MIT 18.02 Multivariable Calculus: Topics, Study Guide, Practice
A study guide for MIT 18.02 (Calculus II): vectors, partial derivatives, multiple integrals and the theorems of Green and Stokes, mapped to free practice.
MIT Updated 3 min read
At a glance
- Course code
- 18.02 (listed in the catalog as Calculus, the Calculus II GIR)
- Units
- 5-0-7 (12 units)
- Offered
- Fall and spring
- Prerequisites
- Calculus I (GIR): 18.01, 18.01A or 18.014
- Covers
- Vectors in 3-space, matrices and determinants, space curves, partial derivatives, gradients, optimization, double, triple, line and surface integrals, Green's, Stokes' and the divergence theorem
- Alternatives
- 18.02A (split across fall and IAP or spring) and 18.022 (more theoretical, fall only)
- Practice here
- Calculus 3
On this page
MIT 18.02 is the second calculus subject in MIT's General Institute Requirements. Because 18.01 already covers single-variable calculus through series, 18.02 goes straight to functions of several variables, which most universities teach as Calculus 3. Nearly every MIT undergraduate takes 18.02 or one of its alternatives. This guide maps each 18.02 topic to free practice with instant feedback.
What MIT 18.02 covers
The MIT catalog describes 18.02 as the calculus of several variables. It opens with vector algebra in three dimensions, along with matrices and determinants, then studies vector-valued functions of one variable and motion in space. The middle of the course is partial differentiation: gradients and optimization. The final third is integration: double integrals and line integrals in the plane, exact differentials and conservative fields, Green's theorem, triple integrals, and line and surface integrals in space, finishing with the divergence theorem and Stokes' theorem.
That is the standard Calculus 3 syllabus, with a short dose of matrices and determinants that many schools save for linear algebra.
18.02 topic map
| 18.02 topic | Practice | Read |
|---|---|---|
| Vectors in 3-space | Vectors and their length | Dot and cross product |
| Dot and cross products, determinants | The cross product | Dot and cross product |
| Lines and planes | Planes in space | Dot and cross product |
| Vector functions and space motion | Curves and motion | Chain rule |
| Partial derivatives | Partial derivatives | Partial derivatives guide, Clairaut's theorem |
| Gradients and directional derivatives | The gradient | Gradient and directional derivatives |
| Optimization | Second-derivative test | Lagrange multipliers |
| Double integrals | Double integrals over rectangles | Double integrals guide |
| Line integrals and conservative fields | Conservative fields | Fundamental Theorem of Calculus |
| Green's theorem | Green's theorem | Green's theorem |
| Triple integrals | Triple integrals | Cylindrical and spherical |
| Surface integrals and flux | Surface integrals and flux | Divergence theorem |
| Stokes' theorem | Stokes and divergence theorems | Stokes' theorem |
18.02 vs 18.02A vs 18.022
| Subject | Format | Emphasis |
|---|---|---|
| 18.02 | One full semester, fall or spring | Computation and applications |
| 18.02A | First half in late fall, second half in IAP or late spring | Same material as 18.02, split in two |
| 18.022 | One semester, fall only | Similar topics with more focus on concepts, including curvature, torsion and change of variables |
The three subjects all satisfy Calculus II, so you take exactly one. Students who finish 18.01A in the first half of fall can move into a Calculus II subject without waiting a full term. If you enjoy proofs and abstraction, read the 18.022 guide.
Example problems at the 18.02 level
1. Angle between vectors. Find the angle between \mathbf{u} = \langle 1, 1, 0 \rangle and \mathbf{v} = \langle 1, 0, 1 \rangle.
2. Partial derivatives. For f(x, y) = x^2 y + e^{xy}:
3. Green's theorem. Find the circulation of \mathbf{F} = \langle -y, x \rangle counterclockwise around the unit circle C. Here \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 1 - (-1) = 2, so
How to do well in MIT 18.02
- Draw before you compute. Sketch every region, surface and vector field. Most wrong limits of integration come from skipping the picture. Practice setting up bounds with double integrals over general regions.
- Keep single-variable skills sharp. Every iterated integral ends as an 18.01 integral, so review u-substitution and the calculus formula sheet.
- Learn the big theorems as one idea. Green's, Stokes' and the divergence theorem all say that the integral of a derivative over a region equals a boundary term, just like the Fundamental Theorem of Calculus.
- Drill under time pressure with practice exams, and check the common calculus mistakes list for sign and orientation errors.
Tip: before using Green's or Stokes' theorem, check the orientation of the boundary. A clockwise curve flips the sign of the answer.
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