MATH 118 Introduction to Functions of Several Variables

Yale University
Spring 2020

Instructor Information

Instructor: Bobby McDonald
Instructor Website:
Office: DL 429
Class meet time: Tu/Th 2:30 - 3:45p
Class meet location: DL 431
Email: (use Piazza, instead)
Office hours (tentative until second or third week):
        Tu/Th: 4:00 - 5:30p, We: 4:00 - 5:30p, (or by appointment)
How to Contact Me: I’d prefer all communication take place on Piazza. For logistical/personal questions, you can send me a private message. For questions regarding in class material, homework, quizzes, or exam study, you can post publicly to the discussion. You can do this anonymously!

Revised (3/19/2020) Syllabus for Online Course


Rather than have a set rigid schedule, we'll try to be flexible and give each topic the time it deserves. I will always give you a heads up as to what the next few topics will be, but in most cases you can safely assume we’ll be doing every section from each chapter, roughly in the following order, and (very) roughly in/around the indicated weeks.

First half (roughly weeks 1-6)

We will start the semester with Linear algebra, from Strang’s “Introduction to linear algebra”, covering chapters 1-4. You do not need to buy this book, but our topic order will be the same. You can find more detail about sections we'll cover and in depth key ideas here.

Chapter 1 (weeks 1, 2) talks about vectors and matrices, and how they relate to systems of linear equations. Remember, for each of these chapters you can find more detail about sections we'll cover and in depth key ideas here.

Chapter 2 (weeks 2, 3) will teach us how to simplify systems of equations, calculate matrix inverses, and factor matrices into simpler ones. This will give us the techniques to solve systems given by invertible matrices, which always have a unique solution.

Chapter 3 (weeks 4, 5) studies vector spaces, concentrating on those associated with matrices. It will allow us to deal with more complicated systems, recognize when solutions may not exist, and when there might be more than one.

Chapter 4 (weeks 5, 6) works with systems that do not have a solution: we will learn how to find the “best answer”, one that does not fit the equations exactly, but it is as close as possible.

Midterm 1 will cover all of the linear algebra we’ve learned. The expected date of this exam is Tuesday, February 25, in class. Really, we'll take it after we finish chapter 4, which is expected around week 7.

Second half (roughly weeks 8, 11-15)

Using what we have learned about matrices. We will cover chapters 12-15 in McCallum’s “Calculus, Multivariable”. You do not need to buy this book, but our topic order will be the same. A more in depth description of the key ideas in each section will be provided later.

Chapter 12 (weeks 7, 8) introduces functions of multiple variables. We will talk about equations of surfaces, draw cross-sections and topographical maps.

  • 12.1 Functions of Two Variables
  • 12.2 Graphs and Surfaces
  • 12.3 Contour Diagrams
  • 12.4 Linear Functions
  • 12.5 Functions of Three Variables

Chapter 13 (week 8) is a review of linear algebra: we will add only the cross product from here, which will allow us to write equations of planes in three dimensions.

  • 13.4 The Cross Product

Chapter 14 (weeks 11, 12, 13) is about derivatives. We will learn how to calculate x and y derivatives, as well as derivatives in other directions (for example along y = x). The techniques are very similar to those of single variable calculus. Tangent lines to curves will be replaced by tangent planes to surfaces, and we will have three different kinds of second order derivatives instead of one; but everything in this chapter should feel familiar.

  • 14.1 The Partial Derivative
  • 14.2 Computing Partial Derivatives Algebraically
  • 14.3 Local Linearity and the Differential
  • 14.4 Gradients and Directional Derivatives in the Plane
  • 14.5 Gradients and Directional Derivatives in Space
  • 14.6 The Chain Rule
  • 14.7 Second-Order Partial Derivatives

Midterm 2: the second exam (not cumulative) will cover the multivariable calculus we’ve learned up through Chapter 14. The expected date of this exam is Tuesday, April 8, in class. Again, we'll really take it after we finish chapter 14, which you should expect around week 13.

Chapter 15 (weeks 14, 15) is about optimization, and finding minima and maxima of multivariable functions. Once again, the material will be quite familiar from single variable calculus, though the second derivative test will be more complicated, as will be the checking of “endpoints” (which will now be entire boundary curves of the region in question). Lastly, we will cover optimization subject to a constraint, using a technique known as Lagrange multipliers.

  • 15.1 Critical Points: Local Extreme and Saddle Points
  • 15.2 Optimization
  • 15.3 Constrained Optimization: Lagrange Multipliers


This is where I'll post notes, handouts, quiz solutions, practice exams and more. I'll post an announcement on Canvas whenever this section gets updated.


