First things first
0.1. The goals
Our goal in this class is threefold:
(1) to obtain a body of knowledge in Advanced Calculus, the basis of the analysis of
real-valued functions of one real variable;
(2) to learn how to communicate ideas and facts in both a written and an oral form;
(3) and, perhaps most importantly, to become acquainted with — indeed, to master
— the process of creating mathematics.
In conducting this class we shall try to model a mathematical community in which
both collaboration and competition are prevalent.
This community is — no, you are —
on the verge of discovering the foundations for a number of rules and recipes which have
been successfully in use for some time. In the process you will recreate a body of knowledge
almost as if you were the first to discover it. However, as we have only nine months to do
this rather than a century or two, there will be some help available to you, most prominently
in the form of these notes which will delineate broadly a path in which discovery will (or
In this course it is allowed and, in fact, required to criticize the person on the board for
flaws or incomplete arguments (you are a scientific community). Criticism has to be leveled
in a professional manner, in particular, it has to be free from any personal insults. At the
same time you have to learn to accept criticism without taking it personally. By learning
to stand up for your ideas (or to accept that you made a mistake) you may get something
out of this course which is of value not only in mathematics.