# UCLA Math 32B Multiple Integrals and Vector Calculus: Guide

> A study guide for UCLA Math 32B: double and triple integrals, change of variables, line and surface integrals, and the theorems of Green, Stokes and Gauss.

Source: https://practicecalculus.com/courses/ucla-math-32b
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 32B
- **Units:** 4 (three hours of lecture and one of discussion per week)
- **Requisites:** MATH 31B and MATH 32A, each with a grade of C- or better (enforced)
- **Covers:** Double and triple integrals, polar coordinates, applications of multiple integrals, change of variables, vector fields, line integrals, conservative fields, parametrized surfaces, surface integrals, and the theorems of Green, Stokes and Gauss
- **Previous course:** MATH 32A (Calculus of Several Variables)
- **Practice here:** [Calculus 3](/learn/calculus-3), multiple integrals and vector calculus units

## Key takeaways

- Math 32B is the second half of multivariable calculus at UCLA: integral calculus of several variables, line integrals and surface integrals.
- It requires both Math 31B and Math 32A with grades of C- or better.
- The first half covers multiple integrals and change of variables, and the second half covers vector fields and surface integrals.
- The course ends with Green's theorem, Stokes' theorem and the divergence theorem.

**UCLA Math 32B** completes UCLA's calculus sequence. After [Math 32A](/courses/ucla-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](/courses/ucla-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

| Math 32B topic | Practice | Read |
| --- | --- | --- |
| Double integrals over rectangles | [Double integrals over rectangles](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals guide](/blog/double-integrals) |
| More general regions | [Double integrals over general regions](/learn/calculus-3/multiple-integrals/general-regions) | [Switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) |
| Triple integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) |
| Polar coordinates | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates) |
| Applications of multiple integrals | [Area, volume, mass and average value](/learn/calculus-3/multiple-integrals/double-integral-applications) | [Center of mass](/learn/calculus-2/improper-integrals-and-applications/center-of-mass) |
| Change of variables | [The Jacobian](/learn/calculus-3/multiple-integrals/change-of-variables) | [U-substitution](/blog/u-substitution) |
| Vector fields | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Line integrals | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Line integrals of functions](/learn/calculus-3/vector-calculus/line-integrals-of-functions) |
| Conservative vector fields | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Parametrized surfaces and surface integrals | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) |
| Green's theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem) |
| Stokes' theorem and the divergence theorem | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem), [divergence theorem](/theorems/divergence-theorem) |

## The UCLA multivariable sequence

| Course | Requisites | Content |
| --- | --- | --- |
| Math 31A | Diagnostic test or Math 1 | Differential calculus, intro to integration |
| Math 31B | 31A | Integration techniques and infinite series |
| Math 32A | 31A | Vectors and differential calculus of several variables |
| Math 32B | 31B and 32A | Multiple integrals, line and surface integrals, vector calculus theorems |

Every Math 32B unit leans on earlier courses: iterated integrals use [31B](/courses/ucla-math-31b) techniques, and line and surface integrals use the parametrizations and gradients from [32A](/courses/ucla-math-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

$$\int_0^1\int_0^x e^{x^2}\,dy\,dx = \int_0^1 x e^{x^2}\,dx = \frac{e - 1}{2}$$

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

$$\int_C \mathbf{F}\cdot d\mathbf{r} = \int_0^1 \langle t^2, t \rangle\cdot\langle 1, 2t \rangle\,dt = \int_0^1 3t^2\,dt = 1$$

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

$$\oint_C \mathbf{F}\cdot d\mathbf{r} = \int_0^1\int_0^1 3y\,dy\,dx = \frac{3}{2}$$

## How to do well in UCLA Math 32B

- **Sketch the region before every integral.** Most mistakes are wrong limits. Practice with [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) and [multiple integrals practice](/learn/calculus-3/multiple-integrals/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](/blog/common-calculus-mistakes) list covers sign errors like these.
- **Finish with mixed review** in the [vector calculus practice](/learn/calculus-3/vector-calculus/vector-calculus-practice) set, keep the [formula sheet](/formulas) handy, and take timed [practice exams](/practice) before the final.

## Frequently asked questions

### What is UCLA Math 32B?

Math 32B, Calculus of Several Variables, is an introduction to integral calculus of several variables, along with line and surface integrals. It covers double and triple integrals, polar coordinates, change of variables, vector fields, conservative fields, surface integrals and the theorems of Green, Stokes and Gauss.

### What do I need before Math 32B?

The catalog enforces Math 31B and Math 32A, each with a grade of C- or better. Integration techniques from 31B and partial derivatives and vectors from 32A are both used constantly.

### Is Math 32B harder than Math 32A?

Many students find 32B harder because setting up integrals over three-dimensional regions and surfaces takes careful geometric thinking. Practicing setups, not just computations, makes the biggest difference.

### What is the difference between Math 32A and Math 32B?

Math 32A is the differential half of multivariable calculus: vectors, curves, partial derivatives and optimization. Math 32B is the integral half: multiple integrals, line and surface integrals, and the big theorems of vector calculus.