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.
UCLA Updated 3 min read
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, vectors and partial derivatives units
On this page
UCLA Math 32A begins multivariable calculus at UCLA. Because its only requisite is Math 31A, it can be taken before 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.
Math 32A topic map
| Math 32A topic | Practice | Read |
|---|---|---|
| Vectors in the plane and in space | Vectors and their length | Dot and cross product |
| Dot product | The dot product | Dot and cross product |
| Cross product | The cross product | Dot and cross product |
| Planes in three-space | Planes in space | Lines in space |
| Parametric equations | Parametric curves | Parametric arc length |
| Vector-valued functions and motion | Curves and motion | Chain rule |
| Functions of several variables and surfaces | Functions of several variables | Quadratic function |
| Partial derivatives | Partial derivatives | Partial derivatives guide, Clairaut's theorem |
| Differentiability and tangent planes | Tangent planes | Linear approximation |
| Gradient and directional derivatives | The gradient | Gradient and directional derivatives |
| Chain rule | The multivariable chain rule | Chain rule guide |
| Optimization | Second-derivative test | Optimization problems |
| Lagrange multipliers | Lagrange multipliers | Lagrange multipliers guide |
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:
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
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 and chain rule you already know, one variable at a time.
- Practice optimization end to end with the partial derivatives practice set, including the Lagrange multiplier setups that close the course.
- Review with the formula sheet, timed practice exams and the common calculus mistakes list.
Practice it now
Vectors and their length
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