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

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

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

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 shows up inside every iterated integral.