Lectures on Advanced Calculus with Applications, I audrey terras Math. Dept., U.C.S.D., La Jolla, CA 92093-0112 November, 2010 Part I Introduction, Motivation, Basics About Sets, Functions, Counting 1Preface These notes come from various courses that I have taught at U.C.S.D. using Serge Lang’s Undergraduate Analysis as the basic text. My lectures are an attempt to make the subject more accessible. Recently Rami Shakarchi published Problems and Solutions for Undergraduate Analysis, which provides solutions to all the problems in Lang’s book. This caused me to collect my own exercises which are included. Exams are also to be found. The main difference between the approach of Lang and that of other similar books is the treatment of the integral which emphasizes the properties of the integral as a linear function from the set of piecewise continuous real valued functions on an interval to the real numbers. Thus the approach can be viewed as intermediate between the Riemann integral and the Lebesgue integral. Since we are interested mainly in piecewise continuous functions, we are really getting the Riemann integral. In these lectures we include more pictures and examples than the usual texts. Moreover, we include less definitions from point set topology. Our aim is to make sense to an audience of potential high school math teachers, or economists, or engineers. We did not write these lectures for potential math. grad students. We will always try to include examples, pictures and applications. Applications will include Fourier analysis, fractals, .... Warning to the reader: This course is to calculus as fixing a car is to driving a car. Moreover, sometimes the car is invisible because it is an infinitesimal car or because it is placed on the road at infinity. It is thus important to ask questions and do the exercises. A Suggestion: You should treat any mathematics course as a language course. This means that you must be sure to memorize the definitions and practice the new vocabulary every day. Form a study group to discuss the subject. It is always a good idea to look at other books too; in particular, your old calculus book. Another Warning: Also, beware of typos. I am a terrible proof reader. Your calculus class was probably one that would have made sense to Newton and Leibniz in the 1600s. However, that turned out not to be sufficient to figure out complicated problems. The basic idea of the real numbers was missing as well as a real understanding of the concept of limit. This course starts with the foundations that were missing in your calculus course. You may not see why you need them at first. Don’t be discouraged by that. Persevere and you will get to derivatives and integrals. We will assume that you know the basics of proofs, sets. Other References: Hans Sagan, Advanced Calculus Tom Apostol, Mathematical Analysis Dym & McKean, Fourier Series and Integrals 1.1 Some History Around the early 1800’s Fourier was studying heat flow in wires or metal plates. He wanted to model this mathematically and came up with the heat equation. Suppose that we have a wire stretched out on the x-axis from x =0to x =1.Let 1 u(x, t) represent the temperature of the wire at position x and time t.Theheat equation is the PDE below,fort>0 and 0 <x<1: ∂u ∂2u = c . ∂t ∂t2 Here c is a positive constant depending on the metal. If you are given an initial heat distribution f(x) onthewireattime 0, then we have the initial condition: u(x, 0) = f(x) also. Fourier plugged in the function u(x, t)=X(x)T (t) and found that to for the solution to satisfy the initial condition he needed to express f(x) as a Fourier series: ∞ 2πinx f(x)= ane . (1) n=−∞ Note that eix = cosx + isinx,wherei =(−1)1/2 (which is not a real number). This means you can rewrite the series of complex exponentials as 2 series - one involving cosines and the other involving sines. Fourier made the claim that any function f(x) has such an expression as a sum of cnsin(nx) and dncos(nx). People took issue with this although they did believe in power series expressions of functions (Taylor series and Laurent expansions). But the conditions under which such series converge to the function were really unclear when Fourier first worked on the subject. Fourier tells us that the Fourier coefficients are 1 −2πiny an = f(y)e dy. (2) 0 If you believe that it is legal to interchange sum and integral, then a bit of work will make you believe this, but unfortu- nately, that isn’t always legal when f is a bad