College of the Holy Cross Mathematics and Computer Science
Mathematics 242 -- Principles of Analysis
Syllabus, Fall 2004
Professor: John Little
Office: Swords 335
Office Phone: 793-2274
email: little@mathcs.holycross.edu (preferred), or jlittle@holycross.edu
Office Hours: MW 1-3pm, TR 9-11am, F 10-11am, and by appointment
Course Homepage: http://math.holycross.edu/~little/Prin04/PrinHome.html
Course Description
In your study of calculus to date, you have approached the subject
from much the same intuitive point of view that Newton, Leibniz,
and their contemporaries and immediate successors
employed. You have learned how to apply the techniques of calculus
to solve many types of problems, but the courses you have taken
have not spent much time addressing exactly why those
methods work (in the cases they do work), or their limitations (the
exact hypotheses needed to make them work). Extremely clever, but intuitive
and rather uncritical uses of infinite series,
derivatives and integrals eventually led to some apparently nonsensical
results that needed to be clarified to ensure the reliability
of calculus in its applications to physics and other areas of
science.
In the education of each mathematician, as in the history of the
subject itself, there comes a time when the
intuitive treatment of calculus must give way to a deeper understanding
of the theoretical foundations. For you as a student of mathematics,
that time is now! In this class we will essentially repeat
some of the historical developments of the 19th century and study the
principles of analysis, the theory of calculus that
emerged at that time. These steps were first taken by Cauchy, Riemann,
Weierstrass and others whose collective work put calculus on a sound
footing and gave complete proofs of the key results.
Text
The text is Analysis With An Introduction to Proof, 3d edition
by S. Lay, published by Prentice-Hall. We will cover most of the material
in Chapters 1 - 8 of the book over the course of the semester.
Topics to be Covered
We will be studying the following topics this term:
- Unit I: (Chapters 1 and 2 of text)
Logic, Proof, and Set Theory (about 7 days)
- Unit II: (Chapter 3 of Lay)
The real number system and its properties (about 6 days)
- Unit III: (Chapter 4 of Lay)
Sequences of real numbers and convergence (about 5 days)
- Unit IV: (Chapter 5 of Lay)
Limits of real-valued functions of a real
variable. Continuity and its consequences. (About 5 days)
- Unit V: (Chapters 6,7 of Lay)
The theory of calculus, with complete proofs. The definitions of the
derivative and definite integral reconsidered. "Interestingly pathological"
examples -- functions that are non-differentiable, non-integrable.
The Mean Value Theorem and its consequences.
The Fundamental Theorem of Calculus. (About 8 days)
- Unit VI: (Chapter 8 of Lay) Infinite series. (About 5 days)
The remaining 3 class days will be devoted to in-class exams.
There is a more detailed provisional day-by-day schedule posted
on the course web site. As always, it may become necessary
to add, subtract, or rearrange topics as we proceed into the
course. Any changes will be announced in class and on the web
site.
Comments About This Course
Together with the Algebraic Structures/Linear Algebra sequence, this
course forms part of the "basic training" (i.e. "boot camp") for mathematics
majors. These intermediate level courses serve as bridges from
the basic mathematics you have seen in high school and in the calculus courses
to the more advanced courses you will normally take in your junior and senior
years.
One of the major goals of these courses is to build your skill
in developing and writing mathematical proofs.
There will almost certainly be times
when the "boot camp" analogy seems painfully apt -- many Mathematics majors
find these courses to be among the most challenging they take at Holy Cross
because the whole way we work may well seem unfamiliar and
baffling at first:
- Many problems we will do will not have a single number
or formula as an answer and more generally, learning ways to
perform calculations will not usually be the main focus of
what we do in the class.
- You will definitely not master this material by learning
rote procedures for solving a set collection of different types of problems.
- A plausible reason for expecting something is true will not be
sufficient.
Instead, in many problems the goal will be first to develop, and
then to explain clearly, the complete logical argument that
establishes the truth of a statement, starting from a clearly
stated set of assumptions and using results established previously.
You will have to think about the logical structure of the subject matter
and understand the definitions of concepts and the statements and proofs of
theorems to get the "skeleton" of the subject firmly in place in your mind.
You will need to commit a collection of key examples to memory and be able to
reason about their properties. You will need to document much more of your
thinking about problems than you probably have done before. While
a good intuition may guide you to correct statements, just making correct
statements will not be enough.
At times, you may ask why we are doing this. The ultimate answer is
that this "abstract" proof-oriented work is the way all mathematics
is communicated and in a sense it is what most of advanced
mathematics is really about. So you have to be prepared for it
if you decide to continue! The concept of mathematical proof is
the unique and distinctive feature of this branch of knowledge;
I think it is no exaggeration to say that it is one of the crowning
achievements of the human intellect. Even applied mathematicians (ones
who work on problems directly inspired by problems in the real world)
must develop new ideas to solve those problems, and then provide
convincing evidence (proofs) that what they claim is true so that others can
follow what they do.