Economics applications (top)

The originators of this course compiled an expository paper with many applications to economics. I'll take some examples from these to use in class, and assign readings out of it.
        Calculus and Linear Algebra Notes - Economics Applications

Useful links/documents (top)

Here are some links and documents that may be usesful, in general, for this course:

  • My old lecture notes from a linear algebra course I taught at UConn. A useful source for extra examples/definitions. The topics are in a different order, and some are skipped. In particular, we'll cover nothing that looks like Sections 1.8 or 1.9, Chapters 3 or 5.
  • An indepth list of key linear algebra topics and sections we'll cover.
  • This website has calculators for many of the matrix operations that we'll be using this semester. It can even show you the steps it uses to reduce a matrix! Try not to rely on it too much, but use it to check your work or if you get stuck.
  • The Essence of Linear Algebra series by 3Blue1Brown has excellent visualizations and explanations for some of the concepts we'll be encountering.

Notes and topics from class (top)

The first two documents are a rough outline (unfilled) of what we expect(ed) to cover in class, with definitions and examples. There are two versions. The rest of the bullets include the just the examples that we covered in class, and completed examples (highlighted in red) that we didn't.

Homework (top)

Exam Resources (top)

Resources for Exam 1

The best place to start. Most of the questions on the exam are inspired by the problems below. They're old homework problems (with solutions) from when I taught a similar course. There are some more (untestible) theoretical questions toward the end of each set, and you can ignore anything about transformations. Start with the topic you are least confortable with.

Old exam problems. Look here for exam problems from previous years. I would do these problems after you've exhausted the problems above.

  • Here are 25 old exam questions I selected from some exams I wrote for a linear algebra course last year. This is a good place to start.
  • Here are more practice problems from 118 last semester. Different material may have been covered. For example, there's nothing on orthogonality here.


Since our class is officially online now, I wanted to give you flexibility in the times that you spend learning the concepts, and use class time to work together practicing them on worksheets. Each class will be a "workshop" with examples and applications of that section's topic. Before each class, there are around 25-30 minutes of video to watch that cover each concept, and an introductory example or two.

NOTE: Video "lectures" will mostly just be introductions to definitions and theorems with and example or two. Most of the extra examples will be done in class on the worksheet. You can look at/print the blank video notes to have all the definitions in one place.

For 3/24/2020: Recap - 3D Geometry, Lines, and Planes

For 3/26/2020: Lecture 3 - Vector Valued Functions

For 3/31/2020: Lecture 4 - Functions of Several Variables

For 4/2/2020: Lecture 5 - Derivatives

For 4/7/2020: Lecture 6 - Directional Derivatives

For 4/9/2020: Lecture 7

For 4/14/2020: Lecture 8 - Local Extrema

  • notes: blank, filled.
  • in class worksheet: blank, solutions

For 4/16/2020: Lecture 9 - Global Extrema

  • notes: blank, filled.
  • in class worksheet: blank, solutions

For 4/21/2020: Lecture 10 - Global Extrema via Lagrange Multipliers

  • notes: blank, filled.
  • in class worksheet: blank, solutions

All videos will be captioned, but I do this myself and it's a time consuming process.
Please don't hesitate to contact me with any accessibility issues.

Extra Help

Resources Available

Offce Hours: Time and location are on the front page (it's best to keep them in one place!). I'll hold hours every week, and you should feel free to drop by to chat about any questions or concerns that you have. You do not need an appointment to come to office hours.

Residential College Tutors: Each college has math and science tutors who hold regular office hours. The schedule can be found at the Center for Teaching and Learning website. Note: You can attend tutor office hours at any college, not just your own.

Study Groups with Peers: You are strongly advised to form study groups with your classmates. Discussion and collaboration are among the best ways to learn mathematics.

General Advice for Success

  • Work outside of class! These topics cannot be learned in 150 = 75 + 75 minutes each week. On average, students (not just in our class, but in general) are supposed to spend an average of two hours outside of class. That would be about six hours per class. That’s why they call it full time!
  • Make mistakes! Particularly on the homework, you might not know what to try first. Just try something. The process of mathematics is full of mistakes. I could fill a book with all of the mistakes I’ve made. They’re an integral part of learning!
  • Work with your peers. Bouncing ideas off of each other is the best way to learn.
  • COME TO MY OFFICE HOURS! It’s free tutoring. From your instructor. You know, the one who writes the exams. Please come, or make an appointment if you can’t!
  • We’ll be assigned a peer tutor, and when we do, I’ll be sure to post relevant information about how to access them on our website. This is the best place to go to get someone else’s take on the material that isn’t me. They might be able to explain something differently.
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