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What is calculus?<br />Calculus originally meant “mathematics”.<br />Derived from the Latin “calx” (counter) – ancient Babylonians would use pebbles to represent units, tens, hundreds, etc, on a primitive abacus.<br />Later, defined as measuring varying rates of change.<br />
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Development of calculus over time<br />Period of anticipation (application without theory)<br />Period of development<br />Period of refinement<br />
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Period of Anticipation<br />Techniques were being used by mathematicians that involved infinite processes to find areas under curves or maximize certain quantities.<br />Applications of what we now call integration appeared long before applications of differentiation.<br />We cannot be sure at which point in time concepts were discovered due to sparse evidence.<br />
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Volume of a frustum pyramid<br />A papyrus dating from c. 1820 BC shows an Egyptian mathematician working on various geometrical problems.<br />Among them is a solution to the calculation of the volume of a pyramidal frustum – a square pyramid with its top cut off.<br />We do not know how the Egyptians arrived at the formula.<br />
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Greek integral geometry<br />Whereas the Egyptians pointed to physical evidence to show that their methods worked, the Greeks relied on geometrical proofs.<br />Archimedes and his people are credited with use of infinitesimals in their work to find areas and volumes. Coming from Latin, infinitesimal means “the ∞th item in a series”.<br />
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Greek integral geometry<br />Although not algebraically sound, the Greeks used infinitesimals because they could supplement it with a geometrical proof.<br />They invented the “method of exhaustion”, which involves finding the area of a shape by inscribing inside and around it polygons – by increasing the number of sides of the polygon, the difference between the areas of the two shapes became “infinitely small”.<br />
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Method of indivisibles<br />Bonaventura Cavalieri (1598-1647) is credited with the discovery of the method of indivisibles.<br />Using a familiar notation, we divide the plane into infinitely many thin rectangles, each of width dx (an infinitesimal) and length f(x).<br />The “area under the curve”<br /> is then the sum of the area <br /> of all the rectangles.<br />We’ll return to infinitesimals<br /> later...<br />
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Greek differential geometry<br />Archimedes noted that the tangent to a curve at a point was equal to the slope of the curve at that point.<br />The Greeks understood the <br /> concept of limits, as shown by<br /> their ability to find the area<br /> between a straight line and a<br /> parabola.<br />Not until the 17th Century did European mathematicians begin discussing the idea of the derivative.<br />
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Indian calculus<br />Bhaskara II had ideas concerning infinitesimal change about what we know as the derivative, and gave a statement that we now call Rolle’s Theorem.<br />Parameshvara did some work on a rough Mean Value Theorem (c. 1400).<br />Yuktibhasa published by Jyesthadeva – unusually it contained proofs and derivations of the stated theorems (c. 1530).<br />
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Period of Development<br />Newton and Leibniz created the foundations of Calculus and brought all of these techniques together under the umbrella of the derivative and integral.<br />The methods they employed made sense, but their arguments were not always logically sound.<br />Other mathematicians built on their ideas, forming the calculus we know today.<br />
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Pre Newton / Leibniz<br />John Wallis, professor of geometry at Oxford, published Arithmeticainfinitorum in 1655, in which he asserted that the equation<br /> held for all rational n (apart from -1), but only proved it for positive integers.<br />This was used to find the area under the curve<br /> and is often regarded as the first general theorem to appear in calculus.<br />
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Pre Newton / Leibniz<br />Pierre Fermat, although best known for his work in number theory, made several important contributions to the field of calculus, including methods of finding maxima and minima on curves.<br />James Gregory proves a restricted version of the Fundamental Theorem of Calculus in 1667, and discovered several Taylor series expansions.<br />
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Isaac Newton<br />1643 – 1727<br />Approached calculus from a physics / geometry background – he used it in his mechanics work, but realised the importance of it as a concept in itself. He previously studied Latin and Greek.<br />Did not publish any definitive papers on the subject of calculus in the field of physics – instead, most of his calculus work was publicised through letters to colleagues, or as a sub-topic in his other works.<br />
