Math 51 logoMath 51 logo

Math 51
Spring 2024

Home Textbook Schedule Homework Enrollment Office hours and help Exams FAQ

  • 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.
  • If you are unable to enroll in the course at all, please attend the lecture/section of your choice and email to be put on the course Canvas site.
  • 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 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.

    First Day Checklist

    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 Monday of the quarter, and discussion sections on TuTh, starting on the first Tuesday 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 on). 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 April 25 and Thursday May 16, both 8-10PM; see details given on the Exam page or by clicking the Exam menu item at the top of this page), the allowable reasons are course-related or competition-related schedule.

      • The final exam is Saturday June 8 from 12:15pm-3:15pm, and all students must take the exam at that time. See all exam details and policies here.

    • Access and Accommodation: Stanford is committed to providing equal educational opportunities for disabled students. Disabled students are a valued and essential part of the Stanford community. We welcome you to our class.

      If you experience disability, please register with the Office of Accessible Education (OAE). Professional staff will evaluate your needs, support appropriate and reasonable accommodations, and prepare an Academic Accommodation Letter for faculty. To get started, or to re-initiate services, please visit

      If you already have an Academic Accommodation Letter, please use this form to upload it and detail the specific accommodations you will need in this course. Letters are preferred by the end of week 2, and at least two weeks in advance of any exam, so we may partner with you and OAE to identify any barriers to access and inclusion that might be encountered in your experience of this course. New accommodation letters, or revised letters, are welcome throughout the quarter; please note that there may be constraints in fulfilling last-minute requests.

      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!

    Class Structure and Assessment

    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:

    • Mondays at 9am (about 5 or 6 questions).
    • Wednesdays at 9am (typically about 4 questions), these are ungraded and optional, intended only for those who wanted some guidance on pre-class reading for the Wednesday class.
    • Fridays at 9am (typically about 4 questions) except on exam weeks (where there is only ungraded optional pre-class reading for the Friday class).
    • Exception: As a "warm-up," in the first week we'll have an ADDITIONAL graded pre-class reading questionnaire, due on the first Wednesday at 8am. (It will be posted on Canvas by the first Monday morning.)
    Grading scheme: The course grade is based on the following components:
    • 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%).
    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.

    Honor code policy

    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, it is forbidden to
    • 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.
    The university is well-aware of "academic educational sites," such as Chegg, Slader, 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.

    Exam proctoring

    This course is participating in the proctoring pilot overseen by the Academic Integrity Working Group (AIWG). The purpose of this pilot is to determine the efficacy of proctoring and develop effective practices for proctoring in-person exams at Stanford. To find more details on the pilot or the working group, please visit the AIWG's webpage.

    Office hours and other resources for help

    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.

    COVID adaptations

    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.

    Spring 2024 -- Department of Mathematics, Stanford University
    Problems with this page? Contact so we can fix the problem.