Advanced Calculus and Linear Algebra I, Spring 2018

Instructor: Andrew Obus
email: obus [at] virginia [dot] edu
office: Kerchof Hall 208
phone: 434-424-4930



TuTh 9:30-10:45, New Cabell Hall 395.  Please ask questions if anything in lecture is unclear.  Lectures will start promptly at 9:30 and run the entire 75 minutes.  I know it is early, but it is essential that you show up on time!

Unlike in other math classes you may have taken, we will not cover all details of the necessary material in lecture --- this course requires some independent reading and learning.  Homework will include reading assignments, and material from these reading assignments will show up on problem sets and exams. 

Discussion Sessions

Discussion sessions will be held Mondays 5:00-5:50 in New Cabell Hall 395 (same as our regular classroom).  The discussion leader is Aron Daw (, who will also be grading your homework.  His office hours are TBA in Kerchof 119. 

The primary purpose of these sessions is to review/flesh out material from the previous week, as well as to correct any pervasive misconceptions arising from the homework.  Attendance at discussion sessions will be extremely useful, and is in any case mandatory

I will also be running a problem session most weeks (see "Office Hours" below).


Main Textbook: Multivariable Mathematics, 4th ed., by Richard E. Williamson and Hale F. Trotter. 

Alternative References:

Linear Algebra: "Linear Algebra and Its Applications" by David C. Lay.

Differential Equations: "Differential Equations: Techniques, Theory, and Applications" by MacCluer, Bourdon, and Kriete.

Office Hours

Tu 11:00-12:00, W 5:00-6:00, Th 2:00-3:00.  The Tuesday and Thursday office hours are standard "drop-in" office hours held in Kerchof 208 (my office).  At the Thursday office hour, priority will be given to students from the Math 3351 class I teach this semester.

The Wednesday office hour will sometimes be a standard drop-in office hour in Kerchof 208 (with priority for Math 3315), and it will sometimes be a problem session where we can work problems that you or I select (room TBA).  Toward the end of the semester, it will be used for student presentations.  The Wednesday sessions are optional until students start presenting (early April), at which point they will become mandatory.   See the section on papers/presentations below.  If you have a conflict between 5:00 and 6:30 on Wednesdays, please let me know immediately.  Generally we will end by 6, but on presentation days we may go later in order to have time for three presentations.

If these times do not work for you, please make an appointment with me.

I will let you know each week whether there is a problem session or a regular office hour that Wednesday.  On Wednesday January 17th (the first week of class), it will be a regular office hour.


This is an integrated course in linear algebra and differential equations.  It is meant to be a challenging course for students interested in mathematics and related sciences.  We will cover more or less all of the material in Math 3250 and Math 3351 (although we will assume knowledge of the basic linear algebra already covered in Math 2315: matrices, vectors, systems of equations, determinants, linear transformations from R^n --> R^m, invertibility, linear independence).  Compared to Math 3250 and 3351, our pace will be faster, our problems will be more challenging, our presentation will be somewhat more conceptual, and there will be a more serious requirement to write basic proofs and justifications.  Completion of this course exempts you from the Math 3250/3351 requirements for the math major or minor.

We will study linear algebra for the first half of the course.  In addition to its applications to multivariable calculus that we have seen in Math 2315, linear algebra is in fact fundamental to physics, computer science, and statistics.  Linear algebra underlies Google's PageRank algorithm, the concept of a best-fit line, and risk models of stock portfolios.  Furthermore, it is only a slight stretch to say that all higher mathematics as it is practiced today (geometry, topology, number theory, analysis, differential equations, etc.) depends fundamentally on linear algebra. 

In particular, linear algebra is fundamental to the study of differential equations, which will be the subject of the second half of the course.  Differential equations are used to model how systems evolve through time and space.  This can be interpreted quite generally (particle motion, fluid mechanics, population dynamics, national economies,...).  We will study various types of differential equations and various methods of solving them, but we will try to keep things as conceptual and related to linear algebra as possible.  We will conclude with a brief study of Fourier Series, and their applications to the heat and wave equations.

In terms of the book, we will cover most of Chapter 3 and Chapters 10-14 (although I expect that you have already seen most of the material in Chapter 10 and the first half of Chapter 14 in your BC Calculus course or equivalent).  I will also be emailing out some (handwritten) lecture notes from Andrei Rapinchuk that we will use as a resource.  We will go somewhat deeper into linear algebra --- especially its applications --- than the book does.

Expected Background: Math 2315, or Math 2310 and some knowledge of linear algebra.

WARNING!!! As you know from Math 2315, this is a very fast-paced class.  It may be the case that you find yourself not having fully understood everything at the end of a lecture.  This is totally normal and nothing to be worried about, but please go over your notes as soon as possible to get up to date (certainly before the next class)!  If you need further help, do not be shy about asking questions at recitation and office hours.  If you fall too far behind in this course, it will be very difficult to catch up --- you do not want to fall into a pattern where each lecture starts to become less understandable than the last.  Please see me promptly if things stop making sense!  Note also the tutoring center below.