guy. This left mathematicians in an uproar in the early 1800’s. And it took at least 50 years to bring some order to the subject. Part of the problem was that in the early 1800’s people viewed integrals as antiderivatives. And they had no precise meaning for the convergence of a series of functions of x such as the Fourier series above. They argued a lot. They would not let Fourier publish his work until many years had passed. False formulas abounded. Confusion reigned supreme. So this course was invented. We won’t have time to go into the history much, but it is fascinating. Bressoud, A Radical Approach to Real Analysis, says a little about the history. Another reference is Grattan-Guinness and Ravetz, Joseph Fourier. Still another is Lakatos, Proofs and Refutations. We will end up with a precise formulation of Fourier’s theorems. And we will be able to do many more things of interest in applied mathematics. In order to do all this we need to understand what the real numbers are, what we mean by the limit of a sequence of numbers or of a sequence of functions, what we mean by derivatives and integrals. You may think that you learned this in calculus, but unless you had an unusual calculus class, you just learned to compute derivatives and integrals not so much how to prove things about them. Fourier series (and integrals) are important for all sorts of things such as analysis of time series, looking for periodicities. The finite version leads to a computer algorithm called the fast Fourier transform, which has made it possible to do things such as weather prediction in a reasonable amount of time. Matlab has a nice demo of the search for periodicities. We modified it in our book Fourier Analysis on Finite Groups and Applications to look for periodicities in LA yearly rainfall. The first answer I found was 12.67 years. See p. 159 of my book. Another version leads to the number 28.75 years. 2 Why Analysis? Some Motivation and a Look Forward Almost any applied math. problem leads to an analysis question. Look at any book on mathematical methods of physics and engineering. There are also many theoretical problems in computer science that lead to analysis questions. The same can be said of economics, chemistry and biology. Here we list a few examples. We do not give all the details. The idea is to get a taste of such problems. Example 1. Population Growth Model - The Logistic Equation. References. I. Stewart, Does God Play Dice? The Mathematics of Chaos, p. 155. J. T. Sandefur, Discrete Dynamical Systems. 2 Define the logistics function Lk(x)=kx(1 − x), for x ∈ [0, 1]. Here k is a fixed real number with 0 <k<4.Let x0 ∈ [0, 1] be fixed. Form a sequence x0,x1 = Lk(x0),x2 = Lk(x1), ··· ,xn = Lk(xn−1), ··· Question: What happens to xn as n →∞? The answer depends on k.Fork near 0 there is a limit. For k near 4 the behavior is chaotic. Our course should give us the tools to solve this sort of problem. Similar problems come from weather forecasting, orbits of asteroids. You can put these problems on a computer to get some intuition. But you need analysis to prove that you intuition is correct (or not). Example 2. Central Limit Theorem in Probability and Statistics. References. Feller, Probability Theory Dym and McKean, Fourier Series and Integrals, p. 114 Terras, Harmonic Analysis on Symmetric Spaces and Applications, Vol. I Where does the bell shaped curve originate? 2 Figure 1: normal curve e−πx , −∞ <x<∞ The central limit theorem is the main theorem in probability and statistics. It is the foundation for the chi-squared test. In the language of probability, it says the following. Central Limit Theorem I. Let Xn be a sequence of independent identically distributed random variables with density f(x) normalized to have mean 0 and standard deviation 1. Then, as n →∞, the normalized sum of these variables X + ···+ X 1 2 1 √ n → the normal distribution with density G(x), where G(x)=√ e−x /2. n n→∞ 2π Here ” → ” means approaches. To translate this into analysis, we need a definition. Definition 1 For integrable functions f and g:R → R, define the convolution f ∗ g ( f "splat" g)tobe ∞ (f ∗ g)= f(y)g(x − y)dy. −∞ 3 Then we have the analysis version of the central limit theorem. Central Limit Theorem II. Suppose that f : R → [0, ∞) is a probability density normalized to have mean 0 and standard deviation 1. This means that ∞ ∞ ∞ f(x)dx =1, xf(x)dx =0, x2f(x)dx =1. −∞ −∞ −∞ Then we have the following limit as n →∞ √ b n b 1 2 (f ∗···∗f)(x)dx → √ e−x /2dx.