UCLA Math 32B Multiple Integrals and Vector Calculus: Guide

UCLA Math 32B completes UCLA's calculus sequence. After Math 32A handles derivatives of functions of several variables, Math 32B handles their integrals and finishes with the central theorems of vector calculus. Because it draws on integration techniques too, it requires Math 31B as well. This guide maps every Math 32B topic to free practice with instant feedback.

What UCLA Math 32B covers

The UCLA General Catalog describes Math 32B as an introduction to integral calculus of several variables, with line and surface integrals. The Mathematics Department's outline starts with integration in two variables over rectangles and more general regions, then triple integrals, polar coordinates and integration in polar coordinates, applications of multiple integrals, and change of variables. The second half moves to vector fields, line integrals, conservative vector fields, parametrized surfaces and surface integrals, and ends with Green's theorem, Stokes' theorem and the divergence theorem.

That is the second half of a standard Calculus 3 course.

Math 32B topic map

The UCLA multivariable sequence

Every Math 32B unit leans on earlier courses: iterated integrals use 31B techniques, and line and surface integrals use the parametrizations and gradients from 32A.

Example problems at the Math 32B level

1. Switching the order of integration. The integral \int_0^1\int_y^1 e^{x^2}\,dx\,dy cannot be done in this order, but the region is 0 \le y \le x \le 1, so

2. A line integral. For \mathbf{F} = \langle y, x \rangle along \mathbf{r}(t) = \langle t, t^2 \rangle, 0 \le t \le 1:

This matches the potential function xy evaluated from (0, 0) to (1, 1).

3. Green's theorem. For \mathbf{F} = \langle -y^2, xy \rangle around the boundary of the unit square [0, 1]\times[0, 1], counterclockwise, Q_x - P_y = y + 2y = 3y, so

How to do well in UCLA Math 32B

• Sketch the region before every integral. Most mistakes are wrong limits. Practice with switching the order and multiple integrals practice.
• Test for conservative fields first. If a field has a potential function, a line integral is just a difference of two values.
• Keep orientation consistent for Green's and Stokes' theorems. The common calculus mistakes list covers sign errors like these.
• Finish with mixed review in the vector calculus practice set, keep the formula sheet handy, and take timed practice exams before the final.