Course Description

Linear Algebra and Multivariable Calculus are two of the most widely used mathematical tools in quantitative work across all fields of study. This course develops conceptual understanding and problem-solving skills in both, highlighting how multivariable calculus is most naturally understood in terms of linear algebra, and addresses a variety of real-world applications.

Our focus is on teaching you skills that underlie a wide array of applications and preparing you for all courses involving advanced quantitative work (across all sciences, engineering, economics, computer science, statistics, and so on).

By the end of this course, you should be able to:
  • relate the algebra of systems of linear equations to the geometry of vectors, acquiring the ability to "see" in dimensions far beyond 3 (thinking is seeing!);
  • analyze the behavior of multivariable functions via partial derivatives, and combine that information with tools of linear algebra (such as matrices and eigenvalues) to solve optimization problems;
  • apply your newly acquired visual skills in high dimensions to gain insight into a variety of real-world applications across data science, natural sciences, and economics.

For a detailed syllabus see the Syllabus page.

First Day Checklist

Enrollment in lectures and sections:

Math 51 students attend lectures on MWF, starting April 1st, and discussion sections on TTh, starting April 2nd. You will enroll in lectures and discussion sections on Axess. Please see the meeting times tab for more information.


See the information at TEXTBOOK tab.


Starting on Wednesday April 3rd, class participation will be recorded via the iClicker2 personal response system (PRS), used for answering questions in class and allowing the instructor to gauge your understanding and helping you actively engage in the learning process. Participation counts for 3% of your course average. You are allotted four lecture absences without penalty, including absences due to late enrollment, illness, oversleeping, dead batteries, forgetting your iClicker2, athletics, job interviews, not registering your PRS device in Canvas, etc. Please note: Bringing a classmate's clicker to lecture and registering responses for him/her is an Honor Code violation, for both you and your classmate. Register your PRS as follows:
  • Please purchase the "iClicker2" from the bookstore or online or used.
  • Login to the Canvas site for your course.
  • Click the i>clicker link in the left-hand side navigation list on Canvas.
  • Enter your iClicker2 registration number following the instructions on the i>clicker Canvas page.
  • If you have already registered your iClicker for some other course, your registration should carry over. However, you should check the registration number against your name on Canvas.


Check for exam conflicts right away and contact us: Except in case of emergency, you must inform us at least two weeks prior to the exam, together with a valid reason for the conflict. For midterm exams (evenings of Thursday April  25 and Thursday May 16, both approx. 7:00 - 9:00 pm), the allowable reasons are course-related or competition-related schedule. The time of the final exam (Friday June 7, from 7-10pm) is set by the University, and all students must take the exam then. See all exam details and policies here.

Students with documented disabilities:

Students who may need an academic accommodation based on the impact of a disability must initiate the request with the Office of Accessible Education (OAE). Professional staff will evaluate the request with required documentation, recommend reasonable accommodations, and prepare an Accommodation Letter for faculty dated in the current quarter in which the request is made. Students should contact the OAE as soon as possible since timely notice is needed to coordinate accommodations. The OAE is located at 563 Salvatierra Walk (phone: 723-1066, URL: You must provide an accommodation letter, dated in the current quarter, at least two weeks prior to an exam, to provide us adequate time to arrange the accommodations.

Affordability of Course Materials

Stanford University and its instructors are committed to ensuring that all courses are financially accessible to all students. If you are an undergraduate who needs assistance with the cost of course textbooks, supplies, materials and/or fees, you are welcome to approach me directly. If would prefer not to approach me directly, please note that you can ask the Diversity & First-Gen Office for assistance by completing their questionnaire on course textbooks & supplies: or by contacting Joseph Brown, the Associate Director of the Diversity and First-Gen Office (; Old Union Room 207). Dr. Brown is available to connect you with resources and support while ensuring your privacy.

ACE section

Additional Calculus for Engineers (ACE) is a supplementary instruction program designed to provide the skills and solid foundation in mathematics to undergraduate students. Despite the name, this program is actually open to students from any major. As part of this program you get more discussion time and an extra problem solving session. On the other hand you must be committed to participation in those sessions and your math 51 course grade will be weighted for your GPA with extra credit. Click here for more information.

Weekly checklist

Math 51 has an "active learning" structure. This means pre-class reading, weekly quizzes and homeworks, combined with daily participation in class activities targeted to specific course objectives.
We will not take up lecture time re-addressing every topic covered the text, leaving extra class time for the material that YOU have voted to be most needing of extra discussion. The goal of this is to make the lecture time more useful for learning the required topics. Discussion sections will encourage group collaboration on problems modeled after those found on homeworks and past course exams.

Preclass reading and quiz.

Every week on Canvas will be posted reading assignment together with about three to five "low-stress" questions. You needn't answer more than one or two sentences per question, and you will get full marks for ANY good-faith answer. These assignments are intended to give us feedback on how the reading has been going; think of them as "creative surveys" in which people are voting for which topics need more discussion in class (and which need less or none). Because we will have to review your feedback in a limited time period, the firm deadlines will be:
  • Thursdays at 11:59pm (typically about 3 questions); and
  • Sundays at 11:59pm (about 4-5 questions).
  • Exception: In the first week we'll have an ADDITIONAL assignment due on Tuesday, April 2 at 11:59pm. (It will be released Monday morning.)

Class sessions:

There are Lectures on Monday, Wednesday and Friday, and discussions on Tuesday and Thursday. Check the page for schedule. The classes will be more interactive than traditional math lectures. Starting from Wednesday April 3rd we record your participation in class activities. For that purpose we use iClicker2. There is no way to "makeup" for your absence, but four missed classes will be penalty-free.


Check the page to read your weekly homework.
Grading scheme: Your grade will be based on the following components:

Spring 2019
Department of Mathematics, Stanford University
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