# UW-Madison MATH 234 Calculus 3: Topics, Study Guide and Practice

> A study guide for MATH 234 at UW-Madison: parameterized curves, partial derivatives, multiple integrals and vector calculus, mapped to free practice.

Source: https://practicecalculus.com/courses/uw-madison-math-234
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 234
- **Credits:** 4
- **Requisites:** MATH 222
- **Covers:** Calculus on parameterized curves, derivatives of functions of several variables, multiple integrals, vector calculus
- **Previous course:** MATH 222 (Calculus and Analytic Geometry 2)
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 234, Calculus: Functions of Several Variables, is the 4-credit third course in the UW-Madison calculus sequence.
- It covers calculus on parameterized curves, derivatives of multivariable functions, multiple integrals and vector calculus.
- The only listed requisite is MATH 222.
- Vector geometry from MATH 222 is the foundation, so review dot and cross products before the first week.

**MATH 234** is Calculus: Functions of Several Variables at the University of Wisconsin-Madison, the third course in the main calculus sequence after MATH 221 and MATH 222. It is a 4-credit course required by most engineering, physics, math and statistics programs. This guide maps every MATH 234 topic to free practice, so you can build each skill by solving problems with instant feedback.

## What MATH 234 covers

The UW-Madison Guide describes MATH 234 as an introduction to the calculus of several variables. It starts with calculus on parameterized curves, moves to derivatives of multivariable functions, then multiple integrals, and finishes with vector calculus.

That is a standard **Calculus 3** course. Because MATH 222 already covers vector geometry in two and three dimensions, MATH 234 can start with curves and motion instead of spending weeks on vectors. Vector calculus (line integrals, Green's theorem, flux, Stokes and the Divergence Theorem) is included, so the course covers the whole multivariable story.

## MATH 234 topic map

| MATH 234 topic | Practice | Read |
| --- | --- | --- |
| Parameterized curves and motion | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Vectors, dot and cross product](/blog/vectors-dot-and-cross-product) |
| Arc length of curves | [Arc length of parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) |
| 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 and tangent planes | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Gradient and directional derivatives | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Optimization | [Critical points and the second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integrals | [Double integrals over general regions](/learn/calculus-3/multiple-integrals/general-regions) | [Double integrals](/blog/double-integrals) |
| Triple integrals and coordinates | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) |
| Vector fields and line integrals | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) |
| Green's theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem) |
| Stokes and divergence | [Stokes and the Divergence Theorem](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem), [Divergence Theorem](/theorems/divergence-theorem) |

## How MATH 234 builds on MATH 221 and 222

| Course | What you bring forward |
| --- | --- |
| MATH 221 | Derivatives, the chain rule, definite integrals |
| MATH 222 | Integration techniques, parametric thinking, vectors, lines and planes |
| MATH 234 | All of it, now with several variables at once |

Every MATH 234 idea is a single-variable idea extended: partial derivatives are ordinary derivatives with other variables held fixed, and a double integral is two single integrals nested. If a step feels shaky, the [MATH 222 guide](/courses/uw-madison-math-222) and the [MATH 221 guide](/courses/uw-madison-math-221) point to the exact lessons to review.

## Example problems at the MATH 234 level

**1. Velocity on a curve.** For $\mathbf{r}(t) = \langle \cos t, \sin t, t \rangle$, the velocity is $\mathbf{r}'(t) = \langle -\sin t, \cos t, 1 \rangle$ and the speed is

$$|\mathbf{r}'(t)| = \sqrt{\sin^2 t + \cos^2 t + 1} = \sqrt{2}$$

**2. A directional derivative.** For $f(x, y) = x^2 + 3y$ at $(1, 2)$ in the direction of $\mathbf{u} = \langle \frac{3}{5}, \frac{4}{5} \rangle$: $\nabla f = \langle 2x, 3 \rangle = \langle 2, 3 \rangle$, so

$$D_{\mathbf{u}}f = 2 \cdot \frac{3}{5} + 3 \cdot \frac{4}{5} = \frac{18}{5}$$

**3. A double integral over a triangle.** Integrate $f(x, y) = x$ over the region $0 \le y \le x \le 1$:

$$\int_0^1 \int_0^x x\,dy\,dx = \int_0^1 x^2\,dx = \frac{1}{3}$$

## How to do well in MATH 234

- **Draw before you integrate.** Most setup errors vanish once the region is on paper. Practice with [switching the order](/learn/calculus-3/multiple-integrals/switching-the-order) to see the same region two ways.
- **Use the gradient as your main tool.** It gives directional derivatives, normal vectors and tangent planes. Our [gradient guide](/blog/gradient-and-directional-derivatives) connects all three.
- **Learn the vector calculus theorems as one family.** [Green's theorem](/theorems/greens-theorem), [Stokes' theorem](/theorems/stokes-theorem) and the [Divergence Theorem](/theorems/divergence-theorem) all trade a boundary integral for an interior one.
- **Keep formulas handy** with the [calculus formula sheet](/formulas), and run timed [practice](/practice) sets before each exam.

> **Tip:** remember the extra factor in non-rectangular coordinates: $r$ in polar and cylindrical, $\rho^2\sin\phi$ in spherical. Forgetting it is one of the most common MATH 234 errors.

## Frequently asked questions

### What is MATH 234 at UW-Madison?

MATH 234 is Calculus: Functions of Several Variables, the multivariable calculus course at the University of Wisconsin-Madison. It introduces calculus on parameterized curves, derivatives of functions of several variables, multiple integrals and vector calculus.

### What are the requisites for MATH 234?

The UW-Madison Guide lists MATH 222 as the requisite.

### Is MATH 234 the same as Calculus 3?

Yes. MATH 234 is the course that plays the role of Calculus 3 at UW-Madison, covering multivariable differentiation, multiple integrals and vector calculus.

### Is MATH 234 hard?

The calculations are often simpler than Calculus 2 integrals, but you have to think in three dimensions and keep track of more variables. Sketching curves, surfaces and regions makes the course much easier.

### How should I prepare for MATH 234?

Review vectors, lines and planes from MATH 222 and polar coordinates, then start partial derivatives. The Calculus 3 path on PracticeCalculus begins with exactly those vector skills.