Calculus II - Honors: MATH 1921-800 Goals for this class Explore calculus using a wider lens E.g., examine the notion of a limit from a wider perspective “Infinite Powers: How calculus reveals the secrets of the universe” by Steven Strogatz 1 / 40 Infinity Principle Infinity Principle centers around 2 themes - mysteries & methodology Methodology: Divide-&-Conquer strategy - Cutting & Rebuilding Cutting - infinitely fine subtraction which is used to quantify the differences between the parts - called Differential Calculus Rebuilding - integrates parts back into the original whole - called Integral Calculus Big challenge - Cope with infinity Mysteries: mystery of curves, mystery of motion, mystery of change Resolution of these mysteries had far reaching consequences on civilization and our everyday lives Figures like squares, cubes, planes proved to be staighthforward Round things were brutal! 2 / 40 Mystery of curves How much area and volume a sphere can hold Circumference and area of a circle were unsurmountable problems Calculus grew out of geometers’ curiosity and frustration with roundness Breakthrough - insisting that curves were actually made of straight pieces Issue - pieces have to be infinitesimally small & infinitely numerious Zoom closely enough on a circle - the portion of it under the microscope begins to look straight & flat Curves arose naturally in parabolic arc of a ball in flight, elliptical orbit of Mars, convex shape of a lens (development of microscopes & telescopes in late Renaissance Europe) Led to fascinations with mysteries of motion on Earth and in the solar system 3 / 40 Mystery of motion Experiments - measured motion of a pendulum, clocked accelerating distance of a ball rolling down a ramp Kepler was in a state of “sacred frenzy” when he found his laws of planetary motion However, math & geometry could not adequately explain the patterns Motion was often not steady. E.g., ball rolling down a ramp, direction of travel kept changing, planets moved faster when they got close to the sun & slowed down when they receded from it Infinity Principle came to the rescue just like it had for curves Changing speed was made up of infinitely many & infinitely similar brief motions at a constant speed. Even the jerkiest driver cannot make the speedometer needle move by much Over an infinitesimal time interval - needle was not moving at all Criticized for playing fast and loose with infinity in mid-1600s 4 / 40 Isaac Newton Found a way to combine symbols of algebra with infinity represented any curve as a sum of infinitely many simpler curves described by powers of x, like x2 , x3 , x4 , etc. Motion of any kind always unfolds one infinitesimal step at a time Handful of differential equations (laws of motion & gravity) explained everything from the arc of a cannonball to orbits of planets Work launched Enlightment - impacted philosophers, poets, underpinned space program & work of African-American mathematician Katherine Johnson (Hidden Figures) 5 / 40 Mystery of change Are the laws of change similar to laws of motion? With experiment & observation, scientists worked out the laws of change and used calculus to solve them & make predictions Examples - Einstein applied calculus to a model of atomic transitions to predict an effect called stimulated emission [ s & e in laser - Light Amplification by Stimulated Emission] Applied medicine to shape 3-drug therapy for HIV patients 6 / 40 Infinity Shaped by everyday concerns such as redrawing boundaries of farmers’ fields after summer flooding of the Nile washed boundaries away Geometers found circles to present challenges - much harder to analyze than triangles, rectangles, etc. Calculus began as an outgrowth of geometry Goal - use infinity to build a bridge between the curved and the straight Infinity helped to find the area of a circle 7 / 40 Area of a circle using a pizza analogy Pizza = perfectly flat and round with an infinitesimally thin crust Want to measure circumference and radius Cut into four quarters and rearrange Cut into 8 slices and rearrange Cut one of the slanted endpieces in half and move that half to the other side Keep increasing number of slices Creating a sequence of shapes that keep getting closer to a rectangle Call rectangle limiting rectangle Area of circle = rC 2 , proved by Archimedes Limit of infinitely many slices gave us the rectangular shape = things became simpler at infinity 8 / 40 Map of Tennessee Get Printable Maps From: Waterproof Paper.com 9 / 40 Lecture 2: Limits & the Riddle of the Wall A limit is like unattainable goal E.g. Riddle of the Wall Walk halfway to a wall Then, walk half remaining distance Then, half of that, and on and on Will you finally get to the wall? Get arbitrarily close to the wall Wall plays the role of a limit Parable of .333 . . . Convert 1/3 into an equivalent decimal 1/3 can be written as 0.333 . . . where the dot-dot-dots mean than the threes repeat infinitely dot-dot-dots = represent the infinitely many threes that we cannot possibly write = completed infinity dot-dot-dots = represent a limit = limit of successive decimals generated during long division on the fraction 1/3 = potential infinity Pretending that process actually terminates & somehow reaches the nirvana of infinity can get us in trouble! 