In thermodynamics, the Gibbs fundamental equation, also called the combined law of thermodynamics, states that the change in energy of a system can always be written as the product of an intensive and an extensive parameter. This takes the form of: [1]

$\Delta U = \sum_{i} X_{i} \Delta Y_{i} \,\!$
where Xi is an intensive quantity, such as pressure or temperature, and Yi is an extensive quantity, such as volume. The expression on the right side of this equation is called a Pfaffian form.

History
The Gibbs equation, was first derived by American engineer Willard Gibbs in the first two pages to his 1873 "Graphical Methods in the Thermodynamics of Fluids" as: [2]

dε = tdη - pdv

where, according to Gibbs, ε is the energy, t the temperature, η the entropy, p the pressure, and v the volume of the body in a state.

Advanced systems
When a system does work or has work done on it the system internal energy is effected, such as when it has a volume dV change due the action of a pressure P, i.e. pressure-volume work (dW = pdV), an expansion of a material section or elongation dl in response to the application of a force F, i.e. elongation work (dW = Fdl), does work (μdn) by transporting a certain number of atoms or molecules dn against a concentration gradient, where μ is the chemical potential, i.e. transport work (dW = μdn), or, among many other possible examples, does work in the action of charge transport in which an amount of charge dq is transported against an electric potential ψ, i.e. electrical work (dW = ψdq), then the Gibbs fundamental equation becomes: [3]

$dU = TdS - PdV + Fdl + \sum_{i=1}^m \mu_i dn_i + \Psi dq\,$

The application of this type of quantification of internal energy of human social systems, e.g. how does one quantify the transport of a singe human molecule across the boundary of a system or how human pressures, temperatures, or volumes are quantified, etc., is one of the most advanced topics in human thermodynamics. [4]

References
1. Schmitz, John E.J. (2007). The Second Law of Life: Energy, Technology, and the Future of Earth as We Know It, (pg. 165). William Andrew Publishing.
2. Gibbs, J. Willard. (1873). "Graphical Methods in the Thermodynamics of Fluids", Transactions of the Connecticut Academy, I. pp. 309-342, April-May.
3. Glaser, Roland. (2000). Biophysics, (pgs. 110-11). Springer.
4. (a) Thims, Libb. (2007). Human Chemistry (Volume One), (preview). (Index: "human thermodynamics", pgs. x, 14, 74, 79, 107, 110, 204, 273, 315). Morrisville, NC: LuLu.
(b) Thims, Libb. (2007). Human Chemistry (Volume Two), (preview), (Ch. 16: "Human Thermodynamics", pgs. 653-702). Morrisville, NC: LuLu.