Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is a central principle in mathematics that connects the concepts of differentiation and integration, two fundamental operations in calculus. This theorem provides a powerful framework for understanding how the two processes are inverses of each other. The key figures associated with its development are Isaac Newton and Gottfried Leibniz, who independently formulated their versions of calculus in the late 17th century. While Newton's work on calculus was published in his influential 1687 treatise, "Principia," Leibniz introduced his findings earlier in 1684, which led to a significant dispute over priority and credit for the discovery of calculus.
The theorem itself establishes that if a function is continuous over a closed interval, then the integral of that function can be computed using its antiderivative. This relationship is essential in various applications, from physics to engineering. Leibniz also contributed to notation in calculus, introducing symbols like dx and dy for differentials, and the integral sign (∫) for integration. The work of both mathematicians laid the groundwork for the modern understanding of calculus and its usage in solving complex problems. This theorem not only highlights the historical significance of calculus but also remains a pivotal tool in mathematical analysis today.
On this Page
Subject Terms
Fundamental Theorem of Calculus
When it comes to understanding the concept of the fundamental theorem of calculus, two names are prominent: Isaac Newton and Gottfried Leibniz. Although either one of these mathematicians can possibly be credited with the development of the fundamental theorem of calculus, it is true to say that neither was the first to do so. The first published work on the fundamental theorem was by James Gregory, known to Newton. Earlier known work was by Pierre de Fermat. By 1666, Newton made free use of algebra and a variety of algorithmic devices and notations in formulating a systematic method of differentiation.
It was while Newton was working on infinite series that he composed, in 1669, "De analysi per aequationes numero terminorum infinitas" (pub. 1711). In that volume, Newton presented the first systematic account of his chief mathematical work: the calculus. Newton’s work here was not far removed from that published by Isaac Barrow (the most important of Newton’s mentors) in 1670. Newton’s notation was close to Barrow’s. Newton circulated De analysi among friends—to Leibniz and Oldenburg—but refrained from publishing it until 1687 in Philosophiae naturalis principia mathematica, the most admired scientific treatise of all times.
The inverse of differentiation is integration. Newton’s prime contribution was in the consolidation of differentiation and integration into a general algorithm applicable to all functions, whether algebraic or transcendental. In the first edition of Principia (1687), Newton admitted that Leibniz possessed a similar method. In the third edition of 1726 (published when Newton was age 83 and following the bitter quarrel between adherents of Newton and Leibniz concerning the independence and priority of the discovery of the calculus), Newton deleted the reference to the calculus of Leibniz. Newton’s discovery came before that of Leibniz (by about ten years), but the discovery by Leibniz was independent of that of Newton. Moreover, Leibniz is entitled to priority of publication because he printed an account of his calculus in 1684 in the Acta Eruditorum (compared to July 5, 1687 edition).
By 1676, Leibniz had arrived at the same conclusion that Newton had reached several years earlier: He possessed a method that was highly important due to its generality. Leibniz therefore developed an appropriate notation, still used today. He fixed on dx and dy for the smallest possible differences (differentials) in x and y. He introduced the symbol ∫—an enlarged letter s for sum—in integral calculus: ∫ y dx. In his 1684 publication, Leibniz introduced the following formulae
for products, quotients, and powers (or roots). These formulae were derived by neglecting infinitesimals of higher order.
In 1716, Leibniz challenged Newton (then age 73) to find the orthogonal trajectories of a one-parameter family of plane curves. Within a few hours, despite his age, Newton solved the problem and gave a general theorem for the task.
Bibliography
Cruz-Filipe Luìs. "A Constructive Formalization of the Fundamental Theorem of Calculus." Lecture Notes in Computer Science 2646 (2003): 108-26.
Körner , Thomas W. Calculus for the Ambitious. New York: Cambridge UP, 2014.
Ross, Kenneth A. Elementary Analysis: The Theory of Calculus. 2nd ed. New York: Springer, 2013.
Sobczyk, Garret, and Omar L. Sànchez. "Fundamental Theorem of Calculus." Advances in Applied Clifford Algebras 21 (2011): 221-31.