Homework will be posted on Collab on Thursdays.  Each homework assignment, in addition to a reading assignment, will consist of two parts:

1) A comprehension quiz that is due in my mailbox in Kerchof by 8:00pm on Mondays.  This will consist of straightforward problems meant to check your basic understanding of the material.  If you find the comprehension quizzes difficult/confusing, that is a sign that you should be coming to office hours/problem sessions/making an appointment with me to discuss!

2) The regular assignment that will be due Thursdays by 5:00pm in Aron's mailbox in Kerchof (you can also hand it to me in class if you wish).   Some of these problems are meant to be difficult --- don't be alarmed if you find them so!

Late homework will never be accepted.  If you know in advance you will be unable to turn in homework when it is due, you should plan to turn it in ahead of time.  I will drop your lowest homework score to allow for missed assignments or for assignments that pose special difficulty.

Homework should be neat, well-organized, and legible. In addition, it must be stapled or paper clipped (no folding over the top-left corner or anything like that). Please write in paragraphs, sentences, and English words (oh my!) when they are called for.  Some problems will require you to write an explanation.  The grader should not have to decipher what you are doing--you should be clear and unambiguous about your methods on a homework problem.

You are encouraged to work together on homework!  But you must write up your own solutions.  I have found that it is helpful if I think about the problems myself first, and then discuss the more difficult questions with others.  It is very important that you truly understand the homework solutions you hand in.  In previous classes I have taught, the students who were the most unpleasantly surprised with their exam grades have been the ones who have "phoned in" their homework.

If you work together on homework, you must write the names of your collaborators on the front.

Homework will be graded and every effort will be made to hand it back promptly.  Grades will be posted on Collab.


There will be a take-home midterm in lieu of a homework assignment due on March 1st at 5:00pm. 

The final exam will also be a take-home exam.  It will be due on Thursday, May 3rd (the day the final exam for our class is scheduled) at 6:00pm.

For both exams, you may use your textbook, course notes, and old homework assignments, but no other sources (including people and internet)!  In particular, you may not work with classmates.


You will be responsible for writing a short (5-8 pages) basic expository paper on a topic in mathematics of your choice (I will give a long list of possible topics, or you can pick your own, subject to my approval).  This is meant to give you an early introduction to mathematical writing.  I will give further guidelines for the paper in an email, but note that it is not expected that your papers display great originality.   Rather, the paper is meant so that you get some practice with the patterns of mathematical exposition while learning about a new topic (this is the "book report" of academic papers).  Papers must be typed.  If you can pull this off in Microsoft Equation Editor, more power to you, but I recommend typesetting your paper in LaTeX (this is what virtually all mathematicians, and many physicists, economists, and computer scientists use to typeset their papers).  There is a bit of a startup cost to learning LaTeX, but the dividends it pays in beauty and clarity are well worth it.  Aron's first Monday session will be a basic LaTeX introduction.  I will also post some LaTeX help documents on Collab.

In addition to your paper, you will give a 20 minute presentation on your topic (plus 5-10 minutes for Q&A) in one of the Wednesday evening sessions toward the end of the semester.  You may use slides if you wish (LaTeX integrates very well with slides using the program "Beamer"), or you may use the chalkboard.

For the paper and presentation, you may work in pairs.  Each pair will write one paper, deliver one presentation, and receive one grade.  If you prefer to work alone, you may do that also, but the expectations for the papers/presentations will be the same regardless of whether you work alone or in pairs.

Your choice of paper topic as well as partner (if you have one) must be presented to me by February 1st.  Presentations will take place during the Wednesday sessions on 4/10, 4/17, 4/24, and possibly 4/3, depending on how many groups there are.  Final papers will be due along with the last homework assignment on April 26th.  Do not leave your writing until the last week --- mathematical writing always takes longer than you think it will! 

Summary of Due Dates

HW#1: 1/25
HW #2, Paper topic: 2/1
HW #3: 2/8
HW #4: 2/15
HW #5: 2/22
Take-Home Midterm: 3/1
HW #6: 3/15
HW #7: 3/22
HW #8: 3/29
HW #9: 4/5
HW #10: 4/12
HW #11: 4/19
HW #12, Final Paper: 4/26
Take-Home Final: 5/3

Final Course Grades

10% Comprehension quizzes
20% Homework
10% Paper
5% Presentation
20% Midterm
35% Final Exam


The University of Virginia Honor Code applies in this class.  You will be asked to sign a statement before each exam acknowledging that you understand this.


All students with special needs requiring accommodations should present the appropriate paperwork from the Student Disability Access Center (SDAC). It is the student's responsibility to present this paperwork in a timely fashion and follow up with the instructor about the accommodations being offered. Accommodations for test-taking (e.g., extended time) should be arranged at least 5 business days before an exam.

Some Useful Links

University of Virginia Undergraduate Math Page

University of Virginia Math Department

Extra Help


If you have (anonymous) comments for me about teaching style or anything related to the course, you can make them on the Collab page for the course.