# Harvard Math 21a Multivariable Calculus: Study Guide, Practice

> A study guide for Harvard Math 21a (Multivariable Calculus): curves, surfaces, partial derivatives, vector fields and Stokes, mapped to free practice.

Source: https://practicecalculus.com/courses/harvard-math-21a
Updated: 2026-11-03

## At a glance

- **Course code:** Mathematics 21A (Math 21a)
- **Credits:** 4
- **Recommended prep:** Math 1b or an equivalent background
- **Covers:** Curves, surfaces and solids in space, vectors, lines and planes, partial derivatives, gradients and optimization, multiple integrals, vector fields, and the theorems of Green, Stokes and Gauss
- **Credit restriction:** Not open for credit after Math 18A or Math 22B
- **Related course:** Mathematics 21B (linear algebra and differential equations)
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- Math 21a is Harvard's general-purpose multivariable calculus course for two and three dimensions.
- It extends limits, derivatives and integrals to functions of several variables and ends with Green's, Stokes' and the divergence theorems.
- The department considers Math 1b, or equivalent background, essential preparation.
- Math 21a and 21b can be taken in either order, though taking 21a first is advantageous.

**Harvard Math 21a** is the multivariable calculus course most Harvard students in the physical sciences, economics, engineering and mathematics take after single-variable calculus. It extends the ideas of [Math 1a](/courses/harvard-math-1a) and [Math 1b](/courses/harvard-math-1b) to functions of two and three variables and is taught in small sections. This guide maps every Math 21a topic to free practice with instant feedback.

## What Harvard Math 21a covers

According to the Harvard Mathematics Department, Math 21a builds geometric reasoning about functions of several variables, with applications in physics, engineering, economics, data science, machine learning and the life sciences. Students learn to describe and picture curves, surfaces and solid regions in three dimensions, and to work with vectors, lines and planes. Partial derivatives, gradients and optimization extend differentiation. Integration moves to regions, curves and surfaces. Vector fields model flow and circulation, and the course ends with Green's theorem, Stokes' theorem and the divergence theorem, which link what happens inside a region to its boundary.

That is a complete **Calculus 3** course.

## Math 21a topic map

| Math 21a topic | Practice | Read |
| --- | --- | --- |
| Points, distance and surfaces in space | [Points and distance](/learn/calculus-3/vectors-and-space/points-and-distance) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Vectors, dot and cross products | [The dot product](/learn/calculus-3/vectors-and-space/dot-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Lines and planes | [Lines in space](/learn/calculus-3/vectors-and-space/lines-in-space) | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) |
| Parametric curves | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Chain rule](/rules/chain-rule) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives guide](/blog/partial-derivatives) |
| Gradients | [The gradient](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Optimization and Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers guide](/blog/lagrange-multipliers) |
| Double and triple integrals | [Double integrals over general regions](/learn/calculus-3/multiple-integrals/general-regions) | [Double integrals guide](/blog/double-integrals) |
| Polar, cylindrical and spherical coordinates | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) |
| Vector fields and line integrals | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) | [Green's theorem](/theorems/greens-theorem) |
| Surface integrals and flux | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | [Divergence theorem](/theorems/divergence-theorem) |
| Stokes' theorem | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem) |

## Math 21a vs 21b vs 19a vs 22

| Course | Focus | Notes |
| --- | --- | --- |
| Math 21a | Multivariable calculus | General purpose, used in many fields |
| Math 21b | Linear algebra and differential equations | Either order with 21a, though 21a first is advantageous |
| Math 19a | Multivariable calculus and differential equations for the life sciences | Fall only, should not be taken in addition to 21a |
| Math 22a and 22b | Linear algebra and multivariable calculus with proofs | 22b roughly matches 21a in content |

If your concentration is in the life sciences, compare this page with the [Math 19a guide](/courses/harvard-math-19a).

## Example problems at the Math 21a level

**1. Equation of a plane.** Find the plane through $(1, 0, 2)$ with normal vector $\langle 2, -1, 3 \rangle$:

$$2(x - 1) - (y - 0) + 3(z - 2) = 0 \quad\Rightarrow\quad 2x - y + 3z = 8$$

**2. A directional derivative.** For $f(x, y) = x^2 + 3y$ at $(1, 2)$ in the direction of $\mathbf{u} = \langle \frac{3}{5}, \frac{4}{5} \rangle$: $\nabla f = \langle 2x, 3 \rangle = \langle 2, 3 \rangle$, so

$$D_{\mathbf{u}} f = 2 \cdot \frac{3}{5} + 3 \cdot \frac{4}{5} = \frac{18}{5}$$

**3. A double integral over a triangle.** Over the region $0 \le y \le x \le 1$:

$$\int_0^1\int_0^x 6y\,dy\,dx = \int_0^1 3x^2\,dx = 1$$

## How to do well in Harvard Math 21a

- **Sketch every object.** The course emphasizes geometric intuition, so draw surfaces, regions and fields before calculating. Our [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) lesson trains reading a region two ways.
- **Practice the big three theorems side by side.** [Green's](/theorems/greens-theorem), [Stokes'](/theorems/stokes-theorem) and the [divergence theorem](/theorems/divergence-theorem) each turn a boundary integral into an integral over a region.
- **Keep single-variable integration fast.** Every iterated integral ends as a Math 1b integral. Review with the [formula sheet](/formulas).
- **Rehearse for the evening midterms** with timed [practice exams](/practice), and scan the [common calculus mistakes](/blog/common-calculus-mistakes) for sign errors.

## Frequently asked questions

### What is Harvard Math 21a?

Math 21a, Multivariable Calculus, takes single-variable calculus into two and three dimensions. It covers curves, surfaces and solids, vectors, lines and planes, partial derivatives and gradients, optimization, multiple integrals, vector fields and the multivariable versions of the Fundamental Theorem of Calculus.

### What is the difference between Math 21a and Math 21b?

Math 21a is multivariable calculus. Math 21b is an introduction to linear algebra in two, three and higher dimensions, with a large part of the course applying it to differential equations. They can be taken in either order, but the department says taking 21a first is advantageous.

### Math 21a vs Math 19a: which should I take?

Math 21a is the general multivariable course used across physics, engineering, economics and data science. Math 19a teaches multivariable calculus and differential equations through life science models and is recommended for life science concentrators. The two should not both be taken.

### Is Math 21a hard?

The calculus is familiar, but visualizing surfaces, regions and vector fields in three dimensions is new for most students. Sketching often and practicing setups helps more than memorizing formulas.

### Can I skip Math 1b and go straight to 21a?

The department warns that skipping Math 1b can hurt later progress and suggests talking to a concentration advisor first. Students placed into the Math 19a to 55 group can start with 21a.