Multivariable Calculus
MATH 21A
Subject & Catalog Number
Course Information
Description
Science relies on understanding quantitative relationships in many different contexts, and many of the most useful relationships are functions of several variables. For this reason, multivariable calculus is a foundational subject with wide-ranging applications in physics, engineering, economics, data science, machine learning, and the life sciences.
In this course, we develop geometric reasoning to build intuition about functions of multiple variables. You will learn to describe and visualize curves, surfaces, and solid regions in space, and to see how familiar ideas from single-variable calculus—such as limits, derivatives, and integrals—extend to these new settings.
You will work with vectors, lines, and planes; explore partial derivatives, gradients, and optimization; and study vector fields that model phenomena such as electricity and magnetism, flow, and circulation. These concepts culminate in the central theorems of vector calculus—Green’s Theorem, Stokes’s Theorem, and the Divergence Theorem—which reveal deep connections between local behavior and global structure.
Throughout, the course emphasizes geometric intuition, visualization, problem solving, quantitative reasoning, and a strong conceptual understanding of the structures underlying multivariable calculus.
Course Notes
Anti-requisite: Cannot be taken for credit if you have passed Math 18A or Math 22B
Class Notes
Math 21a is taught in small sections throughout the day on a Monday/Wednesday/Friday schedule (9:00-10:15 am, 10:30-11:45 am, 12-1:15 pm, 1:30-2:45 pm, and 3:00-4:15 pm with sufficient enrollment). For information about how to register for this course and to rank your time preferences, please see the Canvas site.
Exams for this course will ocur throughout the semester on Thursdays in the block from 6:00 - 9:00 pm. Consult Canvas for more details on the exact Thursdays.
The course will have a mandatory information meeting on the morning of the first day of the semester. Information will be available via Canvas.
Midterms will be scheduled on various Thursdays during the block 6:00 PM - 9:00 PM. You will sign up for a "Quiz Section" to block this time on your schedule.
If there is an unavoidable academic conflict, please email Sarah Chin explaining your conflict and why it is unavoidable. She will arrange an alternative time for you.
Recommended Prep
The first time you enroll in any introductory math course (Math Ma through Math 22a), you are required to take a Math Verification Exam. Your results may determine which introductory course(s) you are eligible to take. For more information please go to this website: https://canvas.harvard.edu/courses/168499. Mathematics 1b, or an equivalent background, is essential preparation for Math 21a. Skipping Math 1b can be detrimental to your academic progress. If your concentration requires Math 21a or Math 21b, you may in fact need the skills, concepts, and ways of reasoning developed in Math 1b. Check with a concentration advisor before skipping Math 1b, as doing so is generally not advisable.
Available for Harvard Cross Registration
NOTE: This course requires additional sections; you will be prompted to choose secondary components during the Add to Cart process
