Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I: (3) Next: Homework Assignment 1
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About This Class

URL: http://www.math.toronto.edu/~drorbn/classes/0304/157AnalysisI/.

Agenda: Understand calculus and its rigorous foundations.

Instructor: Dror Bar-Natan, drorbn@math.toronto.edu, Sidney Smith 5016G, 416-946-5438. Office hours: Thursdays 12:30-1:30.

Classes: Tuesdays 10-11 and Thursdays 9-11 at Mining Building 128.

Tutorials: Mondays at 2-4, in three groups divided by the last non-zero digit of your student number:

digit tutor place Tipu, Hung, Ivanescu
1-3 Vicentiu Tipu, vtipu@math.toronto.edu, SS 4055, 8-4794 RW 142
4-6 Ching-Nam Hung, cnhung@math.toronto.edu, SS 4052, 8-3484 LM 157
7-9 Cristian Ivanescu, cristian@math.toronto.edu, SS 6027, 8-2095 UC 328

Switching Tutorials: Really, I couldn't care less which tutorial you attend, but the tutors may well care because this determines how much grading they will have to do. Thus switches are automatically approved by me but need further approval by the receiving tutor who has the right to say no without any explanation.

Textbook: Michael Spivak's Calculus, 3rd edition (1994).

Lecture Notes: I'll be happy to scan the lecture notes of one of the students after every class and post them on the web. We need a volunteer with a good handwriting!

Course Description: Calculus is one of the glories of modern mathematics. From its distant beginnings with Archimedes, through its systematic formulation by Newton and Leibniz, it has been one of our most powerful tools for understanding the world around us. Nonetheless, real understanding of the concepts of function, limit, and even real number required centuries of work. This hard-won understanding made possible the dramatic mathematical developments of the twentieth century, and is today the starting point for study of mathematics at the university level.
In this course, we develop the theoretical foundations of calculus, emphasizing proofs and techniques, as well as the geometric and physical understanding that underlies them. All results will be proved; our point of view is that we do not understand anything until we can prove it, and the methods of proof themselves lead us to develop techniques for applied problems. We will cover nearly the entire textbook, approximately one chapter per week.

An Unfortunate Prelude: Unfortunately, along with the double cohort Ontario changed its high school math curriculum. This has little bearing with our course as a whole - our course is foundational and we will only care about trigonometry quite late in the year. But your other science classes will care very much about trigonometry very early, so we will have to spend about a week right at the start on that subject, postponing the real beginning of our class to the second week of classes. Prepare for trouble! We're going to jump right in, the pace will be rapid, and we won't follow a textbook. But it's a great topic and if you hold tight and aren't thrown out in one of the sharp turns, the ride will be a thrill.

Problem Sets: There will be about 20 problem sets, largely consisting of problems selected from the textbook. These will be handed out Tuesday in lecture, and will be due into the tutors' mailboxes in the Math Aid Centre, SS 1071, on the following Friday (10 days later) at 2PM. Late submissions will not be graded. I encourage you to discuss the homeworks with other students or even browse the web, so long as you do at least some of the thinking on your own and you write up your own solutions. Remember that cheating is always possible and may increase your HW grade a bit. But it will hurt your exam grades a lot more. Your final HW grade will be the average grade of your best 0.8n assignments, where n is the number of assignments that will be given throughout the year. This means you can skip a small number of assignments at a relatively small penalty.

Tests and Final Exam: We will have four term exams, written in the Monday tutorials, on October 20, December 1, February 9, and March 22. The final exam will be during the final examination period in April-May, 2003, and will cover the entire course. (The exact date will be announced in mid-February.) The last class in Fall semester will be Thursday, Deccember 5; we will not use the Fall examination period.

The Final Grade: I will compute a final numerical score using the weights

This done, I will then apply an appropriate monotone transformation to determine the final grades, which are reported both as a percentage and a letter grade. Thus the absolute percentage scores are not the same as the final grade. (In previous years mean exam scores have been 50% or less, yet not everyone failed.)

Feedback: I'd be very happy to hear from you. There's a link to a feedback form at the top of this class' web site (and here). Anonymous messages are fine, provided they are written with good intent. Though remember that if I don't know who you are I may not be able to address your concern. You will each be required to use this feedback form at least once, on the third week of classes (see below).

Class Photo: To help me learn your names, I will take a class photo on Tuesday of the third week of classes. I will post the picture on the class' web site and you will be required to use the feedback form to identify yourself in the picture.

Advice for Success: The most important thing is to keep up. You learn mathematics by doing problems, by thinking hard, and by discussing. Go to all the lectures, go to all the tutorials, and do not be afraid to ask questions. Some of the smartest mathematicians in history spent their lives without ever understanding the things you are about to learn; it is not surprising if you do not "get" everything the first time you hear it. The problem sets may seem difficult at first, but I promise, if you keep working on them, you will be able to look back on the early ones from later in the course, and be amazed by how simple everything seemed then.
The best way to prepare for the exams is to work lots of problems from the textbook, and to discuss them with your friends, the tutors, and me. The best way to prepare for the final will be to work lots more problems, and also to review the term exams.

Finally, here's our entry at the official UofT Calendar:

Analysis I 78L, 52T

A theoretical course in calculus; emphasizing proofs and techniques, as well as geometric and physical understanding. Trigonometric identities. Limits and continuity; least upper bounds, intermediate and extreme value theorems. Derivatives, mean value and inverse function theorems. Integrals; fundamental theorem; elementary transcendental functions. Taylor's theorem; sequences and series; uniform convergence and power series.
Exclusion: MAT137Y1
Prerequisite: Calc/MCA4U + A&G/MGA4U