Gottfried Wilhelm Leibniz

Gottfried Wilhelm Leibniz and The Calculus Controversy

Gottfried_Wilhelm_von_Leibniz.jpg

General Background

Gottfried Leibniz was born in Leipzig, Germany on July 1, 1646. His father, Friedrich Leibniz, a Professor of Moral Philosophy at the University of Leipzig died when Gottfried was six leaving him a large personal library. By the age of twelve he had taught himself Latin, and had already begun to learn Greek. At age 14 he began his university education at the Universities of Leipzig and Altdorf, mastered his classes and received his doctorate in law by the age of 20. After declining academic employment at Altdorf, Leibniz spent the majority of his life in the service of two noble German families. Leibniz would develop infinitesimal calculus, independently of Newton, and spend much of his life defending his work until his death in 1716.

Leibniz’s Accomplishments

Leibniz was not just a mathematician, he was a scientist, engineer, philosopher and also served as a diplomat. He studied and has influenced many fields, predicting many concepts that were not fully developed until after his time. As a philosopher Leibniz studied metaphysics and proposed the existence of monads (the metaphysical version of atoms). He also believed in logical and symbolic thought, that all actions, human actions included, could be “calculated” using logic. As a scientist and engineer Leibniz opposed Newton by introducing the idea that space, time, and motion are all relative. Einstein would later develop this idea further, but Leibniz had the insight to propose the idea long before he did. He proposed that the core of Earth was molten. In psychology he anticipated the difference between conscious and unconscious states of mind. In sociology he laid the grounds for communication theory. He helped to develop economic policy. Leibniz believed greatly in applying science. He designed wind-driven propellers and water pumps, mining machines, hydraulic presses, lamps, submarines, clocks, invented a steam engine, proposed a method to desalinate water, and more. Leibniz may also have been the first computer scientist and information theorist. Early in his career he developed the binary system, also known as boolean algebra, which all computers are based on today and created a machine capable of calculating the four arithmetical operations; additions, subtraction, multiplication and division (pictured below). Leibniz helped found and was, for the remainder of his life, the first president of the Berlin Academy of Sciences. As a mathematician Leibniz discovered boolean algebra as well as calculus.

Leibniz_machine.jpeg

The Calculus Controversy

Gottfried Leibniz and Isaac Newton developed infinitesimal calculus at about the same time independently from one another. This lead to intense debates on who actually created it. Both scientists developed a similar math using different notation. Newton’s notation was much more difficult to read and use so Leibniz’s is what we use today. Both mathematicians politely acknowledged each other’s claims, but eventually things escalated. Newton wanted to prove that he had invented calculus first. Eventually Leibniz asked the Royal Society for help. Unfortunately by this time Newton was the president of the Royal Society and put his fans on the review committee. He also drafted parts of the report. From 1711 until his death in 1716 Leibniz waged a constant battle against John Keill, Newton, and others, defending himself from accusations that he simply created a new notation after stealing Newton’s ideas.

Leibniz’s Calculus

Leibniz introduced several notations still used today including the integral sign and d which is used for differentials. The product rule of differential calculus is still called Leibniz’s Law and the theorem that explains when and how to differentiate under the integral sign is called Leibniz Integral Rule.

Integral
Given a function f(x) and an interval [a,b], all real, the integral is equal to the area of the region in the x-y plane bounded by the function f(x) and vertical lines of x=a and x=b. Areas below the x-axis are subtracted. This is expressed as;

(1)
\begin{align} % MathType!MTEF!2!1!+- % feqaeaartrvr0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l % bbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0R % Yxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa % caGaaeqabaaaamaaaOqaamaapehabaGaamOzaiaacIcacaWG4bGaai % ykaiaadsgacaWG4baaleaacaWGHbaabaGaamOyaaqdcqGHRiI8aaaa % !3B01! \[ \int\limits_a^b {f(x)dx} \] \end{align}