10 / 40 Limits & the Parable of the Infinite Polygon Take a circle and put three dots on the circle and connect them Board More dots we use =⇒ rounder polygon becomes =⇒ sides get shorter and more numerous As we add more dots, polygons approach the original circle as a limit Infinity is bridging two worlds. That is, taking us from the rectilinear to the round In pizza proof, infinity took us from round to a rectangle Polygon gets closer & closer to being a circle but never truly gets there =⇒ potential infinity 11 / 40 Allure & Peril of Infinity Limiting shape is simpler & more symmetrical Circle simpler that any of the thorny polygons that approach it Pizza proof, rectangle was simpler than the scalloped shapes This is the allure of infinity. That is, everything becomes better there Should we take the plunge and say that a circle is truly a polygon with infinitely many infinitesimal sides? Might be condemned to logical hell (sin of completed infinity)! If circle = infinite polygon with infinitesimal sides, then how long are sides of polygon? Zero length? If so, infinity × zero = circumference of circle! Absolute garbage! Don’t know what infinity × zero is! 12 / 40 Sin of Dividing by Zero Imagine dividing 6 by 0.1. Answer: 60 Imagine dividing 6 by 0.01. Answer: 600 Imagine dividing 6 by 0.0000001. Answer: 60 000 000 The smaller the divisor, the bigger the answer As the divisor approaches zero, the answer approaches infinity That is why we cannot divide by zero. The truth is the answer would be infinite. 13 / 40 Sin of Completed Infinity We thought we could actually reach the limit - that we could treat infinity like any number Greek philosopher Aristotle (4th century BCE) warned that being imprecise with infinity in this way could lead to all sorts of logical trouble Thinking of our pizza proof, potential infinity would mean that the pizza can be sliced into more and more slices, as many as desired but still a finite number of slices and all of nonzero length This leads to no logical difficulties Completed infinity =⇒ infinite number of slices of zero length was forbidden by Aristotle His edict followed for the next 2200 years 14 / 40 Zeno & his Paradox of the Arrow Zeno argued that space and time are discrete, that is, they are composed of tiny indivisible units, say pixels of space and time Paradox: If space and time are discrete, an arrow in flight can never move, because at each instant (pixel in time) the arrow is at some definite place (a specific set of pixels in space). Hence, at any given instant, the arrow is not moving! Not moving between instants because, by assumptions, there’s no time between instants We know from watching movies & videos on our digital devices, motion is very much continuous even when it’s discretized Owing to our perceptual limitations, the motion of the video looks smooth Sometimes our senses do really deceive us If the chopping into pixels is too choppy, we can tell the difference between continuous & discrete For many practical purposes, the discrete can stand in for the 15 / 40 Zeno meets the Quantum Do infinitesimally small things exist in the real world? Realm of Quantum Mechanics = Nature on smallest scales Riddle of the Wall from quantum perspective If walker is an electron, there’s a chance it might walk right through the wall. Effect known as quantum tunneling Hard to make sense of this in classical terms Quantum explanation is that electrons are described by probability waves Waves obey an equation formulated by Austrian physicist, Erwin Schrodinger (1925). Solution to Schrodinger’s eqn show that a small portion of the electron probability wave exists on the far side of the impenetrable barrier =⇒ there is some small nonzero probability that the electron will be detected on the far side of the barrier, as if it had tunneled through the wall With calculus, can calculate the rate at which tunneling events occur & experiments have confirmed predictions Alpha particles tunnel out of uranium nuclei at the predicted rate to produce effect known as radioactivity 16 / 40 Nature on the smallest scale = Quantum Mechanics Tunneling also critical in nuclear-fusion processes that make the sun shine Hard for us to think of a world on an atomic level Calculus has taken the place of intuition Quantum mechanics radical in in many ways but retains the assumption that space & time are continuous Maxwell made same assumption in his theory of electricity & magnetism Newton in his theory of gravity & Einstein in his theory of relativity All