# 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.

Source: https://practicecalculus.com/courses/mit-18-02
Updated: 2026-11-03

## 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](/learn/calculus-3)

## Key takeaways

- MIT 18.02 is the Calculus II subject in the General Institute Requirements, and at MIT Calculus II means multivariable calculus.
- It runs from vector algebra and matrices through partial derivatives to multiple integrals and the theorems of Green, Stokes and Gauss.
- The prerequisite is Calculus I: 18.01, 18.01A or 18.014.
- 18.02A covers the same material split into two halves, and 18.022 covers similar topics with more emphasis on concepts.

**MIT 18.02** is the second calculus subject in MIT's General Institute Requirements. Because [18.01](/courses/mit-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](/learn/calculus-3/vectors-and-space/vectors-and-length) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Dot and cross products, determinants | [The cross product](/learn/calculus-3/vectors-and-space/cross-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Lines and planes | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Vector functions and space motion | [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), [Clairaut's theorem](/theorems/clairauts-theorem) |
| Gradients and directional derivatives | [The gradient](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Optimization | [Second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integrals | [Double integrals over rectangles](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals guide](/blog/double-integrals) |
| Line integrals and conservative fields | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Green's theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem) |
| Triple integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) |
| 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) |

## 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](/courses/mit-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](/courses/mit-18-022).

## Example problems at the 18.02 level

**1. Angle between vectors.** Find the angle between $\mathbf{u} = \langle 1, 1, 0 \rangle$ and $\mathbf{v} = \langle 1, 0, 1 \rangle$.

$$\cos\theta = \frac{\mathbf{u}\cdot\mathbf{v}}{|\mathbf{u}|\,|\mathbf{v}|} = \frac{1}{\sqrt{2}\sqrt{2}} = \frac{1}{2}, \qquad \theta = \frac{\pi}{3}$$

**2. Partial derivatives.** For $f(x, y) = x^2 y + e^{xy}$:

$$f_x = 2xy + y e^{xy}, \qquad f_y = x^2 + x e^{xy}$$

**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

$$\oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_D 2\,dA = 2 \cdot \pi(1)^2 = 2\pi$$

## 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](/learn/calculus-3/multiple-integrals/general-regions).
- **Keep single-variable skills sharp.** Every iterated integral ends as an 18.01 integral, so review [u-substitution](/blog/u-substitution) and the [calculus formula sheet](/formulas).
- **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](/theorems/fundamental-theorem-of-calculus-part-2).
- **Drill under time pressure** with [practice exams](/practice), and check the [common calculus mistakes](/blog/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.

## Frequently asked questions

### What is MIT 18.02?

18.02 is MIT's Calculus II subject: the calculus of several variables. It covers vectors and matrices, functions of several variables, partial derivatives and optimization, and double, triple, line and surface integrals, ending with Green's theorem, Stokes' theorem and the divergence theorem.

### What is the difference between 18.02, 18.02A and 18.022?

All three satisfy Calculus II and exclude credit for each other. 18.02A splits the 18.02 material in two: the first half in the last six weeks of fall, the second half in IAP or the second half of spring. 18.022, offered in fall, treats similar topics with more focus on mathematical concepts.

### What do I need before taking 18.02?

The official prerequisite is the Calculus I GIR, meaning 18.01, 18.01A or 18.014. In practice you want fluent single-variable derivatives and integrals, because every multivariable computation reduces to them.

### Is MIT 18.02 hard?

The calculus itself is mostly single-variable calculus applied one variable at a time. What makes 18.02 feel hard is the geometry: picturing surfaces, regions and vector fields in three dimensions. Sketching every region before you integrate is the habit that helps most.

### Is 18.02 the same as Calculus 3 elsewhere?

Largely, yes. Its topics match a typical Calculus 3 or multivariable calculus course, plus some matrices and determinants that many universities leave to a linear algebra course.