Introduction to Real Analysis MATH 2001

Transcription

1 Introduction to Real Analysis MATH 2001 Juris Steprāns York University September 13, 2010

2 Instructor Instructor: Juris Steprāns My office is N530 in the Ross building. Office hours are Mondays from 4:00 to 5:00 and Wednesdays from 11:00 to 12:00. You are also welcome to make an appointment to see me. My telephone number is (ask for me) or extension However, I check my frequently and often the best way of reaching me is sending an to steprans@yorku.ca You can find a link to my website on the Department site. It is

3 Course description Real analysis is the detailed study of the objects and arguments used in elementary calculus. A first course in calculus is usually focused on the computational techniques that make calculus so useful in analyzing many physical systems. Such a course may overlook several subtle points and the perceptive student will be left wondering why these techniques work. Real analysis provides the justification for the techniques and this analysis often provides precise conditions under which the computational techniques can be applied. Real Analysis I is an introduction the analysis of the arguments behind the computional techniques of Calculus.

4 Important dates The class meets Mondays, Wednesdays and Fridays from 9:30 to 10:30 in Lecture Hall 203 in the North Ross building. The first lecture is on Monday, September 13. The Mid-Term Examination will be held on Friday, November 5. There are no lectures on October 11, 13 and 15. The last date to drop this course without receiving a grade in November 12. The last lecture is on Friday, December 10. The final examination will be held some time between December 12 and December 23.

5 Evaluation The final grade will be determined on the basis of assigned work, a mid-term examination and a final examination according to the following scheme: Graded assignments: 20% Mid-term examination: 30% Final examination: 50% Students of this course will be able to find their scores for the various components of the final grade posted here as they become available.

6 The text The text for the course is: A Radical Approach to Real Analysis, second edition by David M. Bressoud, ISBN Mathematical Association of America. The course will cover Chapters 1 to 5 in detail and cover Chapter 6 according to the time available. Additional resources can be found on the course text web site maintained by David M. Bressoud, the author of the text.

7 Figure: Cover of the textbook.

8 How to read the text The sections of the text will be covered in the same order as presented in A Radical Approach to Real Analysis. You will be told which sections will covered in upcoming lectures and will be expected to have read those sections before the lecture. You will also be expected to re-read them, with greater understanding, after the lecture. In other words, between any two lectures you will re-read the material of the previous lecture and read for the first time the material of the upcoming lecture.

9 Exercises Exercises will be assigned after most lectures. These will also be listed on the course web site with due dates. Not all exercises will be marked; but some will be collected for marking on a random basis. The exercises form an essential part of the course and it is not possible to successfully complete the course without seriously attempting all the exercises. Some assigned exercises may return as questions on the mid-term and final examinations.

10 From the preface to A Radical Approach to Real Analysis This course of analysis is radical; it returns to the roots of the subject. It is not a history of analysis. It is rather an attempt to follow the injunction of Henri Poincare to let history inform pedagogy. It is designed to be a first encounter with real analysis, laying out its context and motivation in terms of the transition from power series to those that are less predictable, especially Fourier s into which even great mathematicians have fallen.

11 More from the preface to A Radical Approach to Real Analysis The traditional course begins with a discussion of properties of the real numbers, moves on to continuity, then differentiability, integrability, sequences, and finally infinite series, culminating in a rigorous proof of the properties of Taylor series and perhaps even Fourier series. This is the right way to build analysis, but it is not the right way to teach it. It supplies little motivation for the early definitions and theorems. Careful definitions mean nothing until the drawbacks of the geometric and intuitive understandings of continuity, limits, and series are fully exposed. For this reason, the first part of this book follows the historical progression and moves backwards. It starts with infinite series, illustrating the great successes that led the early pioneers onward as well as the obstacles that stymied even such luminaries as Euler....

12 Even more from the preface to A Radical Approach to Real Analysis There is an intentional emphasis on the mistakes that have been made. These highlight difficult conceptual points. That Cauchy had so much trouble proving the mean value theorem or coming to terms with the notion of uniform convergence should alert us to the fact that these ideas are not easily assimilated. The student needs time with them. The highly refined proofs that we know today leave the mistaken impression that the road of discovery in mathematics is straight and sure. It is not...

13 Fourier and the heat problem

14 The heat equation The key physical fact is that the rate of cooling (or heating) between two adjacent regions is proportional to the difference in their temperatures. Let z(x, w, t) denote the temperature at the point (x, w) at time t. Given a square with bottom left hand corner at (x, w) and height and width h the amount of heat the square loses at the left hand side from right to left is h z(x, w, t) x while the amount of heat the square loses at the right hand side from left to right is noting the change in sign. h z(x + h, w, t) x

15 The heat equation continued Hence, the total change of heat from left to right is: ( ) h z(x, w, t) z(x + h, w, t) x x where physical constants have been ignored. Similarly, the total change of heat from left to right is: ( ) h z(x, w, t) z(x, w + h, t) x w

16 The heat equation continued On the other hand, the total amount of heat lost or gained should be proportional to the area of the square and the change in temperature and so h 2 z(x, w, t) t h 2 z(x, w, t) = t ( ) h z(x, w, t) z(x + h, w, t) + z(x, w, t) z(x, w + h, t) x x w w and so...

17 The heat equation continued x z(x, w, t) = t ( ) z(x, w, t) z(x + h, w, t) + ( ) z(x, w, t) z(x, w + h, t) h w h Taking the limit of the right hand side as h goes to 0 yields: 2 2 z(x, w, t) = z(x, w, t) + z(x, w, t) t x 2 w 2

18 The heat equation obtained In the steady state, when the temperature is no longer changing it follows that t z(x, w, t) = 0 and hence the variable t can be ignored and the heat equation becomes: 2 2 z(x, w) + z(x, w) = 0 x 2 w 2 This is the equation in which Fourier was interested.

19 Solutions to the heat equation Fourier was interested in finding solutions for z such that z(0, x) = f (x) for some specified function f. The function f specifies the temperature at which the bottom edge of Fourier s plate is maintained.

20 Solutions to the heat equation

21 Solutions to the heat equation Since φ and ψ depend on different variables it must be that φ(w)/φ (w) and ψ(x)/ψ (x) are both constant and, hence φ (w) = Aφ(w) ψ (x) = Aψ(x) and these equations should be familiar from first year calculus.