Math 4035: Advanced Multivariable Calculus

Classroom:  Lockett 113
Time:  TR, 1:30 – 2:50
Office Hours:  Monday 2:00 – 3:00, Thursday 12:30 – 1:30


Classroom


The Classroom
is a space for me to post lectures, videos and other materials related to this and other classes I’m teaching.


Homework Assignments


Communication-Intensive (CxC) Courses

This course will be taught as a Communication-Intensive (C-I) course, which means it meets all of the requirements set forth by LSU’s Communication across the Curriculum (CxC) program, including:

  • instruction and assignments emphasizing informal and formal writing and speaking;
  • teaching of discipline-specific communication techniques;
  • use of draft-feedback-revision process for learning;
  • practice of ethical and professional work standards;
  • 40% of the course grade rooted in communication-based work;
  • a student/faculty ratio no greater than 35:1.

Students interested in pursuing the LSU Distinguished Communicators certification may use this C-I course for credit.  For more information about this student recongnition program, visit CxC.


Textbook: The text for the course will be Analysis on Manifolds, by James Munkres. This is a fantastic textbook that will serve as a lasting resource for anyone interested in the subject. Professor Frank Jones also has a great textbook covering much of the material we’ll discuss.

Prerequisite: The prerequisite for this course is Math 4031.

Course Content

Advanced Multivariable Calculus is an amazing and beautiful subject that benefits tremendously from its close connections to analysis (epsilons and deltas), geometry (lengths and angles), and topology (shapes). Much of the beauty in the subject lies in untangling and exploring these connections. As its name suggests, this course will spend plenty of time on traditionally analytic maters.  There will be no shortage of epsilons, deltas, and inequalities.  Unlike in the first two semesters of analysis, however, we will often see that there are concrete geometric or topological pictures which help to illuminate some of the less clear analytic facts.


Communication-Intensive Activities:

A good mathematician is one who knows the statements to many theorems.  A great mathematician is one who also understands the proofs to most of the theorems they know.  An exceptional mathematician is one who, in addition to knowing a great many theorems and proofs, also knows examples illustrating why each hypothesis in a given theorem are necessary.

This simple statement illustrates a broad distinction between basic mathematical competency and true mathematical fluency.  By this point, I expect that all of you are mathematically competent.  That is, I expect that you understand what a mathematical statement (Theorem, Lemma, Proposition, etc.) is, and roughly what it means to prove a mathematical statement.  By this point in your mathematical careers, you probably all have a decent library of (true) mathematical statements floating around in your head.  Mathematical fluency is an entirely different beast.  It’s not just about memorizing lots of mathematical statements or even their proofs – mathematical fluency is about developing gut-level understandings of what certain mathematical statements are saying, why they are true, and how they fit into the broader mathematical tapestry.

Hand-in-hand with mathematical fluency is the ability to identify and speak appropriately to one’s audience. Who is trying to understand the mathematical statement or proof? Are you trying to convince yourself? A fellow classmate? An anonymous grader? Me? What may look and feel like a complete proof to someone with lots of mathematical training may be utterly unconvincing to a novice. By working hard to distill mathematical statements to as child-like a form as possible and to understand these statements and proofs at a gut-level, we position ourselves to be both better mathematicians and communicators.

Type 1: (writing) Mathematical fluency, thinking and writing

Type 1 (writing) assignments will be similar in spirit to traditional mathematics assignments.  They will center around solving mathematics problems and prove mathematical statements. In these assignments, students will:

  1. develop a basic understanding of why some mathematical statement or fact is true;
  2. identify and write down a formal, step-by-step, complete proof of said fact which is sufficient to convince an anonymous (graduate student) grader that the statement is true;
  3. distill and write down both the mathematical statement and its proof into as “child-like” a form as possible.

Type 2: (writing and speaking) Professor for (part of) a day

Type 2 (writing and speaking) assignments will focus on building the skills necessary for lecturing on mathematics. Though this might sound a bit daunting at first, the exercises should be lots of fun! In a Type 2 assignment, students will be asked to reproduce a segment of the class lecture (roughly 10 minutes worth). You are not expected to generate new mathematics.  You will have already listened to me lecture (hopefully, well!) on the relevant material, and should have somewhere between a basic to thorough understanding of the mathematics itself. Your task is to learn the material, distill it in your own mind until you have a clear understanding of why each statement is true and how they all fit into the broader arc of the class.  Your task is to then rewrite and present your segment of the class lecture. 

Type 3: (speaking) Lightning talks

Type 3 (speaking) assignments will center on developing the skills necessary to speak about interesting discoveries in mathematics.  In these assignments, students will be asked to present one a five-minute “lightning talk” on some theorem or example in complex analysis.  Unlike a traditional lecture, a lightning talk will not have the benefit of dozens of previous lectures laying a rigorous foundation for the topic being discussed. Students are allowed and encouraged to explore topics beyond the scope of lectures (I have lots of suggestions!).  That said, the material presented in each lightning talk must be understandable by a regular class attendee.


Homework and Grades:

There will be several flavors of homework problems assigned throughout the course.

C-I Assignments: There will be (BLANK) many Communication-Intensive assignments

  • Numerous assignments of Type 1 (one or two per week)
  • Two assignments of Type 2
  • One assignments of Type 3

Traditional Problems: There will also be numerous “Traditional” homework problems assigned throughout the semester.  These problems will come in two flavors: “pledged” and “unpledged”.  Unpledged problems are what you’re used to.  Students are allowed to (and encouraged to!) work together on unpledged problems.  Most traditional homework problems I assign will be unpledged.  A few of the problems I assign will be pledged.  Think of a pledged problem like a “take home exam problem”.  When solving these problems, you are free to consult the class notes, the textbook and with me.  You should not, however, discuss any aspect of pledged problems with fellow students.

Homework Policy:  Submitted homework must be turned in by 4:00 PM on its due date. If you know in advance you will be unable to turn in homework when it’s due, you should plan to turn it in ahead of time. Holidays are not an excuse for late homework! If you know that there is an impending holiday, it is your responsibility to make arrangements to submit homework early. Since this is a course which focuses in part on mathematical communication, it is extremely important that submitted work be neat, well-organized, and legible. Ragged margins, multi-column and otherwise over-dense formats, and unstapled sheets are unacceptable. If your handwriting is difficult to read, type your homework. If you tend to scratch out or erase incorrect parts of solutions, do a rough draft or type your homework. Write in paragraphs, sentences, and English words. Use punctuation and conjunctions to indicate your flow of thought rather than arrows or telepathy. Shoot for lucidity rather than terseness.


Final Course Grades: Your grade will be determined by your performance on the various assignments and exams throughout the term. Specifically, your final grade will be determined as follows:

  • Communication-Intensive activities: 40%
  • Traditional problems, unpledged: 20%
  • Traditional problems, pledged: 10%
  • Final 30%

Grading Scale: A – at least 90%; B – at least 80%; C – at least 70%; D – at least 60%.


Ethical Conduct: You are strongly encouraged to talk to others about the material you are learning — this includes fellow students, me and anyone else in the department.  The more engaged you are with the material the better. That said, the work you hand in must be your own. The best way to ensure that the work you turn in accurately represents your own understanding of the material (and to avoid any suspicion of copying) is to write out your homework solutions on your own, not whilst working with a friend. Cheating on exams is unacceptable. Any cheating on homework assignments or during midterms or finals will result in you receiving no credit for the assignment/exam.  Students must abide by LSU’s Code of Student Conduct.


Disability Support: Students who may need accommodations because of a documented disability should meet with me privately within the first week of classes. In addition, students with disabilities should also contact the Office of Disability Services.