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.
UNC Chapel Hill Updated 3 min read
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
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On this page
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 | Vectors and length |
| Vector algebra | Dot product | Dot and cross product |
| Cross product and projections | Cross product | Projections and work |
| Lines and planes | Lines in space | Planes in space |
| Functions of several variables | Functions of several variables | Partial derivatives |
| Partial derivatives | Partial derivatives | Clairaut's theorem |
| Chain rule | Multivariable chain rule | Chain rule |
| Tangent planes and gradients | Tangent planes | Gradient guide |
| Extrema | Second derivative test | Lagrange multipliers |
| Double integrals | Iterated integrals | Double integrals |
| Polar double integrals | Polar double integrals | Polar coordinates |
| Triple integrals | Triple integrals | Cylindrical and spherical |
MATH 233 in the UNC sequence
| Course | Credits | Prerequisite |
|---|---|---|
| MATH 231 Calculus of Functions of One Variable I | 4 | MATH 110 and 130 (C- in 130) or placement |
| 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. Partial derivatives. For f(x, y) = x^3 y^2:
3. A double integral in polar coordinates. The area of the unit disk is
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.
- Do the partial derivatives practice until the rules feel the same as in one variable.
- Try double integral problems and timed practice exams 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.
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Vectors and their length
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