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.
Harvard University Updated 3 min read
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
On this page
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 and 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 | Dot and cross product |
| Vectors, dot and cross products | The dot product | Dot and cross product |
| Lines and planes | Lines in space | Planes in space |
| Parametric curves | Curves and motion | Chain rule |
| Partial derivatives | Partial derivatives | Partial derivatives guide |
| Gradients | The gradient | Gradient and directional derivatives |
| Optimization and Lagrange multipliers | Lagrange multipliers | Lagrange multipliers guide |
| Double and triple integrals | Double integrals over general regions | Double integrals guide |
| Polar, cylindrical and spherical coordinates | Cylindrical and spherical | Polar double integrals |
| Vector fields and line integrals | Vector fields | Green's theorem |
| Surface integrals and flux | Surface integrals and flux | Divergence theorem |
| Stokes' theorem | Stokes and divergence 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.
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. 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
3. A double integral over a triangle. Over the region 0 \le y \le x \le 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 lesson trains reading a region two ways.
- Practice the big three theorems side by side. Green's, Stokes' and the 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.
- Rehearse for the evening midterms with timed practice exams, and scan the common calculus mistakes for sign errors.
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Points and distance in space
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