### George M. Bergman -- undergraduate course materials

Introduction. This webpage contains material of various sorts that has come out of undergraduate courses I have taught; mostly supplementary notes, supplementary exercises, and collections of answers to students' questions from upper division courses.  Most of the handouts are PostScript files; a few are in pdf.  (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 collections of answers to students' questions are in plain text.  I required submission of such questions in my non-giant-lecture courses, so the undergraduate courses for which I have such collections are the upper division ones, and honors calculus.

Non-text-specific handouts.  I begin with the mathematical handouts, ordered roughly according to the level of the course in which it was used.

• Mathematical induction.  An elementary introduction to mathematical induction and why it works.  Written for an honors calculus section, but usable in the non-honors version of that course.  Here is a version without references to the calculus text, which might be used in other courses (e.g., in pre-calculus, or for review in higher-level courses). Last revised Oct. 2014, 5 pp.

• 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 Oct. 2004.

• A useful principle in solving differential equations.  Points out the following approach to solving first-order differential equations:  Find a 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.  This is used to motivate the standard methods of solving first order linear differential equations -- first homogeneous, then non-homogeneous.  Ends with an optional (for Math (H)1B) section on equations of the form  y' = f(y/x).  Recommended for reading just before textbook's development of linear differential equations. Written up Fall 2008 for Math 1B; revised 2010, 2012, 2014 for Math H1B.  Also useful for Math 54.  6 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 four 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 (namely, with or without a 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?  (written to go with Fraleigh's First Course in Abstract Algebra, for Math 113, Spring 2000; 2 pp.) and
The criterion for a finite abelian group to be cyclic  (rewritten to go with Dummit and Foote's Abstract Algebra for Math H113, Spring 2009, 1p.).
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.  Both the above texts prove this using the structure theorem for finitely generated abelian groups, a result I don't try to squeeze into Math (H)113; so I have prepared these handouts proving the cyclicity criterion directly (and with the hypothesis on solutions to  an=e  weakened to apply only to prime values of  n).  The writeups follow the texts' notations; in particular, in the version for Fraleigh, the image of a set  X  under a map  f  is denoted  f[X].

• 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).  The above version was written for Math 113, taught Spring 1997 from Fraleigh, revised in Spring 2009 when I prepared; this version, for Math H113, taught from Dummit and Foote.  1p.

• Mathematical Problem Seminar -- class log.  We used to give a "problem solving seminar", Math H117; this has since been replaced by Math 191, the "Putnam Preparation" course.  In these notes from the last time I taught Math H117, I list, for each day, the new problems that were suggested, and the progress the class had made on old problems.  I recommend this for students interested in tackling challenging questions.

Here is some non-mathematical material relevant to mathematics students:

• The Greek alphabet. 1p.

• 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.
• Text-specific handouts.    These are mostly additional exercises that I have put on homework sheets when teaching from the text in question.  Some of the exercises are comparable in difficulty to those in the text, but fill a gap or give some interesting perspective on the material.  Others I have not assigned, but have noted on the homework sheets that "students interested further interesting and/or in more challenging problems" might like to look at them; these might be appropriate for an honors course in the same material.

Here are the collections for lower-division courses:

The next few items concern texts for upper-division courses, and along with supplementary exercises, they include files labeled "Answers to students questions".  In these courses, I require every student to submit a question on each day's reading.  I incorporate the answers to some of these into my lectures; others I answer by e-mail.  I maintain cumulative web-pages of answers sent by e-mail on points that seemed important enough that other students in the class might want to refer to them.  (See the handout Mathematical symbols in e-mail above for the conventions I follow regarding such symbols in these answers.)  Some of the answers in these files have been edited retrospectively to improve clarity etc.
Unfortunately, answers to the questions asked by the greatest numbers of students may not get into these files, since they are answered in class.  Files of the same sort for my graduate courses (only one so far) can be found at the end of my page of graduate course materials