Math 282 Spring 2022

Instructor: Robert Lipshitz

Basic course information

Time: MTWF 1:00–1:50 p.m.
Place: University 102.
Textbook: Multivariable Calculus by James Stewart, 8th edition.
Office hours: M 9-10 a.m., W 4-5 p.m., F 4-5 p.m. by Zoom. Subject to change.
Final exam: Per the university Final Exam Schedule
Midterm exams: April 22 and May 16. Subject to change if necessary.
There is also WeBWorK homework and a Canvas webpage for tracking grades.


Math 281 (first quarter multivariable calculus).

Description and goals

While Math 281 focused on derivatives in several variables, this course focuses on integrals. We study two different kinds of integrals: integrals of functions from Rn to R, which generalize the notion of the area under a curve, and integrals of vector fields, that is, functions from Rn to Rn. (We will focus on the cases n=2 and n=3). The former are related to volumes and densities, and are important both for physics (and related areas like physical chemistry) and for probability (and related areas like machine learning). The latter are closely related to fluid flows (and more abstract flows like electric and magnetic fields); classical applications again come from physics.

Specific “learning outcomes” include:

  • Being able to evaluate double, triple, and higher integrals, over regions in the plane, 3-space, and beyond.
  • Being able to integrate in polar, cylindrical, and spherical coordinates.
  • Being able to use iterated integrals for applications like computing the center of mass and first moments.
  • Being able to integrate functions and vector fields along curves and surfaces.
  • Being able to compute the curl and divergence, and understanding the relationship to conservative vector fields.
  • Understanding and being able to apply various multi-variable versions of the Fundamental Theorem of Calculus, including Gauss's Theorem and Stokes' Theorem.

A course is more than its “learning outcomes”: the goal is understanding, not the ability to perform specific manipulations.



Online homework 15%
Quizzes 15%
Midterm 1 20%
Midterm 2 20%
Final 30%

Scores will be curved before averaging the different components, in case some component is unexpectedly hard, but a combined raw score of 90% will receive at least an A-, a combined raw score of 80% will receive at least a B-, and so on.

Homework and quizzes

  • The course will have both written and online homework.
  • Written homework is due (most) Wednesdays. You will not turn it in, and it will not be graded, but...
    • I will post solutions, and...
    • Most Fridays there will be a quiz with one or two problems, at least one of which is identical to a written homework problem except with different numbers (except in week 1).
  • Online homework, via WebWorks, is due on Mondays, except as noted. (Due dates may change.) There will be written and online homework assignments due during Week 10.
  • The lowest two online homework scores and lowest two quiz scores will be dropped, as a uniform way of handling illnesses, family crises, and other events that might interfere with the course. If you are sick for more than two weeks, contact me to discuss grades / makeup work.
  • You may use any resources you like on the homework, and may work together on it, but be careful not to get too much help.
  • Late homeworks will not be accepted, but the lowest written homework score and lowest online homework score will be dropped. Due to limited resources, only part of the homework will be graded carefully.
  • You may not use the book, your notes, your phone, a calculator, or any other resources on the quizzes (or exams).


All exams will be given in class, and there will typically not be makeup exams. If you know in advance that you will miss an exam, contact me immediately to make arrangements. If you miss an exam because of an emergency, contact me as soon as possible to discuss your options.

All exams are closed-note, closed-book, and without electronic assistance (including calculators and cell phones). Using any notes or electronic device or communicating with anyone except me during an exam constitutes cheating. Cheating on a quiz or exam will be reported to the university, with a recommended punishment of failing the course.

Students with disabilities

I, and the University of Oregon in general, are committed to an inclusive learning environment. If you have a disability which may impact your performance on exams, please contact the Accessible Education Center to discuss appropriate accommodations. If there are other disability-related barriers to your participation in the course, please either discuss them with me directly or consult with the Accessible Education Center.

Written homework

Remember that there is also online homework!


This schedule is tentative, and may change during the quarter.

Week Date Topic Sections
1 3/28 Introduction to the class. Review of 281. Ch. 12-14
  3/29 Double integrals over rectangles 15.1
  3/30 Double integrals over general regions 15.2
  4/1 Quiz. Then more examples and practice
2 4/4 Double integrals in polar coordinates 15.3
  4/5 More polar double integrals. Applications of double integrals. 15.3, 15.4
  4/6 More on applications of double integrals 15.4
  4/8 Quiz. Then more examples and practice  
3 4/11 Surface area 15.5
  4/12 Triple (and higher) integrals 15.6
  4/13 Triple integrals in cylindrical and spherical coordinates 15.7, 15.8
  4/15 Quiz. Then more examples and practice  
4 4/18 More examples with cylindrical and spherical coordinates  
  4/19 General change of variables formula 15.9
  4/20 Review
  4/22 Midterm 1
5 4/25 Vector fields 16.1
  4/26 Line integrals of functions 16.2
  4/27 Line integrals of vector fields 16.2
  4/29 Quiz. Then more examples and practice  
6 5/2 Fundamental theorem of line integrals 16.3
  5/3 Conservative vector fields 16.3
  5/4 Green's Theorem 16.4
  5/6 Quiz. Then more examples and practice  
7 5/9 Curl and divergence 16.5
  5/10 Surface area (again) 16.6
  5/11 More examples and practice
  5/13 Quiz. Then review
8 5/16 Midterm 2
  5/17 Surface integrals 16.7
  5/18 More surface integrals  
  5/20 Quiz. Then more examples and practice.  
9 5/23 Stokes' Theorem 16.8
  5/24 More Stokes' Theorem
  5/25 Gauss's Theorem (a.k.a. the Divergence Theorem) 16.9
  5/27 Quiz. Then more examples and practice.  
10 5/30 Memorial day holiday
  5/31 More Gauss's Theorem
  6/1 More examples and practice, or something fun (decided by a vote)
  6/3 Quiz. Then review. (Last day of classes.)  



Handouts will be posted here, in case you lost the physical copy.


Reading mathematics. You are expected to read the sections in the textbook before coming to class. It's usually only a few pages, so read it carefully. Note down the questions you have; I would expect you to have at least one per page. Read the section again after class. See which questions you now understand. Think about the remaining questions off and on for a day. See which you now understand. Ask someone (e.g., me) about the questions you still have left.

Getting help. If you're having trouble, get help immediately. Everyone who works seriously on mathematics struggles, but if you don't get help promptly you will soon be completely lost. The first places to look for help are my office hours. The Teaching and Learning Center also facilitates individual and small-group tutoring, and the Math Library offers drop-in help, though this course is at the boundary of what either can accommodate.

Teaching to learn. The best way to learn mathematics is to explain it to someone. You'll find that, particularly in office hours, I'll try to get you to explain the ideas. You should also try explaining the material to each other. The person doing the explaining will generally learn more than the explainee. Another thing to try is writing explanations to yourself, in plain English or as close as you can manage, of what's going on in the course. File them somewhere, and then look back at them a few days later, to see if your understanding has changed.