# CMU 21-259 Calculus in Three Dimensions: Topics and Practice

> A study guide for Carnegie Mellon 21-259 Calculus in Three Dimensions: vectors, partial derivatives, multiple integrals and vector calculus, with practice.

Source: https://practicecalculus.com/courses/carnegie-mellon-21-259
Updated: 2026-11-08

## At a glance

- **Course code:** 21-259
- **Units:** 10 (per the department's sample schedule)
- **Offered:** Not stated on the department pages
- **Prerequisites:** Not stated on the department topic list; it follows 21-122 in the department's sequence
- **Covers:** Vectors, lines, planes and quadric surfaces, vector-valued functions and curvature, multivariable limits and partial derivatives, gradients, optimization and Lagrange multipliers, multiple integrals, vector fields, line and surface integrals, Green's, Stokes' and the Divergence Theorem
- **Next course:** Not stated on the department pages
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- 21-259 is Carnegie Mellon's multivariable calculus course.
- The official topic list runs from vectors and quadric surfaces to Stokes' Theorem and the Divergence Theorem.
- It includes curvature and the unit tangent, normal and binormal vectors, which some Calculus 3 courses skip.
- A placement exam is offered, and passing it gives a prerequisite waiver rather than course credit.

**CMU 21-259**, Calculus in Three Dimensions, is Carnegie Mellon's multivariable calculus course. The Department of Mathematical Sciences publishes a detailed topic list for its 21-259 placement exam, and it reads like a full Calculus 3 syllabus. This guide walks through it and links each topic to free practice with instant feedback.

## What 21-259 covers

The department's topic list opens with vector geometry: vector arithmetic in $\mathbb{R}^2$ and $\mathbb{R}^3$, dot and cross products and their geometric meaning, projections, lines and curves in $\mathbb{R}^3$ (including parallel, intersecting and skew lines), planes and distances, and quadric surfaces and cylinders.

Next come vector-valued functions: componentwise derivatives and integrals, arc length, reparametrization by arc length, curvature, and the unit tangent, normal and binormal vectors.

Then multivariable differential calculus: contour maps, limits and continuity including path arguments, partial derivatives and Clairaut's Theorem, the multivariable chain rule and implicit differentiation, directional derivatives and the gradient, tangent planes and linear approximation, differentiability, critical points and the second derivative test, absolute extrema with the Extreme Value Theorem, and Lagrange multipliers.

The list finishes with integration: double integrals with Fubini's Theorem, polar coordinates and change of variables with Jacobians, triple integrals in Cartesian, cylindrical and spherical coordinates, mass and centers of mass, vector fields, divergence and curl, potential functions, line integrals and the Fundamental Theorem of Line Integrals, Green's Theorem, surface integrals and flux, Stokes' Theorem and the Divergence Theorem.

## 21-259 topic map

| 21-259 topic | Practice | Read |
| --- | --- | --- |
| Dot and cross products | [Dot product](/learn/calculus-3/vectors-and-space/dot-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Projections | [Projections and work](/learn/calculus-3/vectors-and-space/projections-and-work) | [Cross product](/learn/calculus-3/vectors-and-space/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) |
| Vector-valued functions and 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) |
| Contour maps and functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives guide](/blog/partial-derivatives) |
| Partial derivatives and Clairaut's Theorem | [Second partial derivatives](/learn/calculus-3/partial-derivatives/second-partial-derivatives) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Chain rule | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Chain rule](/rules/chain-rule) |
| Gradient and directional derivatives | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Tangent planes | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Extrema | [Critical points and the second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Extreme Value Theorem](/theorems/extreme-value-theorem) |
| Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers guide](/blog/lagrange-multipliers) |
| Double integrals and Fubini | [Switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) | [Double integrals](/blog/double-integrals) |
| Polar coordinates and Jacobians | [Change of variables](/learn/calculus-3/multiple-integrals/change-of-variables) | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) |
| Triple integrals | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) |
| Mass and centers of mass | [Double integral applications](/learn/calculus-3/multiple-integrals/double-integral-applications) | [Double integrals practice](/practice-problems/double-integrals) |
| Divergence, curl and potentials | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) |
| Line integrals | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Green's theorem](/theorems/greens-theorem) |
| Surface integrals, Stokes and Divergence | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem), [Divergence theorem](/theorems/divergence-theorem) |

## Example problems at the 21-259 level

**1. A cross product.** $\langle 1, 0, 0 \rangle \times \langle 0, 1, 0 \rangle = \langle 0, 0, 1 \rangle$, a unit vector perpendicular to both.

**2. A Lagrange multiplier problem.** Maximize $f = xy$ on $x + y = 10$. From $y = \lambda$ and $x = \lambda$ we get $x = y = 5$ and a maximum of $25$.

**3. Spherical coordinates.** The volume of the unit ball is

$$\int_0^{2\pi}\int_0^{\pi}\int_0^1 \rho^2 \sin\phi\,d\rho\,d\phi\,d\theta = \frac{4\pi}{3}$$

## How to do well in 21-259

- **Make vector geometry automatic.** Planes, projections and cross products appear in almost every later topic.
- **Sketch every region.** Fubini, change of variables and triple integrals all depend on getting the limits right.
- **Know when a field is conservative.** Checking the curl first can turn a hard line integral into a one-line answer.
- **Review single-variable techniques.** Integration from [21-122](/courses/carnegie-mellon-21-122) shows up inside every iterated integral.

## Frequently asked questions

### What is CMU 21-259?

21-259 Calculus in Three Dimensions is Carnegie Mellon's multivariable calculus course. Its topic list covers vectors and 3D geometry, vector-valued functions, partial derivatives, optimization, multiple integrals and vector calculus.

### How many units is 21-259?

The Department of Mathematical Sciences sample schedule lists 21-259 at 10 units.

### Does 21-259 cover Stokes' Theorem?

Yes. The topic list ends with surface integrals, flux, Stokes' Theorem and the Divergence Theorem.

### Can I place out of 21-259?

The department offers a 21-259 placement exam. Passing it gives a prerequisite waiver, not course credit, and no calculators or other technology are allowed.

### What should I review before 21-259?

Integration techniques and parametric curves from 21-122, plus derivative rules from 21-120. Every iterated integral is built from single-variable integrals.