Isaac NewtonPortrait by Sir Godfrey Kneller,Wikipedia/Wikimedia Commons

Gottfried Wilhelm von LeibnizPortrait by Andreas Scheits, Wikipedia/Wikimedia Commons

2.2.1 INTRODUCING THE LEIBNIZESThe Leibniz supplements will show you how more sophisticatedmathematics, and in particular calculus, can be used in economicmodels. You don’t need to use them to understand our models, butthey may help you if you are taking more advanced ormathematics courses. In this first Leibniz, we explain where thename comes from, and introduce some basic notation.

Who invented calculus?Arguably the most famous scientific controversy of all time was betweenSir Isaac Newton and Gottfried Leibniz over who invented calculus.

Sir Isaac Newton (1642–1726) was an English mathematician andphysicist who is recognized as one of the most influential scientists whoever lived. As well as inventing calculus he discovered the law of gravity,laid the foundations of classical mechanics, made major contributions tothe theory of optics and formulated a law of cooling. As Master of the Mintunder three monarchs, Newton founded the gold standard, which was thecore of the international monetary system for almost 200 years.

Newton first used calculus methods in a manuscript published in 1666.The methods were used in his book Mathematical Principles of NaturalPhilosophy, which was published in 1687. He completed his book oncalculus, Method of Fluxions, in 1671, but did not publish it until 1736.

Gottfried Wilhelm von Leibniz (1646–1716) was a Germanmathematician and philosopher. In 1675 he used integral calculus to findthe area under a curve and introduced the elongated S, written , that weuse to represent an integral, and for differential. His work on philosophyfocused on the principle of optimism, according to which God had createdthe best of all possible worlds, although his treatise Theodicee on the topicwas lampooned by Voltaire in his novel Candide.

Newton’s supporters accused Leibniz of plagiarism in his work oncalculus. By the time of his death, his reputation was in decline and he diedin poverty. His reputation has subsequently been rebuilt by bothmathematicians and philosophers.

Modern historians accept that Newton and Leibniz invented calculusindependently, at about the same time. Therefore, to decide whom to namethe calculus supplements after, we tossed a coin. Leibniz won.

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Notation and conventions

Functions of one variable

function of one variable, where is the argument and is the output

first derivative of

alternative notation for the first derivative of

second derivative of

alternative notation for the second derivative of

Integration

function of one variable, where is the argument and is the output

indefinite integral of

definite integral of from to

Functions of two variables

function of two variables, where and are the arguments and is theoutput

partial derivative of with respect to , treating as a constant

partial derivative of with respect to , treating as a constant

second derivative of with respect to , treating as constant

second derivative of with respect to , treating as constant

mixed partial derivative; first derivative of with respect to

mixed partial derivative; first derivative of with respect to

mixed partial derivative when and are equal

Many of the Leibnizes containrecommendations for furtherreading, which in most cases referto selected passages of: MalcolmPemberton and Nicholas Rau.2015. Mathematics for economists:An introductory textbook, 4th ed.Manchester: Manchester UniversityPress.

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