rely on assumption of continuity 17 / 40 Does time lose its continuous character? We don’t know what it’s like on such a small scale No consensus on how to visualize space & time on these small scales Agreement on how small the scales may be Forced upon us by three fundamental constants of nature G = gravitational constant ~ = reflects strength of quantum effects (appears in Schrödinger’s wave equation of quantum mechanics) c = speed of light (speed limit for the universe) Planck - father of quantum theory - combine these constants to produce a scale of length p Planck length = ~G/c3 When we plug in the values of the constants, Planck length = 10−35 metres Corresponding time to travel this distance = 10−43 seconds 18 / 40 What kind of numbers are we talking about? Numbers put a bound on how fine we can slice space or time Take largest distance possible = estimated diameter of the universe Divide by Planck’s length This extreme ratio is a number with only 60 digits in it That is the most we need to express distance in terms of time Using way more digits than that would be colossal overkill! Yet, in calculus we use infinitely many digits all the time In the idealized world of calculus, we pretend that everything can be split finer & finer without end Whole story of calculus built on that With infinity, things are made simpler 19 / 40 The Man Who Harnessed Infinity = Archimedes Archimedes His use of Infinity Principle & his approach of blending mathematics with physics Estimation of π Let’s think of a circle as track Say we walked around the track in x steps - the steps would be represented by small straight lines around the circle Can estimate length of track by counting the number of steps & multiplying by the length of a step 20 / 40 Archimedes - Estimation of π Archimedes proceeded similarly Began with taking 6 straight steps along the circle Divide hexagon into 6 equilateral triangles C > 6r Define π as the ratio of the circumference to the diameter π = C/d = C/2r > 6r/2r = 3 ← lower bound for π Went from 6 steps → 12 steps → 24 steps → 48 steps → . . . → 96 steps Got progressively harder to calculate step length Applied Pythagorean theorem and computed square roots by hand Ultimately, with his 96-gon, he showed that 3 + 10/71 < π < 3 + 10/70 3 + 10/70 reduces to 22/7 ← famous approximation No mention of π in Euclid’s Elements (a generation or two before) 21 / 40 Tao of Pi π gave rise to the notion of irrational numbers Around the same time, the Greeks realized that the diagonal of a unit square was also not a whole number! As of now, twenty-two trillion digits of π have been computers by some of the fastest computers Yet twenty-two trillion digits is nothing compared to the infinitude of digits that define the actual pi Where are these digits?? They don’t exist in the material world, as of now π is defined as the unattainable limit of a never-ending process. Unlike the sequence of polygons approaching a circle, there is no end in sight for π, no limit we can ever know 22 / 40 Archimedes & the Quadrature of the Parabola Arc of a three-point shot in basketball approximates a parabola To Archimedes, a true parabola is obtained by slicing through a cone with a plane Steeper cut produces an ellipse A cut that has the same slope as the cone itself produces a parabola Archimedes wanted to compute the quadrature of a parabolic segment Find quadrature = express area between parabola and line segment in terms of the area of a square/rectangle/triangle/rectilinear figure 23 / 40 Quadrature of the Parabola 24 / 40 Quadrature of the Parabola 25 / 40 Quadrature of the Parabola 26 / 40 Achimedes - The Method The two slats balance each other Same is true for every vertical slice All the ribs from the parabola end up at S They balance all the slats from the huge outer triangle ACD Since those slats have not moved, this means all the parabolic mass shifted to S balances the huge triangle right where it is Next, Archimedes replaces the infinitely many slats of the huge outer triangle with an equivalent point of their own, called the triangle’s center of gravity 27 / 40 Achimedes - The Method Huge triangle acts as if its entire mass were concentrated at that single center of gravity Archimedes shows that this point lies on the line F C at a point precisely three times closer to the fulcrum F than S is Since entire mass of the triangle sits three times closer to the pivot point, the parabolic segment must weigh a third as much as the huge triangle in order for them to balance =⇒ area of parabolic segment = 1/3 huge outer triangle ACD Outer triangle has four times of inner triangle ABC Archimedes deduces that the parabolic segment must have 4/3 area of triangle ABC inside it! Problem with argument! 