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Gottfried Leibniz<br />1646 – 1716<br />Prior to mathematics, Leibniz had studied fields as varied as law, politics and metaphysics.<br />In 1672, Christiaan Huygens convinced him to study mathematics, as it was a “rational subject”.<br />Leibniz was very aware of good notation, and put much thought into the symbols he used.<br />
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Newton’s work<br />Newton’s university career at Cambridge was largely based around Law. While looking at some astronomy books at the local fair in 1663, he found that he did not understand the mathematics in them – and so began his mathematical career.<br />The astronomy led Newton to trigonometry, which he also was not able to grasp.<br />Newton’s entry into calculus came from his work on the binomial theorem and infinite series.<br />
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Newton’s work<br />He graduated from Cambridge in 1664, the same year as the bubonic plague swept across London and the country. The university colleges remained closed for two years, and during this time he returned home.<br />In these two years, from 1665 to 1666, Newton formulated the law of gravity, developed the fundamental laws of motion, and invented calculus.<br />
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Newton’s work<br />During this period, Newton wrote the (unpublished) De Analysi per AequationesNumeroTerminorumInfinitas, in which he outlines his early calculus ideas.<br />Newton computed and tabulated the area under the curves <br /> for varying integer n.<br />
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Newton’s work<br />Using the generalised binomial theorem, we can work with rational n, for example, n = ½. He obtained the series<br />Newton obtained an infinite series, and Newton noticed that he did not have to start work with integrals at all; he could interpolate the series directly.<br />This led him onto his development of calculus.<br />
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Newton’s work<br />To compute the area under a curve,<br />Newton’s method stated:<br />Start with an indefinitely small triangle whose area is a function of x and y.<br />An infinitesimal increase in the value of x, changing it to x + o, will create a new formula for the area.<br />Using the binomial theorem as described before, recalculate the area.<br />Remove all quantities of o, as these terms “multiplied by it will be nothing in respect to the rest”, leaving an algebraic expression.<br />y<br />x<br />x + o<br />
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Newton’s work<br />However, there was one big problem with Newton’s<br />work, which Newton himself pointed out.<br />He admitted the logical limitations – no matter<br />how small the o terms were, disregarding them wasn’t<br />mathematically sound!<br />
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Newton’s work<br />Newton was not happy with his method, in no small part due to the infinitesimals that he strived so hard to avoid.<br />Newton redefined his work in terms of continual flowing motion in his new compilation MethodusFluxionum et SerierumInfinitarum. He attempted to form his calculations on ratios of changes.<br />
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Newton’s work<br />Newton developed his theory of fluxions, the rate of generated change over time. He used the notation for the fluent (in flux) x. He used this theory to make many contributions to physics, including calculations involving the ellipses of planets’ orbits.<br />Newton wrote mainly for himself, never intending to publish. The notation and symbols he used were often thought up “on the day”.<br />
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Leibniz's work<br />Leibniz focussed on the tangent problem. Like Newton, he attempted to view the tangent as a ratio.<br />Unlike Newton, he viewed it as the ratio between the x and y of the (x,y) co-ordinates.<br />Therefore, he saw the integral as the sum of the x values for infinitesimal intervals in the y values – i.e. a sum of infinitely many rectangles.<br />
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Leibniz's work<br />Leibniz's method was similar<br /> to the method of indivisibles<br /> employed by Cavalieri some<br /> time earlier.<br />Leibniz also realised the existence of an inverse to the integral – the slope of the tangent. This was the first formal notion that the two were connected in a strong way.<br />
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Leibniz's work<br />Leibniz viewed dx as the “next point on the line” after x. This didn’t make sense... but somehow it worked.<br />With this view, he found that<br /> reasoning that dxdywas<br /> infinitely small, and could be ignored.<br />Leibniz introduced the symbols and .<br />dy<br />x dy<br />y<br />y dx<br />x<br />dx<br />
