sign in sign up
<<
Home , Fluent (mathematics)

Calculus CURVES Types of Curves

• In the second book Descartes divides curves into two classes, namely, geometrical and mechanical curves. He defines geometrical curves as those which can be generated by the intersection of two lines each moving parallel to one co‐ordinate axis with ``commensurable'' velocities; by which terms he means that dy/dx is an algebraical , as, for example, is the case in the ellipse and the cissoid. • He calls a curve mechanical when the ratio of the velocities of these lines is ``incommensurable''; by which term he means that dy/dx is a trancendental function, as, for example, is the case in the cycloid and the quadratrix.

• From `A Short Account of the History of ' (4th edition, 1908) by W. W. Rouse Ball. Example: Cycloid

• A cycloid is generated by a point on the circumference of a rolling circle. Example: Quadractrix

B C

E

F D' C'

A D Newton –Some Biographical Details

• Born “seriously premature” on Christmas Day, 1642. His father had died in October, and his mother remarried when he was 3, leaving him to be raised by his grandmother. • But his father did see to it that he had an education, and he attended Trinity College, Cambridge, in 1661. • Cambridge wasn’t in good shape. A lot of political appointees as faculty, a lot of drinking among the students. • Newton was pretty much on his own. Newton –Some Biographical Details

• He kept a notebook of his sins (he was raised in the Puritan tradition), including failure to pray often enough and being inattentive in church. Also, "Threatening my [step‐]father and mother Smith to burn them and the house over them." Newton –Some Biographical Details

• Conducted experiments on light, color, and vision. For one, he stared at the sun and recorded the effects on his vision for several days afterward. • For another, he took a stick, or “bodkin” and pushed it behind his eye… • Well, let’s let him tell the story. Newton –Some Biographical Details

• He pushed the stick “betwixt my eye and ye bone as neare to ye backside of my eye as I could, and pressing my eye with ye end of it ... There appeared severall white, darke, and coloured circles, which circles were plainest when I continued to rub my eye with a point of ye bodkin . . .” • This was accompanied in his notebook by a very nice diagram showing the stick sliding under and behind his distorted eyeball, labeled with letters from a through g. Newton –Some Biographical Details

• As has been noted in some books, Newton could be “impossibly single minded.” • He read Descartes’ Geometrie, recalling that he would read a few pages, become stumped, and start over at page one, each time making it a little farther. • Generally, he read what he wanted, taught himself, and pursued his own interests. Newton –Some Biographical Details

• By about 1664, he had prepared himself to step forward into new territory. Using a four‐ year stipend as a Master’s student, he began an extremely fruitful period of his life. • Two of those years were spent back home in Lincolnshire, because Cambridge was closed due to the Plague. Newton – Output from the 4 years:

• The generalized • Method of (differential ) • Method of inverse fluxions ( calculus) • Theory of colors • Theory of gravitation • Probably more…. Generalized Binomial Theorem

• ⁄ • This formula proved to be very useful in the development of calculus. Fluxions

• Newton thought of curves as being generated by moving “points” that he called fluents, denoted by x, y, z, u, v, etc. The velocities of fluents were fluxions, and were denoted by , and so on. • Moments of fluents were infinitely small amounts by which fluents increased over an infinitely small time period, . They were denoted by . Fluxions

• The basic procedure for calculating fluxions was to replace any fluent by , simplify the resulting equation, divide by o,and then ignore any terms with a second or higher order power of o. • “But whereas o is supposed to be infinitely little, that it may represent the moments of quantities, the terms that are multiplied by it will be nothing in respect to the rest; I therefore reject them…” Fluxions ‐ Example Fluxions – Notes:

• Fluxions have fluxions, e.g. , etc. • Notice that the is built into Newton’s . • With these tools, Newton found , , , points of inflection, concavity, and even arc lengths. Notes:

• Later, Newton replaced his “the terms that are multiplied by [o] will be nothing in respect to the rest; I therefore reject them” argument with a rudimentary argument. Fluents

• The inverse problem for Newton was, given a , to find the associated fluent. In some sense this is the problem of integration, but in Newton’s case, it might be more accurate to think of it as solving a . • Typically he used the “antiderviative” approach, and often resorted to representations and term‐by‐term “integration” when that approach didn’t work. That’s why the generalize binomial theorem and the associate work was so useful. Newton: One Last Theorem

• By combining his ability to take areas under curves by finding fluents, his mastery of the generalized binomial theorem, and his knowledge of geometry, Newton was able to find an approximation of π accurate to 16 decimal places. • “I am ashamed to tell you to how many places of figures I carried these computations, having no other business at the time.” Newton

• Newton went on to publish Philosophiæ Naturalis Principia Mathematica in which he laid out his principles of physics, light, gravitation – basically all of what we call Newtonian physics. • Went on to run the royal mint, become an important social and political figure. • Buried in Westminster Abbey. LEIBNIZ Leibniz – Some Biographical Details

• Born in Leipzig in 1646. • Bachelor’s degree from University of Leipzig at age 17. • Doctorate in Philosophy from University of Altdorf in 1667, aged 19. • Entered the political and governmental service, and from 1672‐1676, while on a diplomatic mission to France, became very interested in, and prolific in, mathematics. Leibniz – Some Biographical Details

