George M. Bergman -- undergraduate course materials
Most of the items below are handouts I use in various undergraduate
courses.
These can be viewed or downloaded as PostScript files.
(The source files are in locally enhanced troff, so I can't
provide TEX files, but here is a
link
to software that can be used in viewing PostScript on a Windows system.)
The next few items contain mathematical material related
to one or more courses, but not to a particular text.
I have put them roughly in the order of the level of the course.
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The idea of a matrix.
Discussion of an mxn matrix as representing a linear
transformation from n-tuples of real numbers to m-tuples
of real numbers.
Written for our two sophomore calculus courses, Math 53
(multivariable calculus without linear algebra)
and Math 54 (linear algebra and differential equations).
I also give it out as a review sheet in the upper-division abstract
algebra course, Math 113. 1p., last revised Spring 2000.
-
A general principle in solving differential
equations.
Points out that methods used for solving several special sorts
of differential equations, including equations
of the forms y' = f(y),
y' = f(y/x),
y' = P(x)y,
and y' = P(x)y + Q(x),
can all be derived from one general principle: Find
a 1-parameter family of transformations of the plane that must take
solutions of the given differential equation to other solutions,
and make a change of coordinates so that these transformations
become vertical translations.
The differential equation then reduces to an integration.
Optional handout for differential equations material in
lower-division courses Math 1B and Math 54;
written up Fall 1999 for Math 1B. 1.5 pp.
-
Some notes on sets, logic, and mathematical
language.
Basic set-notation, logical connectives, quantifiers; how
changing order of quantifiers changes
the meaning of statements; meanings of some phrases
such as ``well-defined'' and ``without loss of generality''.
For use as a supplement in any of the basic upper-division courses.
12pp.; last revised July 2008.
The
version you get by clicking above uses
"blackboard bold" symbols for integers, real numbers, etc..
You can get a similar version with regular
boldface instead (which also notes
the existence of the blackboard bold notation), and three
versions that are tailored for use with specific texts, and that
note points about those authors' notation: One
for Rudin's Principles ...,
one for
Beachy and Blair's Abstract Algebra,
one for
Friedberg, Insel and Spence's Linear Algebra,
and one for
Rotman's First course in Abstract Algebra.
The last two of these use ":" rather than "|" for "such that" in
set-brackets.
My source file is set up so that if you want a version with a
particular combination of notations for integers/reals/etc.,
for subsets and supersets (with or without bar on bottom),
and for "such that", which does not refer specifically to one
of the abovementioned texts, I can fairly easily create one for you.
I also have a short file of
Answers to students' questions
on the above handout, accumulated over several semesters.
-
Proof that the group An is
simple for all n>=5.
Written Spring 2000 for Math 113. 3pp.
-
When is a finite abelian group cyclic?
To show that a finite subgroup of the multiplicative group of a field
is cyclic, one needs to know that an abelian group which, for
each n, has at most n elements a satisfying
an=e is cyclic.
Fraleigh's First Course in Abstract Algebra (or at least,
the edition thereof that I most recently taught from) proves this using
the structure theorem for finitely generated abelian groups, but
that is too heavy a result to fit into Math 113; so I have prepared
this handout proving the cyclicity criterion directly
(and with the hypothesis on solutions to an=e
restricted to prime values of n).
The writeup follows Fraleigh's notation; in particular, the image
of a set X under a map f is denoted f[X].
Written Spring 2000 for Math 113. 2pp.
-
Sketch of my favorite proof of the First Sylow
Theorem.
This is the "orbit counting" proof, but with a twist: The
fact that a certain binomial coefficient is not divisible by
p, needed for the proof, is proved not by number theory,
but by running the proof backwards in the case of a particular
group of the same order which we know has a Sylow subgroup
(namely, a cyclic group).
Written Spring 1997 for Math 113. 1p.
Finally, some non-mathematical material relevant to
mathematics students:
-
Mathematical symbols in e-mail.
Because of a requirement which I generally make in my upper division
and graduate courses, that students must submit a question about
the reading on each day when there is a reading assignment,
I have a lot of e-mail correspondence with the class.
In this handout I note that mathematicians today generally
express symbols in their e-mail in TEX (and give
an example), then recommend some conventions for use in my classes,
which borrow some features from TEX, but don't
look so technical. 1p. Last revised Fall 2001 for Math 113.
-
On Incomplete grades. Rules and
procedures regarding the grade of ``Incomplete'' in Berkeley courses.
The above is in html; to get it to print on one page, you need
to fiddle with the "Print preview" option on your browser.
Here is a PostScript
version which does so without extra work.
-
The Greek alphabet. 1p.