Math 281 Fall 2023

Instructor: Robert Lipshitz

Basic course information

Time: MTuWF 2:00–2:50 p.m.
Place: Straub 252.
Textbook: Multivariable Calculus by James Stewart, Daniel Clegg, and Saleem Watson, 9th edition.
Office hours: In person: M 3:00-3:50 p.m., F 4:00-5:00 p.m.; Zoom: Tu 7:00 - 8:00 p.m.. Subject to change.
Final exam: Per the university Final Exam Schedule.
Midterm exams: October 23 and November 13. Subject to change if necessary.
There is also WeBWorK homework and a Canvas webpage for tracking grades.


Math 253 (third-quarter single-variable calculus), though most of the material we will use is from 251.

Description and goals

This course has three, connected parts. In the first, we develop some basic language for studying Rn and describing lines, planes, and other surfaces in it. We focus especially on the case of R3, both because it is useful in physics and engineering and because developing an intuition is easier there. In the second part, we use these tools to apply calculus to curves in R<sup>n</sup>, developing notions like continuity and derivatives (velocity) there. These generalizations turn out to be fairly easy, though that allows us to touch on some deeper topics, like curvature. Curves are functions where the values (co-domain) lie in Rn. The third part of the course focuses on functions of several variables, i.e., functions where the domain is Rn. Such functions come up in physical situations, but also in almost any setting that involves data. Again, we discuss continuity and derivatives of these functions; both turn out to be much more subtle than the 1-dimensional case. In the last few weeks, we discuss optimization problems for such functions (much of which is called "machine learning" these days) and constrained optimization.

Specific “learning outcomes” include:

  • Compute and understand the geometry of the dot and cross product, and be able to use them to compute angles, projections, distances, and normal vectors.
  • Understand how to describe lines, planes, as well as other curves and surfaces, via parametric and implicit equations, and how to identify curves and surfaces from equations.
  • Determine continuity, limits, and derivatives of vector-valued functions (curves), and the relationship with velocity and acceleration.
  • Use derivatives and integrals to describe basic geometric properties of curves, including arclength, TNB frames, and curvature.
  • Determine whether functions of several variables are continuous and find limits of functions of several variables.
  • Compute partial derivatives, directional derivatives, and the gradient of functions of several variables, and understand their geometric meanings.
  • Find tangent planes to graphs of functions of two variables, and use them to approximate the function.
  • Find maxima, minima, and saddle points of functions of several variables, including with constraints.

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) Mondays. You will not turn it in, and it will not be graded, but...
    • I will post solutions, and...
    • Most Wednesdays 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. (In week 1, the quiz will be about material from 251.)
  • Online homework, via WeBWorK, 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. The university no longer permits me to make accommodations on a case-by-base basis.
  • 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.
  • You may not use the book, your notes, your phone, a calculator, or any other resources on the quizzes (or exams).
  • Homework solutions, quizzes, and exams may not be posted anywhere on the internet, including Chegg and other tutoring sites. Re-posting them is a violation of both course policy and copyright law, and will be reported for disciplinary action.


With limited exceptions, the university now requires that students who miss an exam be treated the same, independent of the reason. That is, students who miss the exam because of being hospitalized for illnesses must be treated the same as students who skip an exam because they decide they want more time to study. So:

  • If you have a conflict with a midterm exam and you alert me at least 10 days in advance, you will have the opportunity to take a version the exam a few days early, without penalty. In particular, this is the mechanism for accommodating cases 3 and 4 in the UO Attendance and Engagement policy.
  • If you miss one midterm exam, I will compute your midterm exam score by taking a weighted average of your score on the other midterm exam and the final (after normalizing using the class means and standard deviations).
  • If you miss both midterm exams, you will have the opportunity to take a makeup midterm 2 within two weeks of midterm 2, at a 15% linear penalty. That is, whatever score you get on the makeup exam will be multiplied by 0.85. If you do not take the exam within two weeks, your midterm exam score will be computed as zero.
  • If you miss the final exam and are otherwise passing the class, you will receive an incomplete in the class and have the option to take a makeup exam in the first two weeks of the winter quarter, with a 5% linear penalty. If you do not take the exam in that time, I will compute your grade as if you received a zero on the final exam.
  • If you miss the final exam but were otherwise failing the class, you will not have an opportunity to re-take the exam, and will receive and F in the class.

(If you find it dehumanizinag having this all spelled out, rather than being treated fairly on an individual basis, I completely agree.)

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, supported by the University of Oregon, am 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 9/26 Course overview. Coordinate systems, vectors. 12.1-12.2
  9/27 Quiz. Then: more vectors, dot product. 12.2-12.3
  9/29 More dot product, correlation, review. 12.3
2 10/2 Cross product. WeBWorK 1 Due. 12.4
  10/3 Equations for lines and planes. 12.5
  10/4 Quiz. Then: more on equations for lines and planes. 12.5
  10/6 More examples and practice  
3 10/9 Quadric surfaces, cylinders. WeBWorK 2 Due. 12.6
  10/10 Vector functions and space curves 13.1
  10/11 Quiz. Then: derivatives and integrals of vector functions. Velocity and acceleration. 13.2, 13.4
  10/13 More examples and practice.  
4 10/16 Arc length, curvature. WeBWorK 3 Due. 13.3
  10/17 More arc length, curvature, TNB frames. 13.3
  10/18 Quiz. Then: More practice and examples.  
  10/20 Midterm review.
5 10/23 Midterm 1. 14.1
  10/24 Functions of several variables. WeBWorK 4 Due. 14.2
  10/25 Limits and continuity. (No quiz this week.) 14.2
  10/27 More limits and continuity.  
6 10/30 Partial derivatives. WeBWorK 5 Due. 14.3
  10/31 Tangent planes and linear approximation. 14.4
  11/1 Quiz. Then: The chain rule. 14.5
  11/3 More examples and practice.  
7 11/6 Directional derivatives and the gradient. WeBWorK 6 Due. 14.6
  11/7 More gradient. Then practice and examples. 14.6
  11/8 Quiz. Then: midterm review.
  11/10 Veteran's Day Holiday - no class.  
8 11/13 Midterm 2. WeBWorK 7 Due.  
  11/14 Optimization 14.7
  11/15 Quiz. Then: More optimization 14.7
  11/17 More examples and practice.  
9 11/20 More optimization. WeBWorK 8 Due. 14.7
  11/21 Quiz. Then: more Lagrange multipliers. 14.8
  11/22 Review  
  11/24 Thanksgiving -- no class.  
10 11/27 More Lagrange multipliers. 14.8
  11/28 More Lagrange multipliers. WeBWorK 9 Due.  14.8
  11/29 Quiz. Then more Lagrange multipliers, calculus on manifolds. 14.8, +
  12/1 Review. (Last day of classes.)  



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

3D Models


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.