensive variables are the temperature τ, the chemical potential μ, and the pressure p. Compare to extensive variables.
The Legendre Transform is a mathematical tool that we employ to change variables in expression of then energy, such as defining F = U - τσ in order to change variables from σ to τ in the energy.
The Maxwell Relations give relationships between the partial derivative of one variable with respect to a variable in a different pairing and the corresponding cross partial derivative of the other variable in the second pairing with respect to the other variable in the first.
The pressure is defined such that: p = - 
, and is one of the important intensive variables in thermodynamics.
The Thermodynamic Identity relates the energy U to the 6 variables we have discussed:
