If you cannot enroll in the course, please be patient and keep checking for open spots. Enrollment for a specific section time is first-come first-served.
Students currently enrolled in a section that conflicts with their other courses may attend any Math 51 sections during Weeks 1-2, but will eventually need to attend the in-person classes in which they are officially enrolled for the rest of the quarter; so please use the first two weeks of the quarter to switch your enrollment appropriately.
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 the course text 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 schedule see
the Schedule page.
Students are expected to be familiar with the material of Math 19, 20, and 21.
Welcome to Math 51! This syllabus site details the course's policies, schedules, and expectations,
including for assignments and grading calculation.
Per University policy, your decision to take the course implies that you agree to
these requirements and to the grading policies spelled out here; so be sure to read
everything on these pages.
- Enrollment in lectures and sections:
Math 51 students attend lectures on MWF, starting on the first Wednesday of the quarter, and discussion sections on TuTh, starting on the
first Thursday of the quarter.
Enroll in lectures and discussion sections on Axess.
Please see the enrollment tab for more information.
- Required materials:
- The textbook has been specially created by the Stanford Math department in consultation with colleagues in many other departments; it is free and electronic-only. To get the book using your SUNet ID, visit the textbook page by clicking here or selecting the Textbook menu item at the top of this page.
The book contains much more than is covered in the course. It also includes many fully worked examples, helpful for studying. We hope it will be a useful resource for topics that you may encounter in later coursework. On the second page of the introduction, you will find the e-mail address for reporting any corrections, typos, etc. The authors of the text are very eager to hear from you.
- Calculators are not required for Math 51, and they are not permitted for any exams (where we keep the numbers simple). There are a small number of homework problems for which any basic scientific calculator (even a free online one) is useful to convert some expressions into decimal approximations (and such calculations never arise on exams). There is no programming anywhere in the course.
- Check for exam conflicts right away and contact us: Except in case of emergency, you must inform us of exam conflicts at least two weeks prior to the exam, together with a valid reason for the conflict. The allowable reasons are course-related or competition-related schedule.
- There will be two 2-hour midterm exams and one 3-hour final exam. For midterm exams
(Thursday October 19 and Thursday November 9, both 8-10PM;
see details given on the Exams page or by clicking the Exams
menu item at the top of this page), the allowable reasons are course-related or
- The final exam is Tuesday December 12 from 12:15pm-3:15pm, and all students must take the exam at that time.
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: 1-650-723-1066).
You must provide
an accommodation letter, dated in the current quarter, by filling out the ACCOMMODATIONS & FLEXIBILITY FORM at least two weeks prior to
an exam, for us to have adequate time to arrange the accommodations. Renewing OAE
accommodation is NOT equivalent to alerting the staff of Math 51!
Math 51 has an
"active learning" structure; research
has shown that pre-class reading, combined with daily participation in class activities targeted to specific learning goals, improves student
learning outcomes in math and science courses. Furthermore, active learning increases student performances and narrows achievement gaps for
historically underserved students. Here's what this means for us:
Both MWF class sessions and TuTh discussion sections are more interactive than traditional math classes:
- Twice each week (exception on weeks with exams) there will be a modest amount of reading in the course text to introduce
some of the motivation behind the topic(s) to be discussed in class, along with an associated questionnaire
on Canvas to be completed before class. We strongly urge you to watch the instructional videos on How to read the course textbook before the first Pre-Class Reading Questionnaire. In addition, there will be one check-in question on material from the previous chapter and lecture to re-enforce your learning; this check-in question will be graded for accuracy, you can think of it as a practice exam question. The responses to the rest of the questionnaire are not
graded for correctness, just for a good-faith effort, to inform
how the instructor organizes the classroom time around the learning goals for that day.
- The TuTh discussion sections focus on small-group collaboration with worksheets consisting of
problems designed around the learning goals and themes in the homework and exam questions.
The goal is to engage with the new skills and concepts, and to learn from your peers as well as from the guidance of a graduate student
who answers questions. The work in discussion sections is aimed at giving practice with the material
recently learned in the course; it is not graded, and complete solutions are provided later in the day for each TuTh worksheet.
Canvas questionnaire assignments on the twice-weekly pre-class reading: a typical questionnaire consists of 1 check-in question (always the first question) and 3 to 5 "low-stress" questions. Except for the check-in question, you needn't answer more than one or two sentences per question, and you get full marks for ANY good-faith answer. These assignments are intended to give the instructor feedback on how the reading went and how the course is going; think of them as surveys in which students are voting for which topics need more motivation in class (and which need less or none). Because we will have to review your feedback in a limited time period, the firm deadlines are:
The course grade is based on the following components:
- Mondays at 9am.
- Wednesdays at 9am; these are ungraded and optional, intended only for those who wanted some guidance on pre-class reading for the Wednesday class.
- Fridays at 9am except on exam weeks (where there is only ungraded optional pre-class reading for the Friday class).
- Exception: In the first week, there will be only one questionnaire, due Friday at 9am. (It will be posted on Canvas by the first Thursday morning.)
Note that per University regulations, if you are enrolled in the course on the credit/no credit grading basis, then the minimum letter-grade equivalent to "CR" is C-minus.
- 70% for exams, with the breakdown of 20% for both midterms and 30% for the final
(see exams page for dates, policies, and practice exams);
- 20% for weekly written homework assignments (total points earned divided by 80% of total possible points, not to exceed 100%);
- 10% for pre-class reading questionnaires on Canvas (total points earned divided by 80% of total possible points, not to exceed 100%).
By Math Department policy, any student found to be in violation of the Honor Code on any assignment or exam in this course will receive a final course letter grade of NP.
You are fully responsible to adhering to the requirements of the Honor Code document. In particular, the following are not permitted:
The university is well-aware of "academic educational sites," such as Chegg, CourseHero, etc. Their use in connection with a course assignment or exam is an Honor Code violation that is taken very seriously at Stanford.
More information about the Stanford Honor Code can be found here.
- Collaborating with another student or any other person on an exam.
- Copying from another's homework or exam, or allowing another student to copy your work.
- Communicating with a person other than the teaching staff via email, text messaging, Google, any form of social media, messenger, chat rooms, message boards, etc., about anything related to the exam.
- Plagiarism of material that you did not create, such as copying parts of posted solutions or text wholesale from anywhere, including the internet. The work that you submit must be your own. This also includes representing another's work as your own.
- Sharing the exam questions or anything in your solutions with any other person for any reason. The restrictions on sharing exam content applies until 11:59PM on the exam date.
You are encouraged to attend the office
hours provided by the
instructors and teaching assistants.
You may attend the office hours of any teaching staff member in-person or online. No appointment is ever necessary
for virtual office hours,
just drop in to the posted Zoom link at the scheduled time
with your questions!
For in-person office hours, instructors and TAs may at their discretion impose office capacity limit and meet with
students on a first-come first-served basis; a sign-up sheet will be posted outside the office to facilitate
these in-person meetings.
The scheduled in-person and online office hours for any given week can be found on the office hours page. Note that they might change slightly from week to week, so it's always a good idea to double-check this calendar frequently.
The office hours page also lists some other help resources.
While cell phones are not prohibited in in-person classes, recording or taking pictures in class is strictly prohibited without the consent of your instructor. Please ask before doing so.
Consistent with university policy, face coverings are recommended in classrooms and office hours.
Students should attend the in-person lecture and discussion section to which they are officially assigned. This ensures that classrooms do not exceed official room capacity, and supports prompt notification should a specific section need to move online at short notice. Course messages are sent out via Canvas, please ensure that your Canvas notifications are on so that you receive any announcements promptly.
As standard practice, lectures and discussion sections in Mathematics courses are taught in-person. As such, Zoom links will not be provided. Additionally, in-person lectures and discussion sections will not be recorded.
Students who miss class due to illness (including COVID-19) should make arrangements to obtain lecture notes from other students in the class. As standard practice, there are no make-up exams or remote exams. If you will miss an exam due to illness, please reach out to your instructor for more information.