Spring 2022: Math 361, Advanced Calculus


Key info

Lectures: TR, 10:15am-11:45am

  • Lecture notes (on canvas)

Instructor: Truong-Son Van,   Email: tsvan+361@sas.upenn.edu,   Office: DRL 3N8C

Office Hours (Instructor): M: 5-6pm (zoom), Th: 4-5pm (in person).

TA: Leonardo Ferreira Guilhoto,   Office Hours (TA): M: 10-11am, F: 4-5pm

Prerequisites: Math 360

Penn’s COVID-19 guidance

  • Masks are required indoors in public and shared spaces for ALL, including those who are fully vaccinated.
    • Exceptions to the masking requirement include single occupancy offices and shared spaces where 6ft distancing can be maintained, with roommates in our college house suites/rooms, and by permission in instructional settings for academic reasons.

Important dates

  • Midterm 1: Feb 15
  • Midterm 2: Mar 22
  • Final: TBD
  • First day of class: Jan 12
  • MLK Jr. Day (no classes): Jan 17
  • Course selection period ends: Jan 25
  • Drop period ends: Feb 21
  • Spring break: March 5-13
  • Last day to withdraw: Mar 28

Textbook(s) and References

I will use a combination of the following books:

  • Real Mathematical Analysis [Online Access via Penn Library]. This book is very good and clear. We will first cover Chapter 4 of this book.

  • Basic Analysis II, by Jiri Lebl [Buy, Free]. This book has a nice pace and cover a lot of the materials in this course.

  • Introduction to Analysis in Several Variables: Advanced Calculus, by Michael Taylor [Buy, Free]. This is a nice book written by a renowned expert. If you were to need to read the book in paper, I encourage you to buy it. We will study near the end of the class differential forms from this book.

Course description

This is the second course in the undergraduate analysis sequence. We will rigorously study the theory of n-dimensional calculus. Topics to be discussed are (but not limited to): function space, Arzela-Ascoli theorem, Stone-Weierstrass theorem, continuity, derivative, implicit function theorem, inverse function theorem, path integral, multivariable integral, surface integral, k-form, Green’s theorem, Stoke’s theorem.

Learning objectives

Through out this course, student will

  1. continue to develop their skills will epsilon-delta proofs in general setting, rather than one dimensional setting as in MATH 360,

  2. learn the basics of differential geometry such as curves and surfaces,

  3. learn multivariable integral, Fubini Theorem, surface integral,

  4. learn Green’s and Stoke’s theorem and their applications.

Class Policies (subject to change)


  • Lectures will be recorded and livestreamed but the recordings will not be readily available (see attendance section for more details).
  • If you must sleep, please don’t snore. (Thanks Gautam Iyer for this amazing policy!)
  • Please be respectful to your classmates.


  • Attendance is strongly encouraged, either in person or on a live Zoom meeting. If you have to be absent for any reason, please submit a Course Absence Report. Only those students who submit the reports will have access to lecture recordings in case then want to catch up.


  • Homework must be turned in by 23:59 p.m. ET on the due date.
  • All homework must be scanned and submitted electronically (I will NOT take homework slipped under my door).
  • 2 extra points for typed homework.
  • Collaboration for homework is strongly encouraged but you MUST write up your own work. Word-to-word copying is plagiarism.
  • Generously credit all of the people who you collaborate with at the beginning of your work.
  • If you use outside sources (internet, books, friends, etc.) for a particular problem, acknowledge them at the beginning of the problem. You will NOT be penalized for consulting outside sources as long as you credit them.
  • Late homework policy:
    • Late homework wll NOT be accpeted. However, the two worst homeworks will be dropped.
  • Advice:
    • Eat well and get enough sleep.
    • Start early. One problem per day is more pleasant than seven problems in one night.
    • Try to understand the materials rather than rote memorization. This will show in exams.
    • Try to write clearly and demonstrate clarity of thoughts.


There are 2 midterms and 1 final.

Please make sure to adhere to the following points:

  • NO collaboration
  • NO calculator

Grading (subject to change)

Homework is 70% and each exam is 10% of the total grade.

A: 93% or above, B: 83% or above, C: 73% or above, D: 63% or above


In general, take good care of your health. You’re a human being first, before a student. Your academinc performance will be affected if you are not in good health. If you experience mental health issues, please consider counseling at Penn’s Counseling & Psychological services. It is NOT a weakness to seek help. I do that from time to time.

  • 24/7 mental health hotline (CAPS): 215-898-7021.

Accomodations for Students with Disabilities

If you have a disability and have a letter from the Student Disabilities Services office, please meet and discuss with me as early as possible so I can make appropriate accomodations for you.

Academic Integrity

Please read the Code of Academic Integrity carefully.

Cheating will NOT be tolerated and will be reported to the Office of Student Conduct. In the worst case, it can result in expulsion.

That said, make sure you keep the following points:

  • Discussing homework is not cheating and strongly encouraged.
  • You need to write up your own solutions after discussions. Word-by-word copying is cheating.