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\centerline{Mathematics 42--Principles of Analysis}
\centerline{Course Information--Spring 1997}
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\leftline{{\bf Professor:}  John Little}
\leftline{{\bf Office:} Swords 335}
\leftline{{\bf Office Phone:}  793-2274}
\leftline{{\bf Office Hours:}  TBA, and by appointment}
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{\bf Course Description}
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The early part of the history of a mathematical subject is
often characterized by explosive growth of knowledge, as new methods
and tools are introduced that are capable of solving problems that were
previously difficult or even impossible.  Frequently, though, those
methods that clearly ``work'' are really
only partially understood.  This was true especially of the
development of the calculus by Newton, Leibniz, and their successors
in the 1600's and 1700's.  Extremely clever, but intuitive and 
uncritical uses of the rules for derivatives and integrals eventually
led to some apparently nonsensical
results.  One big feature of the mathematics of the 1800's was 
a reconsideration of the theoretical foundations of calculus and 
various (ultimately successful)
attempts by Cauchy, Riemann, Weierstrass and many others 
to put the subject on a sound footing and give complete
proofs of the key results.  (It is important to note
that this was anything but a case of pursuing  
abstraction for its own sake. The concern over the theory of calculus was 
motivated largely by a desire for {\it reliability\/} in the 
{\it application\/} of calculus to physics and other areas of science.) 

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In your study of calculus to date, you have approached the subject 
from much the same intuitive point of view that Newton and Leibniz
employed.  You have learned how to apply the techniques of calculus
to solve many types of problems, but the courses you have taken 
have probably not spent much time addressing exactly why those 
methods work (in the cases they do work), or their limitations (for
instance, the exact hypotheses needed to make them work).

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In the education of 
each mathematician, there comes a time when the intuitive treatment
of calculus must give way to a deeper understanding of the theoretical
foundations of the subject.  For you, that time is {\it now\/}!
In this class we will, in a sense, repeat some of the 
historical developments of the 1800's and consider the principles
of analysis, the {\it theory of calculus\/}.

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The topics we will study this semester are:
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\itemitem{Unit I:}  (Chapter 1 of Bilodeau and Thie) 
The ``setting'' for calculus -- the natural numbers 
(${\bf N}$), the rational numbers (${\bf Q}$), and the real numbers 
(${\bf R}$). (About 5 days)
\itemitem{Unit II:} (Chapter 2 of B+T) 
Sequences of real numbers. (About 8 days) 
\itemitem{Unit III:} (Chapter 3 of B+T)
  Properties of real-valued functions of a real
variable.  Continuity and its consequences.  (About 8 days)
\itemitem{Unit IV:} (Chapters 4,5 of B+T)
 The theory of calculus. The definitions of the 
derivative and definite integral reconsidered.  ``Pathology'' -- functions
that are non-differentiable, non-integrable.  The Mean Value Theorem and
its consequences -- how information about $f'$ yields information
about $f$.  The Fundamental Theorem of Calculus. (About 10 days)
\itemitem{Unit V:} (Chapter 6 of B+T) Infinite series. (About 7 days) 
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The remaining two days will be devoted to in-class exams.  See below for 
tentative dates.  As
always, it may become necessary to add, delete, or rearrange topics as we
progress through the course.
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{\bf Text}
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The text is {\it An Introduction to Analysis\/}
by G.~Bilodeau and P.~Thie (both faculty members at ``that other Jesuit
college in Chestnut Hill'' -- don't hold that against them, though!)
We will cover most of the material in Chapters
1 - 6 of the book over the course of the semester. 
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{\bf Course Format}
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In order for a student to get as much as possible out of this or any
course, regular active participation and engagement with the ideas
we discuss are necessary. 
To get you more directly involved in the subject matter of this course,
regularly throughout the semester the class will break down into groups
of 3 or 4 students for one or more days, and each group will work individually
for (a portion of those) class periods on a group discussion exercise.
I will be responsible for designing and preparing these exercises, and 
 I will be available for questions
and other help during these periods.  At the conclusion of some
discussions, the class as a whole will reconvene to talk about what has
been done, to sum up the results, to hear short oral reports from each group,
etc.  Each group will keep a written record of their
observations, results, questions, etc. which will be handed in.  I will
return them with comments.  Some of the meetings of the class will also 
be structured as lectures when that seems appropriate.
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{\bf Grading}
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The assignments for the course will consist of:
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\item{1.} Two midterm exams  
together worth $40\%$ of the course grade.  Tentative dates:
\itemitem{a.}  Wednesday, February 26.
\itemitem{b.}  Friday, April 18.
\item{2.} Final exam worth $30\%$ of the course grade.  The final exam for 
this course will be held at 2:30 p.m. on Wednesday, May 7.  
\item{3.} Individual problem sets, worth $15\%$ of the course grade.
\item{4.} Group reports from discussion days, worth $15\%$ of the course grade.
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If you ever have a question about the grading policy, or about your 
standing in the course, please feel free to consult with me.
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