Differential
Closely related to derivatives, a differential is an infinitesimally small change in a variable. A small change in a variable is represented by δx, not Δx. Differentials specifically are written dx, where dx is the change in x. A key property of a differential is that if y is a function of x then the differential dy of y is related to dx by the formula;

(2)
\begin{align} dy =\frac{dy}{dx}dx \end{align}

where dy/dx is the derivative of y with respect to x. Unfortunately differential notation is not inherently mathematically precise, but it can be made so by four methods;
-differentials as linear maps
-differentials as nilpotent elements of commutative rings
-differentials in smooth models of set theory
-differentials as infinitesimals in hyperreal number systems

Leibniz’s Law (product rule of differential calculus)
Leibniz’s Law governs the differentiation of products of differentiable functions. Stated in Leibniz notation as;

(3)
\begin{align} \frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx} \end{align}

This equations shows the change of uv if u(v) and u(x) are two differentiable functions of x and there is an infinitesimal change in x. Proof of this rule can be found here;
http://en.wikipedia.org/wiki/Product_rule

Leibniz’s Integral Rule (differentiation under the integral sign)
This rule tells us that if we have an integral of the form;

(4)
\begin{align} % MathType!MTEF!2!1!+- % feqaeaartrvr0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l % bbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0R % Yxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa % caGaaeqabaaaamaaaOqaamaapehabaGaamOzaiaacIcacaWG4bGaai % ilaiaadMhacaGGPaGaamizaiaadMhaaSqaaiaabMhadaWgaaadbaGa % aeimaaqabaaaleaacaqG5bWaaSbaaWqaaiaabgdaaeqaaaqdcqGHRi % I8aaaa!3EA7! \[ \int\limits_{{\rm y}_{\rm 0} }^{{\rm y}_{\rm 1} } {f(x,y)dy} \] \end{align}

then for

(5)
\begin{align} % MathType!MTEF!2!1!+- % feqaeaartrvr0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l % bbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0R % Yxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa % caGaaeqabaaaamaaaOqaaiaadIhacqGHiiIZcaGGOaGaamiEaWGaaG % imaOGaaiilaiaadIhamiaaigdakiaacMcaaaa!39C9! \[ x \in (x0,x1) \] \end{align}

the derivative of this integral can be expressed;

(6)
\begin{align} % MathType!MTEF!2!1!+- % feqaeaartrvr0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l % bbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0R % Yxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa % caGaaeqabaaaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG4b % aaamaapehabaGaamOzaiaacIcacaWG4bGaaiilaiaadMhacaGGPaGa % amizaiaadMhacqGH9aqpaSqaaiaadMhacaaIWaaabaGaamyEaiaaig % daa0Gaey4kIipakmaapehabaWaaSaaaeaacqGHciITaeaacqGHciIT % caWG4baaaaWcbaGaamyEaiaaicdaaeaacaWG5bGaaGymaaqdcqGHRi % I8aOGaamOzaiaacIcacaWG4bGaaiilaiaadMhacaGGPaGaamizaiaa % dMhaaaa!52D6! \[ \frac{d}{{dx}}\int\limits_{y0}^{y1} {f(x,y)dy = } \int\limits_{y0}^{y1} {\frac{\partial }{{\partial x}}} f(x,y)dy \] \end{align}

if f and δfx are both continuous over a region in the form;

(7)
\begin{align} % MathType!MTEF!2!1!+- % feqaeaartrvr0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l % bbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0R % Yxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa % caGaaeqabaaaamaaaOqaaiaacUfacaWG4bGaaGimaiaacYcacaWG4b % GaaGymaiaac2facqGHxdaTcaGGBbGaamyEaiaaicdacaGGSaGaamyE % aiaaigdacaGGDbaaaa!3F7B! \[ [x0,x1] \times [y0,y1] \] \end{align}

Proof can be found here;
http://en.wikipedia.org/wiki/Leibniz_integral_rule

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