College of the Holy Cross Mathematics and Computer Science
Mathematics 242, section 2 -- Principles of Analysis
Syllabus, Spring 2014
Professor: John Little
Office: Swords 331
Office Phone: 793-2274
email: little@mathcs.holycross.edu, or jlittle@holycross.edu
Office Hours: MF 9 - 10am, 11 - 12noon; T 10 - 12noon, W 3 - 5pm, R 3 - 4pm, and by appointment
Course Homepage: http://mathcs.holycross.edu/~little/Principles14/PrinHome.html
(also accessible through Moodle)
The Theory of Calculus
In your study of calculus to date, you have learned how the techniques of calculus
and how to apply them to solve many types of problems. You have approached the subject
in much the same way that Newton, Leibniz, their contemporaries and immediate successors
in the 17th and 18th centuries did. But the courses you have taken
probably have not spent much time addressing exactly why those
methods work (in the cases they do work), or their limitations (the
situations where they do not work, or the exact hypotheses needed to make them
work).
In the historical development of the subject, part of what happened
after this initial exploratory period was that extremely clever, but
intuitive and rather uncritical uses of infinite series, derivatives and integrals
eventually led to some apparently nonsensical results. The situation had to
be clarified to ensure the reliability of calculus in its applications
to physics and other areas of science. And even after these apparently
nonsensical results were explained or clarified, unexpected and strange properties
of sets of numbers and functions continued to emerge. Mathematicians
wanted to be certain that what they were discovering was really true
and not just a result of confusion or error.
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 the mathematicians
Cauchy, Riemann, Weierstrass and others. Their
collective work put calculus on a sound footing and gave mathematicians
the tools needed for complete proofs of the key results.
Text
The text for this course is An Introduction to Analysis, 2nd edition
by Gerald Bilodeau, Paul Thie, and G.E. Keough, published by Jones and Bartlett
(BTK in the following). We will cover most of the material
in Chapters 1 - 6 of the book over the course of the semester.
Topics to be Covered
We will be studying the following topics this term:
- Unit I: (Chapter 1 of BTK)
The real number system and its properties (about 7 class days)
- Unit II: (Chapter 2 of BTK)
Sequences of real numbers (about 8 days)
- Unit III: (Chapter 3 of BTK)
Functions and Continuity (about 6 days)
- Unit IV: (Chapter 4 of BTK)
Derivatives (about 5 days)
- Unit V: (Chapter 5 of BTK)
Integrals and the Fundamental Theorem (about 5 days)
- Unit VI: (Chapter 6 of BTK)
Infinite series (about 6 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 (think "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 sequence
- 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 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.
- 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. On a deeper level, it is even 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 (those
who work on problems directly inspired by questions 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! Moreover, I think you will find our textbook to be
an excellent resource, especially for the motivation behind the
topics we study. 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
Most classes will follow a lecture/discussion format. I will pause
frequently and ask questions to make sure the class is following
the development of the topic for the day and to keep you involved.
To get you more directly involved in the subject matter of this course,
several times during the semester, I may break the class down into groups
of 3 or 4 students for one or more days to work together on a
group discussion exercise.
Grading
The assignments for the course will consist of:
- Three in-class midterm exams, together worth 45% of course grade.
The (tentative) dates are
- Friday, February 21,
- Friday, March 28, and
- Friday, May 2.
Since everyone can have a bad day occasionally, your lowest in-class exam score will
be weighted less than the other two (9%,18%,18%).
- Final exam worth 30% of the course grade. The final
exam for this course will be given at the established time for MWF 10am classes (to
be announced later in the semester by the Registrar). Please do not make
any travel plans for the summer break before this is determined!
- Weekly individual problem sets, worth 15% 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 count for
5% of your course grade.
- The remaining 5% of your course average -- class participation and
possible group discussion write-ups.
I will be keeping your course average in numerical form throughout
the semester, and only converting to a letter for the final course
grade. The course grade will be assigned according to
the following conversion table (also see Note below):
- A -- 94 and above
- A- -- 90 - 93
- B+ -- 87 - 89
- B -- 84 - 86
- B- -- 80 - 83
- C+ -- 77 - 79
- C -- 74 - 76
- C- -- 70 - 73
- D+ -- 67 -- 69
- D -- 60 - 66
- F -- 59 and below.
Note: Depending on how the class as a whole is doing, some
downward adjustments of the above letter grade boundaries may be made.
No upward adjustments will be made, however. (This means, for
instance, that an 85 course average would never convert to a letter
grade of B- or below, but it might convert to a B+ in some cases.)
If you ever have a question about the grading policy, or about your
standing in the course, please feel free to consult with me.
Advice On How To Succeed In This Class
A good "work ethic" is key but it may not be enough. Contrary
to what you might believe, mathematics professors definitely want you to learn and
do well in their courses! But you will also need to do your part by
working hard. That is certainly necessary to succeed
in this course. Unfortunately, though, hard work and
dedication alone might not be sufficient.
Mathematics at this level is not an easy subject for anyone.
It requires a certain kind of very careful and logical thinking based
on a deep knowledge of definitions of terms and basic examples
that is simply not everyone's "cup of tea."
Come to class. Unless you are deathly ill, have a genuine family
emergency, have to be away at a College athletic event, etc.
plan on showing up here at 10:00am every Monday, Wednesday, and Friday
this semester. If attending class wasn't important, all college courses
would be run as MOOCs or by correspondence, and your tuition would be much lower!
Read the textbook. Don't just use it
to look for worked problems similar to ones on the problem sets. (In this
course, there won't be many of those in any case!)
You will find alternate explanations of concepts that may help
you past a "block" in your understanding. Reading a math book
is not like reading a novel, though. You will need to read very
carefully, with pencil and paper in hand, working through
examples in detail and taking notes. Make a list of questions to
ask in office hours or at the next class. One thing to bear in mind
while reading your text is that the result of an example is often
secondary to the process used in obtaining the result.
Take notes and use them. This may seem obvious, but in my
experience too many students diligently copy down everything on the board,
and then never look at their notes again. Used intelligently, your notes
can be a valuable resource as you work on problem sets and prepare for exams.
One method that works for some people is to create a second, polished
set of notes for the course by recopying the notes you take in
class, filling in details or providing other explanations as you go.
You will need to do this carefully, though. The idea of this method
is for you to think about what you are recopying as you do it, not
just to act as a stenographer.
Set up a regular study schedule and work at a steady pace. It's
not easy to play catch-up in a mathematics course, since every day builds
on the previous one. You should expect to budget at least 6 hours in a typical week
for work outside of class. The best way to use your time is to
do a few problems, a little reading from the
book, and reviewing or recopying of class notes every day.
Most importantly, if you are having difficulty learning something,
get help as soon as possible. You can do this by asking questions during
class (any time something isn't clear), or by seeing me during office hours.
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 the possible 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.