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

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

How MATH 234 builds on MATH 221 and 222

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 and the MATH 221 guide 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

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

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

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 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 connects all three.
• Learn the vector calculus theorems as one family. Green's theorem, Stokes' theorem and the Divergence Theorem all trade a boundary integral for an interior one.
• Keep formulas handy with the calculus formula sheet, and run timed 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.