Lest this sound too grim, be aware that you will have lots
of chances to develop and practice these new skills, and I will always be
willing to give you the benefit of my experience working with this kind
of mathematics! Even if you find this difficult at first, persistence
and openness to a different way of thinking will usually pay off in time.
Course Format
To get you more directly involved in the subject matter of this course,
several times during the semester, the class will break down into groups
of 3 or 4 students for one or more days, and each group will work together
for a portion of those class periods on a group discussion exercise.
The exercises will be made up by me. I will be present and available for
questions and other help during these periods. At the conclusion of some
of these discussions, at times the class as a whole may
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 be
responsible for a write-up of solutions for the questions from each discussion
day, and those will be graded and and returned with comments.
Grading
The assignments for the course will consist of:
- Three in-class midterm exams, together worth 40% of course grade.
The (tentative) dates are
- Friday, October 1,
- Friday, October 29, and
- Friday, December 3.
- Final exam worth 25% of the course grade. The final
exam for this course will be given Wednesday, December 15, at 8:30 a.m..
- Individual problem sets, worth 20% of the course grade.
- Group reports from discussion days, worth 10% of the course grade.
- Weekly ``definitions quizzes'' -- given the first five minutes
of each Friday class when an hour exam is not scheduled.
The average of the best 5 out of the 9 quizzes will be count for
the remaining 5% of your course grade.
If you ever have a question about the grading policy, or about your
standing in the course, please feel free to consult with me.
Departmental Statement on Academic Integrity
Why is academic integrity important?
All education is a cooperative enterprise between teachers and
students. This cooperation works well only when there is trust and
mutual respect between everyone involved.
One of our main aims as a department is to help students become
knowledgeable and sophisticated learners, able to think and work
both independently and in concert with their peers. Representing another
person's work as your own in any form (plagiarism or ``cheating''),
and providing or receiving unauthorized assistance on assignments (collusion)
are lapses of academic integrity because they subvert the learning process
and show a fundamental lack of respect for the educational enterprise.
How does this apply to our courses?
You will encounter a variety of types of assignments and examination
formats in mathematics and computer science courses. For instance,
many problem sets in mathematics classes and laboratory assignments
in computer science courses are individual assignments.
While some faculty members
may allow or even encourage discussion among
students during work on problem sets, it is the expectation that the
solutions submitted by each student will be that student's own work,
written up in that student's own words. When consultation with other
students or sources other than the textbook occurs, students should
identify their co-workers, and/or cite their sources as they would for
other writing assignments. Some courses also make use of collaborative
assignments; part of the evaluation in that case may be a rating of each
individual's contribution to the group effort.
Some advanced classes may use take-home
examinations, in which case the ground rules will usually allow no
collaboration or consultation.
In many computer science classes, programming projects are
strictly individual assignments; the ground rules
do not allow any collaboration or consultation here either.
What are the responsibilities of faculty?
It is the responsibility of faculty in the department to
lay out the guidelines to be followed for specific assignments in
their classes as clearly and fully as possible, and to
offer clarification and advice concerning those guidelines
as needed as students work on those assignments.
The Department of Mathematics and Computer Science upholds the
College's policy on academic honesty.
We advise all students taking mathematics or computer science courses
to read the statement in the current College catalog carefully and
to familiarize themselves with the procedures which may be
applied when infractions are determined to have occurred.
What are the responsibilities of students?
A student's main responsibility is to follow the guidelines laid down
by the instructor of the course. If there is some point about the
expectations for an assignment that is not clear, the student is responsible
for seeking clarification. If such clarification is not immediately available,
students should err on the side of caution and follow the strictest possible
interpretation of the guidelines they have been given.
It is also a student's responsibility to protect his/her
own work to prevent unauthorized use of exam papers, problem solutions,
computer accounts and files, scratch paper, and any other materials used in
carrying out an assignment. We expect students to have the integrity to say
``no'' to requests for assistance from other students when offering that
assistance would violate the guidelines for an assignment.
Specific Guidelines for this Course
In this course, all examinations and quizzes will be closed-book and given in-class.
No sharing of information with other students in any form will
be permitted during exams and quizzes. On group discussion write-ups,
close collaboration is expected. On the problem sets, discussion of the
questions with other students in the class, and with me during office
hours is allowed, even encouraged.
BUT, if you do take advantage of any of these
options, you will be required to state that fact in a "footnote"
accompanying the problem solution. Failure to follow this rule
will be treated as a violation of the College's Academic
Integrity policy.