# UNC MATH 233 Calculus III: Syllabus, Topics and Practice

> A study guide for UNC Chapel Hill MATH 233, Calculus of Functions of Several Variables: vectors, partial derivatives and multiple integrals.

Source: https://practicecalculus.com/courses/unc-chapel-hill-math-233
Updated: 2026-11-08

## At a glance

- **Course code:** MATH 233
- **Credits:** 4
- **Prerequisites:** MATH 232
- **Covers:** Vector algebra, solid analytic geometry, partial derivatives, multiple integrals
- **Gen Ed:** IDEAs in Action FC-QUANT
- **Honors version:** Available
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 233 is the 4-credit multivariable calculus course at UNC Chapel Hill, and the third course in the calculus sequence.
- The official description lists vector algebra, solid analytic geometry, partial derivatives and multiple integrals.
- The prerequisite is MATH 232.
- MATH 233 is a prerequisite for many upper-level UNC math courses, including MATH 521 Advanced Calculus I.

**UNC MATH 233** (Calculus of Functions of Several Variables) is the third course in the UNC Chapel Hill calculus sequence and the one that takes calculus into two and three dimensions. It carries 4 credits, meets the FC-QUANT requirement, and lists MATH 232 as its prerequisite. This guide maps each topic to lessons you can practice with instant feedback.

## What UNC MATH 233 covers

The UNC mathematics department describes MATH 233 as: vector algebra, solid analytic geometry, partial derivatives, multiple integrals. An honors version is available.

In plain terms, that means working with vectors, dot and cross products, and equations of lines, planes and surfaces in space (solid analytic geometry), then extending derivatives to functions of several variables and integrals to regions in the plane and in space. The official description does not name line integrals, Green's theorem or Stokes' theorem, so treat those as optional unless your syllabus lists them. A schedule and textbook are not part of the listing.

## MATH 233 topic map

| MATH 233 topic | Practice | Read |
| --- | --- | --- |
| Points and distance in space | [Points and distance](/learn/calculus-3/vectors-and-space/points-and-distance) | [Vectors and length](/learn/calculus-3/vectors-and-space/vectors-and-length) |
| Vector algebra | [Dot product](/learn/calculus-3/vectors-and-space/dot-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Cross product and projections | [Cross product](/learn/calculus-3/vectors-and-space/cross-product) | [Projections and work](/learn/calculus-3/vectors-and-space/projections-and-work) |
| Lines and planes | [Lines in space](/learn/calculus-3/vectors-and-space/lines-in-space) | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Chain rule | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Chain rule](/rules/chain-rule) |
| Tangent planes and gradients | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Extrema | [Second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integrals | [Iterated integrals](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |
| Polar double integrals | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates) |
| Triple integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) |

## MATH 233 in the UNC sequence

| Course | Credits | Prerequisite |
| --- | --- | --- |
| [MATH 231](/courses/unc-chapel-hill-math-231) Calculus of Functions of One Variable I | 4 | MATH 110 and 130 (C- in 130) or placement |
| [MATH 232](/courses/unc-chapel-hill-math-232) Calculus of Functions of One Variable II | 4 | C- in MATH 231 or placement |
| MATH 233 Calculus of Functions of Several Variables | 4 | MATH 232 |

## Example problems at the MATH 233 level

**1. The equation of a plane.** The plane through $(1, 0, 2)$ with normal vector $\langle 2, -1, 3 \rangle$ is

$$2(x - 1) - (y - 0) + 3(z - 2) = 0 \quad\Longrightarrow\quad 2x - y + 3z = 8$$

**2. Partial derivatives.** For $f(x, y) = x^3 y^2$:

$$f_x = 3x^2 y^2, \qquad f_y = 2x^3 y$$

**3. A double integral in polar coordinates.** The area of the unit disk is

$$\int_0^{2\pi} \int_0^1 r\,dr\,d\theta = 2\pi \cdot \frac{1}{2} = \pi$$

## How to do well in MATH 233

- **Sketch in 3D often.** Planes, cylinders, paraboloids and spheres come up constantly, and a quick sketch catches sign errors.
- **Keep the geometry of vectors in mind:** the dot product measures angle, the cross product gives a perpendicular vector and an area.
- **Set up bounds carefully** before integrating. Draw the region and decide the order of integration first, as in [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order).
- **Do the [partial derivatives practice](/learn/calculus-3/partial-derivatives/partial-derivatives-practice)** until the rules feel the same as in one variable.
- **Try [double integral problems](/practice-problems/double-integrals)** and timed [practice exams](/practice) before tests.

> **Tip:** when you take a partial derivative with respect to $x$, treat every other variable as a constant. Most errors come from forgetting that.

## Frequently asked questions

### What is MATH 233 at UNC?

MATH 233 Calculus of Functions of Several Variables is UNC Chapel Hill's Calculus 3 course. The department describes it as vector algebra, solid analytic geometry, partial derivatives and multiple integrals. An honors version is available.

### What is the prerequisite for MATH 233?

MATH 232 Calculus of Functions of One Variable II.

### Does MATH 233 cover Green's theorem and Stokes' theorem?

The official course description does not mention line integrals or the big theorems of vector calculus. It lists vector algebra, solid analytic geometry, partial derivatives and multiple integrals. Your instructor's syllabus will say whether any vector calculus is included.

### Can high school students take MATH 233?

The UNC math department runs a Math 233 for High School Students program. Check the department website for current eligibility and how to apply.

### Which courses need MATH 233?

Several upper-level courses list MATH 233 (or MATH 235) as a prerequisite, including MATH 521 Advanced Calculus I.