• Returned to Germany and worked for the House of Hanover for the next 40 years. • Created a calculating machine using ball‐ bearings and binary notation. • Tried to convert all of China to Christianity • Had plans to reunite all Christian churches. • Extremely versatile, he contributed to many areas over his lifetime. Sums and Differences

• Whereas Newton saw motion as the driving force behind his version of calculus, the foundation of Leibniz’ version of the calculus was what he noticed about combinations of sums and differences. Sums of Differences

• If you have a sequence of numbers and from it form the sequence of differences such that , then: • The sum of the differences is just the difference between the last and first elements of the original sequence:

Sums of Differences – Visual

• We’ll let our sequence be ordinates of points that divide up a region under a curve:

8

6

4

2

5 10 15 Sums of Differences – Visual

• Then the sequence of differences is just the sequence of red segments in the diagram.

8

6

4

2

5 10 15 Sums of Differences – Visual

• It’s easy to see that the difference between the endpoints of any sub‐sequence here is just the corresponding sums of differences.

8

6

4

2

x 5 10 x 15 Differences of Sums

• If you have a sequence of numbers and from it form the sequence of partial sums where , then: • The sequence of differences of partial sums is exactly the original sequence. That is, for each m, . • This isn’t hard to see, since clearly . Differences of Sums – Visual

• If you add up all the ordinates to a certain point to find a partial sum

8

6

4

2

5 10 15 Differences of Sums – Visual

• Then take away the previous partial sum,

8

6

4

2

5 10 15 Differences of Sums – Visual

• It’s pretty clear what’s left:

8

6

4

2

5 10 15 Leibniz

• In the words of C. H. Edwards, “These considerations planted in Leibniz’ mind a vivid conception that was to play a dominant role in his development of the calculus –the notion of an inverse relationship between the operation of taking differences and that of forming sums of the elements of a sequence. (p. 238, The Historical Development of the Calculus, © 1979) Taking it to the

• About this time, Leibniz was exposed to some work by Huygens and Pascal involving the summing up of “indivisibles” and the differential or characteristic triangle. This led him to the conclusion that the rules for sums and differences held for infinite sequences or collections as well. Taking it to the Infinitesimal

• He created his own symbols to discuss differences and sums in this context. The infinitesimal differences he called differentials and denoted the operation of taking these differences by d. The operation of taking infinite sums he denoted by , an elongated “S” from the Latin word summa. Taking it to the Infinitisimal

• In Leibniz’ notation, the fact that the difference of the sums gives the original sequence back can be stated as , which is reminiscent of our statement of the Fundamental Theorem of Calculus. Thus for Leibniz, integration and differentiation were inverse operations pretty much from the start. Example

In trying to find of a curve, consider a point with a line. t ds Form the point of dy a tangency, go over an dx infinitesimal amount dx, b 5 10 find a point on the curve. Then the distance between the points is ds, and the vertical distance up is dy. Example

This forms a differential triangle or in Leibniz’ terms, a characteristic t ds dy a triangle. It is “similar” to dx the larger triangle formed by the tangent line and b 5 10 segments of length b and a. Example

Thus, , and so . Taking the t “infinite sum,” we have ds dy a . To Leibniz, dx this meant that the problem b 5 10 of determining arclength could be reduced to a simpler problem of finding area under a simpler curve. Leibniz’ Differentiation

• It looks much like Newton’s, in fact: Differentials

• “…since can be omitted as being infinitely small in comparison with .” • So both Newton and Leibniz depend on the ability to “ignore” the infinitesimal or infinitely small in certain circumstances. Leibniz’ Rules:

• – Proof: . Subtracting uv from both sides and ignoring since it is much smaller than everything else, the result follows. • • Comparison

Newton Leibniz Notation that worked for him. Notation that captured the conceptual essence of calculus Methods giving concrete results that can General methods that can be applied to a be generalized wide variety of specific problems Fairly explicit need for a limit concept The limit concept is more hidden in the notation as a single entity is fundamental dx and dy are separate entities, with only a geometrically significant quotient Integral is indefinite integral, an inverse Integral is an infinite sum of differentials rate of change Infinite series crucial to results Preferred closed form solutions Both used entities that behaved sometimes like O and sometimes not; for both, the inverse relationship between and integral were vital; both developed a wide variety of rules and applications that made calculus a general tool. Newton vs. Leibniz

• Newton and Leibniz got involved in a war of precedence –who deserved the credit for inventing calculus? • It is now very clear they developed their ideas independently. Leibniz’ development was later (1672‐1676) than Newton’s (1664‐1666). But Leibniz published first. Newton vs. Leibniz

• Newton’s followers eventually accused Leibniz of stealing key ideas without giving credit to Newton. Leibniz appealed to the Royal Society for redress against these charges of plagiarism. • However, Newton happened to be President of the Royal Society. It is no surprise that the charges were upheld. Newton appeared to be much more the aggressor in this battle. Newton vs. Leibniz

• Newton was a national hero, while Leibniz died pretty much alone and unrecognized. • Nevertheless, the superiority of his notation and the British refusal to use it or his methods, led to about a century of stagnation in mathematics for the British Empire, while the center of mathematical development moved to the continent. • Leibniz lost the battle, but won the war.