As the operational approach to elucidate the history of the formulation and development of the concept of energy, it is important to first trace its mathematical origin, particularly, in its relation to the development of differential calculus and its fundamental concept of time derivative: the time rate of change of certain quantities. Differential calculus was rightfully invented by Sir Isaac Newton in the early or middle 17th century but was not published until the end of that century. This delay generated priority concerns between Leibniz who invented integral calculus during the same period. The fundamental concept of integral calculus is the antiderivative. However, by the fundamental theorem of calculus (study of the infinitesimals of continuous change) it is now accepted by most mathematicians that derivative and antiderivative are inverse dual of each other by the mathematics of functional transformation. The first is the ratio of differences of function undergoing infinitesimal change while the second is the sum of the product of a function and its infinitesimal variation. The first relates to slope of a function, the second, infinite series expansion of same function. The first requires its limit must exist, the second requires convergence of the series.

Concurrently, the mathematical development of optimizing principles in physics started earlier and continued into the calculus of variations. One was Fermat’s principle of least time which has its origin in the emergent science of optics as an offshoot of the invention of the telescope. This principle states that light always travel along the path requiring least time. By the early 18th century, mathematicians had several proofs that nature always attempts to maximize or to minimize certain important physical quantities. The final outcome is Maupertuis’s Principle of Least Action of 1744. This is the basic axiom of quantum mechanics and the constant of action is no other but Planck’s constant found in the quantum theory of light and energy. Later, quantum field theories use the same principle to develop the Lagrangian energy function formalism which at the infinitesimal level of continuous change is a true equivalent of functional differential of energy. Nonetheless, this differential energy is simply a guise of directed energy, a vector quantity. That is a quantity with both magnitude and direction.