MIT 18.022 Calculus: Topics, Study Guide and Practice

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, 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 vs 18.02 vs 18.02A

All three require Calculus I, such as 18.01 or 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

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

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

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 and multiple integrals practice to stay fast.
• See the theorems as one pattern. The Fundamental Theorem of Calculus, Green's theorem and Stokes' theorem are versions of one statement, which is how 18.022 presents them.
• Use the formula sheet and timed practice before exams, and skim the common calculus mistakes for sign errors in cross products.