# UCLA Math 32A Calculus of Several Variables: Study Guide

> A study guide for UCLA Math 32A: vectors, curves, partial derivatives, gradients, the chain rule, optimization and Lagrange multipliers, with free practice.

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

## At a glance

- **Course code:** MATH 32A
- **Units:** 4 (three hours of lecture and one of discussion per week)
- **Requisite:** MATH 31A with a grade of C- or better (enforced)
- **Covers:** Vectors, dot and cross products, planes, parametric equations, vector-valued functions, arc length and curvature, quadric surfaces, limits, partial derivatives, tangent planes, gradients, the chain rule, optimization and Lagrange multipliers
- **Next course:** MATH 32B (Calculus of Several Variables), which also requires MATH 31B
- **Practice here:** [Calculus 3](/learn/calculus-3), vectors and partial derivatives units

## Key takeaways

- Math 32A is the first half of multivariable calculus at UCLA: differential calculus of several variables and vector field theory.
- It requires only Math 31A, so it can be taken before Math 31B.
- The department outline runs from vectors and curves in space to partial derivatives, gradients, optimization and Lagrange multipliers.
- Math 32B, the integration half, requires both Math 32A and Math 31B.

**UCLA Math 32A** begins multivariable calculus at UCLA. Because its only requisite is [Math 31A](/courses/ucla-math-31a), it can be taken before [Math 31B](/courses/ucla-math-31b). It moves calculus from curves in the plane to vectors, surfaces and functions of several variables. This guide maps every Math 32A topic to free practice with instant feedback.

## What UCLA Math 32A covers

The UCLA General Catalog describes Math 32A as an introduction to differential calculus of several variables and vector field theory. The Mathematics Department's course outline begins with vectors in the plane and in space, the dot and cross products, and planes. It then covers parametric equations, vector-valued functions and their calculus, arc length, speed and curvature, and motion in space. The second half turns to functions of two or more variables, quadric surfaces, limits and continuity, partial derivatives, differentiability and tangent planes, the gradient and directional derivatives, the chain rule, optimization and Lagrange multipliers.

That is the first half of a standard **Calculus 3** course. Multiple integrals and vector calculus come in [Math 32B](/courses/ucla-math-32b).

## Math 32A topic map

| Math 32A topic | Practice | Read |
| --- | --- | --- |
| Vectors in the plane and in space | [Vectors and their length](/learn/calculus-3/vectors-and-space/vectors-and-length) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Dot product | [The dot product](/learn/calculus-3/vectors-and-space/dot-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Cross product | [The cross product](/learn/calculus-3/vectors-and-space/cross-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Planes in three-space | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) | [Lines in space](/learn/calculus-3/vectors-and-space/lines-in-space) |
| Parametric equations | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) | [Parametric arc length](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) |
| Vector-valued functions and motion | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Chain rule](/rules/chain-rule) |
| Functions of several variables and surfaces | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Quadratic function](/functions/quadratic-function) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives guide](/blog/partial-derivatives), [Clairaut's theorem](/theorems/clairauts-theorem) |
| Differentiability and tangent planes | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Gradient and directional derivatives | [The gradient](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Chain rule | [The multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Chain rule guide](/blog/chain-rule) |
| Optimization | [Second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Optimization problems](/blog/optimization-problems) |
| Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers guide](/blog/lagrange-multipliers) |

## Math 32A vs Math 32B

| Course | Units | Requisites | Content |
| --- | --- | --- | --- |
| Math 32A | 4 | 31A | Vectors, curves and differential calculus of several variables |
| Math 32B | 4 | 31B and 32A | Multiple integrals, line and surface integrals, Green, Stokes and Gauss |

Students who take 32A before 31B should plan to finish 31B before enrolling in 32B.

## Example problems at the Math 32A level

**1. A cross product.** For $\mathbf{u} = \langle 1, 2, 0 \rangle$ and $\mathbf{v} = \langle 0, 1, 3 \rangle$:

$$\mathbf{u}\times\mathbf{v} = \langle 2\cdot 3 - 0\cdot 1,\ 0\cdot 0 - 1\cdot 3,\ 1\cdot 1 - 2\cdot 0 \rangle = \langle 6, -3, 1 \rangle$$

This vector is perpendicular to both $\mathbf{u}$ and $\mathbf{v}$.

**2. The multivariable chain rule.** If $z = x^2 y$ with $x = t^2$ and $y = 3t$, then

$$\frac{dz}{dt} = 2xy\cdot 2t + x^2\cdot 3 = 2(t^2)(3t)(2t) + 3t^4 = 15t^4$$

which matches differentiating $z = 3t^5$ directly.

**3. A critical point.** For $f(x, y) = x^2 + y^2 - 4x + 6y$, setting $f_x = 2x - 4 = 0$ and $f_y = 2y + 6 = 0$ gives $(2, -3)$. Since $f_{xx} = f_{yy} = 2$ and $f_{xy} = 0$, the discriminant is $4 > 0$, so $(2, -3)$ is a local minimum with $f = -13$.

## How to do well in UCLA Math 32A

- **Visualize in three dimensions.** Sketch vectors, planes and level curves for every problem. Most errors come from misreading the geometry.
- **Lean on single-variable rules.** Partial derivatives use the [product rule](/rules/product-rule) and [chain rule](/rules/chain-rule) you already know, one variable at a time.
- **Practice optimization end to end** with the [partial derivatives practice](/learn/calculus-3/partial-derivatives/partial-derivatives-practice) set, including the Lagrange multiplier setups that close the course.
- **Review with the [formula sheet](/formulas)**, timed [practice exams](/practice) and the [common calculus mistakes](/blog/common-calculus-mistakes) list.

## Frequently asked questions

### What is UCLA Math 32A?

Math 32A, Calculus of Several Variables, introduces differential calculus of several variables and vector field theory. It covers vectors, curves in space, functions of several variables, partial derivatives, gradients, the chain rule, optimization and Lagrange multipliers.

### Can I take Math 32A before Math 31B?

Yes. The catalog lists Math 31A with a C- or better as the only enforced requisite for 32A. You will need both 31B and 32A before Math 32B.

### Is Math 32A hard?

The derivatives are familiar single-variable skills applied one variable at a time. The new challenge is three-dimensional geometry, so practice visualizing vectors, planes and surfaces.

### What comes after Math 32A?

Math 32B, which covers multiple integrals, line and surface integrals and the theorems of Green, Stokes and Gauss. It requires Math 31B and Math 32A with grades of C- or better.