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.
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 and 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, so work them timed in the weeks before each midterm.
Today
Today's MATH 21A plan
What a Fennie Daily Plan looks like for MATH 21A. Yours is built from your own syllabus and adapts every day to your deadlines and progress.
First plan free, no card required. Fennie is independent and unaffiliated with your school.
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.
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