Instructor: Ari Stern (stern@wustl.edu)

Office Hours: Tue 6-7 pm, Thu 12-1 pm Central Time (via Zoom, accessed
through Canvas)

Assistant
(AI): Alberto
Dayan
(alberto.dayan@wustl.edu)

Office Hours: Mon 6-7 pm, Wed 9-10 am Central Time (via Zoom, accessed
through Canvas)

Although I hope it won't be necessary, it's possible that some aspects of the course may need to be adjusted to respond to changing circumstances of the COVID-19 pandemic during the semester. If that is the case, I will communicate any changes to you via email and/or Canvas announcements, and by updating this syllabus.

The Honors Mathematics I-II sequence (203-204) aims to give students a rigorous understanding of single-variable and multivariable calculus and linear algebra, together with their theoretical underpinnings. Along the way, students are introduced to the language and methods of modern mathematics, with a strong emphasis on proofs.

This first semester begins with an introduction to logic, set theory, and methods of proof, which we then use to formulate an axiomatic treatment of the real numbers. Next, we will use these foundations to develop the theory of single-variable calculus, including the Riemann-Darboux integral, limits and continuity, differentiation, and sequences and series (of real numbers and of functions). Since Calculus BC or equivalent is a prerequisite, I assume that students have already mastered many of the mechanical/computational techniques of calculus; this will let us move relatively quickly through the material, focusing on the theoretical aspects generally not encountered in a first calculus course. In the last few weeks of the semester, we will begin the study of linear algebra that will carry us forward into Math 204, where linear algebra will play an essential role in the development of multivariable calculus.

By the conclusion of this course, I expect students to have a firm grounding in the key theoretical ideas and techniques (e.g., epsilon-delta proofs) of single-variable calculus, as well as some of the basic concepts of linear algebra. As the term progresses, I also expect students to become comfortable reading, writing, and analyzing proofs, and to get a general appreciation for how abstraction and rigor (e.g., moving from an intuitive understanding of real numbers to a formal one) play a central role in higher mathematics.

Synchronous remote lectures will be held Monday-Thursday,
11:00-11:50 am Central Time, live via Zoom. These lectures will
also be recorded to the cloud for later viewing. Live and recorded
lectures will both be accessed
through Canvas using
the Zoom link in the navigation bar. Synchronous attendance at
lectures is *strongly encouraged* but not mandatory.

Synchronous remote discussion sections, led by the AI, will be held on Fridays from 6:00-6:50 pm Central Time, also via Zoom. In these sessions, students will work together on problems in small groups, discuss them, and submit their written solutions afterwards.

*Participation in discussion sections is mandatory*. If
you cannot attend a particular meeting due to illness or other
unforeseen circumstances,
email the AI at
least one hour before the start of that meeting to request an
alternative way to receive credit for discussion problems. This
is at the discretion of the AI, and you may need to provide
documentation for an excused absence. (This policy is intended
for rare emergencies. If you already know that you will be
unable to attend discussion section meetings, please contact the
instructor and AI as soon as possible.)

*Scheduling Note:* Discussion sections were originally
scheduled for 11:00-11:50 am, but due to time zone issues, this
would have required about 1/3 of the class to attend in the
middle of the night. After surveying students to determine an
acceptable alternative time, we chose one that both US and
international students could attend synchronously.

Problem sets will be posted
to Canvas
approximately weekly, and students will submit their solutions
electronically by uploading them before the specified due date and
time. You are encouraged to discuss the homework with your fellow
students, but *your final write-up must be your
own*. Please make sure that your solutions are written clearly
and legibly.

In lieu of exams, each student will present an individual final project on a mathematical topic of their choice related to the course. Midway through the term, students will submit a short written proposal of the topic they plan to cover; this proposal is due Thursday, November 5. During the last week of classes (December 14-18), students will give a brief remote presentation on this topic to the class and hand in a written report.

Grades will be based on a weighted average of homework (50%, lowest score dropped), discussion section problems (25%, lowest score dropped), and the final project (25% = 5% proposal + 10% final presentation + 10% final report).

The corresponding letter grade will be assigned using the following scale:

A |
A- |
B+ |
B |
B- |
C+ |
C |
C- |
D |
F |
---|---|---|---|---|---|---|---|---|---|

≥90% | ≥85% | ≥80% | ≥75% | ≥70% | ≥65% | ≥60% | ≥55% | ≥50% | <50% |

The scale may be adjusted upward based on the class's cumulative averages. However, it will not be adjusted downward, so your letter grade is guaranteed to be at least that corresponding to your score on this scale. The grade of A+ is given at the instructor's discretion.

Pass/Fail policy: You must earn at least a letter grade of C- to get a P.

The required text for this course is *Calculus*,
vol. I (second edition), by Tom M. Apostol, published
by Wiley.

The following schedule of topics is preliminary and may be adjusted during the term, since some topics may take more or less time to cover than predicted.

Weeks 1-2: Logic, sets, functions, numbers, and methods of proof (Apostol I)

Weeks 3-4: Integration and applications (Apostol 1-2)

Week 5: Limits and continuity (Apostol 3)

Weeks 6-7: Differentiation, fundamental theorems, and transcendental functions (Apostol 4-6)

Week 8: Sequences and series of real numbers (Apostol 10)

Week 9-10: Sequences and series of functions, Taylor expansion (Apostol 11 and part of 7)

Week 11: Vector algebra in ℝ^{n} (Apostol 12-13)

Week 12: Abstract vector spaces (aka linear spaces, Apostol 15)

Week 13: Linear transformations and matrices (Apostol 16)

Week 14: Student presentations

All students are expected to adhere to high standards of academic integrity, as specified in the undergraduate student academic integrity policy. Since this course is offered through the College of Arts & Sciences, any violations of this policy will be referred to the College’s Academic Integrity Officer. Violations of this policy include, but are not limited to:

- homework collaboration exceeding that permitted in the Homework Assignments section above (e.g., preparing your final write-up side-by-side with another student rather than independently, copying another student's solutions);
- submitting the work of others as your own (e.g., copying homework solutions found online, including outside material in your final project without citing the source).

If you have any questions, or are unsure about what is permitted/prohibited by this policy, please ask me.

*In many cases, academic integrity violations are the result
of getting behind in coursework and making bad decisions under
pressure. Keep up with your assignments, ask questions when you
are unsure what is expected of you, and do not give in to the
temptation to cut corners.*