Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I: | (3) |
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**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 | |

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

- Term exams: 5%, 15%, 10%, and 10%, respectively.
- Homework: 20%.
- Final Exam: 40%.

**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:

MAT157Y1

Analysis I 78L, 52TA 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