28 / 40 Completed infinity used in “The Method” Archimedes argument deals with Completed Infinity! Outer triangle = made up of all parallel lines inside itself Outer triangle = Completed infinity of slats Likewise, parabolic segment = made up of all the parallel lines drawn inside the curve Archimedes went on to apply “The Method” to many other problems about curved shapes 29 / 40 Surface area & volume of sphere Figure: Sphere in a cylindrical hatbox Using the Method, Archimedes found that the sphere had 2/3 the volume of the hatbox & 2/3 of its surface area (assuming tob and bottom lids are also counted). Archimedes Palimpsest first came to light in 1899 in a Greek Orthodox library in Constantinople Now lives in the Walters Art Museum in Baltimore 30 / 40 The Method - From Computer Animation to Facial Surgery Archimedes legacy lives on today Characters seem lifelike due to Archimedes insight - any smooth surface can be convincingly approximated by triangles The more triangles used and the smaller we make them, the better the approximation becomes 31 / 40 From Computer Animation to Facial Surgery What’s true for mannequins is equally true for ogres, clownfish & toy cowboys Like Archimedes, modern-day animators at DreamWorks created Shrek’s round belly and his trumpet-like ears out of tens of thousands of polygons Even more such polygons were required for a tournament scene in which Shrek battled local thugs - each frame of that scene took over forty-five million polygons No trace of the polygons anywhere in the finished movie! 32 / 40 Archimedes - The Method - in Avatar Figure: Avatar by DreamWorks Infinity Principle teaches us that the straight and the jagged can impersonate the curved and the smooth In Avatar, level of polygonal detail became more extravagant Director James Cameron insisted that animators use about 1 million polygon to render each plant in the imaginary world of Pandora It cost 300 million dollars to produce! First movie to use polygons by the billions 33 / 40 The Method in computer-generated movies Toy Story used far fewer polygons Took an animator a week to sync an eight-second shot Whole film took 4 years and 800 000 hours of computer time to complete 34 / 40 Geri’s Game Figure: Geri’s Game First computer-animated film with a human main character Geri was built from angular shapes Animators at Pixar fashioned Geri’s head from a complex polyhedron, a three-dimensional gem-like shape that consisted of about 4500 corners with flat facets in between them Animators subdivided those facets repeatedly to create an increasingly detailed depiction 35 / 40 Geri’s Game Subdivision process took up much less memory than earlier methods and allowed for much faster animations Revolutionary advance in computer animation However, it channeled Archimedes To estimate π, Archimedes walked around a circle in 3 steps (triangle) He increased his number of steps until he got to a 96-gon Animators approximated Geri’s wrinkly forehead, nose and folds of skin in his neck by repeatedly subdividing a polyhedron (Geri’s Game) DreamWorks took the next steps forward in realism and emotional expressiveness in Shrek 36 / 40 Facial Surgery Surgeries for patients with overbites and malformations from birth Applied mathematicians Deuflhard, Weiser, & Zachow used calculus & computer modeling to predict the outcomes of complex facial surgeries Built an accurate map of a patient’s facial-bone structure Biomechanically accurate model of the material properties of skin and soft tissues such as fat, muscle, tendons, blood vessels With computer model, surgeons could perform operations on virtual patients Computer calculated how the virtual soft tissue behind the face would move and reconfigure itself in response to stresses produced by the face’s new bone structure Soft tissues posed its own geometrical challenges. The team represented the three dimensional volume between the skull and behind the face by hundreds of thousands of tetrahedrons Their model successfully predicted the position of 70% of the patient’s facial skin 37 / 40 Archimedes & Mystery of Motion Figure: Archimedes Legacy - first principled use of infinite processes to quantify the geometry of curved shapes Did not address challenges having to do with motion 38 / 40 Upcoming Presentations Chapter 3 - Discovering the Laws of Motion: Group 1 Chapter 4 - Dawn of Differential Calculus: Group 2 Chapter 5 - Crossroads: Group 3 Chapter 6 - Vocabulary of Change: Group 4 Chapter 7 - Secret Fountain: Group 3 Chapter 8 - Fictions of the Mind: Group 4 Chapter 9 - Logical Universe: Group 2 Chapter 10 - Making Waves: Group 1 Group 1 - Phoebe, Renie, Hallie, Christian Group 2 - Allison, Callie, Chao, Jacob W. Group 3 - Destin Harris, Alex, Jackson Hanchek, Dillon Williams Group 4 - Luke, Blaine, Jacob D., Jackson Hughes 39 / 40 Chapter 3 - Important Concepts Galileo & retrograde motion Misconception about sun rotating around the earth Issues with Catholic Church First practitioner of the scientific method - ball on inclined plane Differential calculus & Galileo 40 / 40
