# MIT 18.022 Calculus: Topics, Study Guide and Practice

> A study guide for MIT 18.022, the concept-focused Calculus II: vectors, matrices, curvature, vector fields, multiple integrals and Stokes, with free practice.

Source: https://practicecalculus.com/courses/mit-18-022
Updated: 2026-11-03

## At a glance

- **Course code:** 18.022 (Calculus, satisfies the Calculus II GIR)
- **Units:** 5-0-7 (12 units)
- **Offered:** Fall
- **Prerequisites:** Calculus I (GIR): 18.01, 18.01A or 18.014
- **Covers:** Vector algebra, matrices and determinants, the multivariable derivative, curves with arc length, curvature and torsion, gradient, curl and divergence, multiple integrals with change of variables, line and surface integrals, Stokes' theorem
- **Alternatives:** 18.02 and 18.02A (credit is granted for only one of the three)
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- 18.022 is a fall-only Calculus II subject at MIT that covers the topics of 18.02 with more focus on mathematical concepts.
- It adds curvature and torsion of space curves and change of variables in multiple integrals to the usual multivariable syllabus.
- It ends with Stokes' theorem in one, two and three dimensions, presenting the fundamental theorems as one family.
- 18.022, 18.02 and 18.02A exclude credit for each other, so students take one of them.

**MIT 18.022** is the Calculus II option for MIT students who want multivariable calculus with a stronger conceptual thread. It is offered in the fall and counts toward the same requirement as [18.02](/courses/mit-18-02), so students choose one or the other. This guide lists what the catalog says 18.022 covers and links every topic to free practice with instant feedback.

## What MIT 18.022 covers

According to the MIT catalog, 18.022 treats the calculus of several variables with the topics of 18.02 but with more focus on mathematical concepts. It starts with vector algebra, the dot product, and matrices and determinants. It then defines continuity, differentiability and the derivative for functions of several variables, and studies parametrized curves through arc length, curvature and torsion. Vector fields come next, with gradient, curl and divergence. The integration half covers multiple integrals and change of variables, then line and surface integrals, and closes with Stokes' theorem in one, two and three dimensions.

Compared with a standard **Calculus 3** course, 18.022 covers the same core plus curvature, torsion and a more general view of the fundamental theorems.

## 18.022 topic map

| 18.022 topic | Practice | Read |
| --- | --- | --- |
| Vectors and the dot product | [The dot product](/learn/calculus-3/vectors-and-space/dot-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Matrices and determinants | [The cross product](/learn/calculus-3/vectors-and-space/cross-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Limits, continuity and differentiability | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives guide](/blog/partial-derivatives) |
| The derivative and linear approximation | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Chain rule and gradient | [The multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Parametrized curves, arc length | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Parametric arc length](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) |
| Vector fields, curl and divergence | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) | [Divergence theorem](/theorems/divergence-theorem) |
| Multiple integrals | [General regions](/learn/calculus-3/multiple-integrals/general-regions) | [Double integrals guide](/blog/double-integrals) |
| Change of variables | [The Jacobian](/learn/calculus-3/multiple-integrals/change-of-variables) | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) |
| Line and surface integrals | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Green's theorem](/theorems/greens-theorem) |
| Stokes' theorem | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem) |

## 18.022 vs 18.02 vs 18.02A

| Subject | Offered | Style |
| --- | --- | --- |
| 18.022 | Fall | More focus on concepts, adds curvature, torsion and change of variables |
| 18.02 | Fall and spring | The standard Calculus II subject |
| 18.02A | Split: late fall, then IAP or late spring | 18.02 content in two halves |

All three require Calculus I, such as [18.01](/courses/mit-18-01) or [18.01A](/courses/mit-18-01a). Pick 18.022 if you enjoy asking why a theorem is true, and 18.02 if you want the most direct route through the computations.

## Example problems at the 18.022 level

**1. Curvature of a helix.** For $\mathbf{r}(t) = \langle \cos t, \sin t, t \rangle$, $\mathbf{r}'(t) = \langle -\sin t, \cos t, 1 \rangle$ and $\mathbf{r}''(t) = \langle -\cos t, -\sin t, 0 \rangle$. Then $|\mathbf{r}' \times \mathbf{r}''| = \sqrt{2}$ and $|\mathbf{r}'| = \sqrt{2}$, so

$$\kappa = \frac{|\mathbf{r}' \times \mathbf{r}''|}{|\mathbf{r}'|^3} = \frac{\sqrt{2}}{2\sqrt{2}} = \frac{1}{2}$$

**2. Divergence and curl.** For $\mathbf{F} = \langle x^2, xy, z \rangle$:

$$\nabla\cdot\mathbf{F} = 2x + x + 1 = 3x + 1, \qquad \nabla\times\mathbf{F} = \langle 0, 0, y \rangle$$

**3. Change of variables.** The area of the disk $x^2 + y^2 \le 4$ in polar coordinates, where the Jacobian is $r$:

$$\int_0^{2\pi}\int_0^2 r\,dr\,d\theta = 2\pi \cdot 2 = 4\pi$$

## How to do well in MIT 18.022

- **Know the definitions cold.** Concept questions often hinge on exactly what differentiable means. Rewrite each definition in your own words and test it on an example.
- **Keep computing.** Conceptual courses still grade computations. Use [partial derivatives practice](/learn/calculus-3/partial-derivatives/partial-derivatives-practice) and [multiple integrals practice](/learn/calculus-3/multiple-integrals/multiple-integrals-practice) to stay fast.
- **See the theorems as one pattern.** The [Fundamental Theorem of Calculus](/theorems/fundamental-theorem-of-calculus-part-2), [Green's theorem](/theorems/greens-theorem) and [Stokes' theorem](/theorems/stokes-theorem) are versions of one statement, which is how 18.022 presents them.
- **Use the [formula sheet](/formulas)** and timed [practice](/practice) before exams, and skim the [common calculus mistakes](/blog/common-calculus-mistakes) for sign errors in cross products.

## Frequently asked questions

### What is MIT 18.022?

18.022 is one of MIT's three Calculus II subjects. It studies the calculus of several variables, following the topics of 18.02 but with more attention to concepts: the derivative of a multivariable function, parametrized curves, vector fields, multiple integrals and Stokes' theorem.

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

Both satisfy Calculus II and cover multivariable calculus. 18.02 is offered every fall and spring and leans toward computation. 18.022 is offered in fall only and puts more weight on mathematical concepts, including curvature, torsion and change of variables.

### Is 18.022 harder than 18.02?

It asks for more comfort with abstraction and reasoning about definitions, which many students find more demanding. The computations are the same kind, so strong multivariable practice helps in either subject.

### What do I need before 18.022?

The prerequisite is Calculus I: 18.01, 18.01A or 18.014. You should be fluent with single-variable derivatives and integrals, and it helps to have seen vectors before.