Harvard MATH 21A: Multivariable Calculus

Math 21A covers multivariable calculus — vectors, partial derivatives, multiple integrals, and vector fields — and is the standard math course for Harvard students in sciences, economics, and pre-med tracks who arrive with single-variable calculus done.

Fennie is independent and not affiliated with Harvard University. This is an unofficial study guide.

Build my MATH 21A study plan

What makes it hard

Visualizing in three dimensions is the recurring obstacle: setting up double and triple integrals over odd regions, and keeping gradient, divergence, and curl conceptually distinct. The course is taught in many small sections, so exam difficulty is set by common exams that can outpace what your individual section emphasized.

What you'll cover

• Vectors and surfaces
• Partial derivatives and gradients
• Optimization and Lagrange multipliers
• Double and triple integrals
• Vector fields and line integrals
• Green's and Stokes' theorems

The MATH 21A study guide

How to study for Harvard MATH 21A, step by step.

1
Practice setting up integrals from region descriptions
Given a region in words, write the bounds — in multiple orders, in multiple coordinate systems. Setup is where Math 21A exam points live, and the skill decays fast without weekly reps.
2
Sketch everything in three dimensions
Surfaces, level curves, regions of integration — draw them, badly if necessary. The students who struggle most in 21A are the ones who try to do multivariable calculus purely symbolically.
3
Keep gradient, divergence, and curl distinct
Write a one-line plain-English description of what each measures and revisit it weekly. Conceptual confusion among the three is a reliable source of common-exam errors.
4
Study to the common exams, not just your section
Exam difficulty is set course-wide and can outpace what your individual section emphasized. Past common exams are the calibration tool — work them timed in the weeks before each midterm.
5
Keep the rotation going with Fennie
Upload the Math 21A syllabus and Fennie's Daily Plan keeps integral setups in steady practice rotation paced to the common exam dates, with setup-focused quizzes generated from your actual course materials. Free to start.
Start my MATH 21A plan free

How Fennie helps with MATH 21A

Daily Plans keep Math 21A's integral setups in steady practice rotation — the skill that decays fastest without reps. Chat through how to slice a region or why Lagrange multipliers work, and drill practice problems on the integral setups that dominate the common exams.

FAQ

Is Math 21A hard?

Moderately — the concepts are visual and the common exams are fair but thorough. Students who practice setting up integrals from region descriptions do well; those who memorize formulas don't.

Do pre-meds need Math 21A?

Many Harvard pre-meds take 21A to satisfy quantitative requirements, though requirements vary by medical school. It's also required or expected for several science concentrations.

Math 21A or 21B first?

They're largely independent — 21A is multivariable calculus, 21B is linear algebra and differential equations — and Harvard allows either order for most students.

Pass MATH 21A with a plan, not a cram

Upload your MATH 21A materials and Fennie generates a Daily Plan paced to your deadline — plus chat, flashcards, and quizzes built from the actual course content.

Get started free

More Harvard courses

MATH 55 — Studies in Algebra and Group Theory / Real and Complex Analysis

Math 55 (55A in fall, 55B in spring) is Harvard's legendary honors freshman math sequence, covering proof-based abstract algebra, group theory, and real and complex analysis at extraordinary speed and depth. It's widely described as the hardest undergraduate math class in the country.

MATH 1A — Introduction to Calculus

Math 1A is Harvard's introductory calculus course — limits, derivatives, integrals, and the Fundamental Theorem of Calculus — for students who didn't take or don't have credit for calculus before college. It leads into Math 1B and the rest of the math sequence.

MATH 21B — Linear Algebra and Differential Equations

Math 21B covers linear algebra — systems, matrices, eigenvalues, orthogonality — and applies it to differential equations and Fourier series. It's a workhorse requirement for applied math, economics, CS, and science concentrators.

MATH 25 — Theoretical Linear Algebra and Real Analysis

Math 25 (25A in fall, 25B in spring) is Harvard's honors first-year sequence in proof-based linear algebra and real analysis — the rigorous track below the legendary Math 55. It's the standard landing spot for strong students who want real mathematics at a survivable pace.