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Leibniz's work<br />Because his use of infinitesimals seemed to make calculus work, they were widely adopted. However, they provided conceptual difficulties.<br />Much of mathematics in later years attempted to rigorize what Newton and Leibniz had discovered – more precisely, how did calculus work without having to depend on very small quantities – sometimes zero, other time positive but small?<br />
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The debate<br />Both Newton and Leibniz seem to have invented calculus independently, using different methods, but converging on the same results.<br />Newton and Leibniz argued over the ownership of the work. Newton claims that he thought of the ideas first, but the fact that Leibniz was the first to publish his findings meant that his ideas were the first to be brought to the attention of the public. <br />
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The debate<br />There have been many “conspiracy theories” as to what happened:<br />Did Newton visit Leibniz and copy his work, when he reached a dead-end in his research?<br />Did Leibniz get his original ideas from Newton?<br />Not much can be said, due to the lack of copyright system available for mathematics at that time.<br />Within the UK, the prevailing opinion in the 18th Century was that Leibniz had plagiarised Newton. However, in modern times it is believed that they each independently invented calculus.<br />
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Post – Newton / Leibniz<br />In 1691, Michael Rolle, a self-educated mathematician, published Rolle’s Theorem, a special case of the Mean Value Theorem.<br />In 1715, Brook Taylor invented integration by parts and gave a formal definition for Taylor series.<br />
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Leonhard Euler<br />In Euler’s three most famous textbooks, he “translated” Newton’s ideas into Leibniz’s superior notation.<br />He calculated the two trigonometric power series.<br />(These were taken with a pinch of salt due to the notion of convergence having not been rigourised.)<br />
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Leonhard Euler<br />Euler popularised the symbols for π, e and i.<br />He was able to show:<br /> , leading to the famous formula<br />
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Period of Refinement<br />Mathematicians like Cauchy and Weierstrass expanded a great deal on the calculus already in place.<br />They set down precise definitions of all the work that had gone before, to ensure that everything that they were working on was logically accurate.<br />
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Augustin-Louis Cauchy<br />Cauchy, who worked briefly as a military engineer, saw the importance of rigourisly defining the previous works of calculus.<br />His work includes definitions of continuity, convergence, the derivative and the integral.<br />Cauchy’s definition of a limit of a sequence:<br />“When the values successively attributed to a particular variable approach indefinitely a fixed value so as to differ from it by as little as one wishes, this latter value is called the limit of the others.”<br />
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Augustin-Louis Cauchy<br />Introduced the f(x+h) method for evaluating derivatives.<br />He also discovered the Cauchy Convergence Criterion, a sufficient condition for an infinite series to converge.<br />
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Karl Weierstrass<br />Often cited as the “father of modern analysis”.<br />Provided a rigorous treatment of calculus using arithmetic and inequalities, replacing Cauchy’s vaguely worded definitions.<br />Weierstass’s definition of a limit L of a sequence (an):<br />
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Karl Weierstrass<br />Many mathematicians incorrectly believed at that time that a continuous function must be differentiable at most points (all but finitely many points). Weierstrass provided a counter-example:<br />Due to his highly rigorous approach to the definitions, he was able to go on to prove previously unproven theorems, including the Intermediate Value Theorem and the Bolzano-Weierstrass Theorem.<br />
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Bernhard Riemann<br />Up until now, the process of integration had been seen as a mixture of the opposite of differentiation and the “area under the curve”.<br />Riemann introduced the notion of the Riemann integral in the 19th Century, in terms of a series. From this, he proved that integration is the inverse of differentiation.<br />
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Summary<br />We’ve come a long way: from finding methods that “just worked” like the volume of a frustum, to formalising the relationship between the derivative and the integral.<br />I haven’t covered all of calculus up to the present day – there’s still complex calculus, different definitions of the integral (Riemann / Lebesgue), others...<br />I haven’t touched on modern applications, but there are many!<br />
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