Waterloo MATH 237 Calculus 3: Topics, Study Guide and Practice

MATH 237 is Calculus 3 for Honours Mathematics at the University of Waterloo, the multivariable calculus course that follows MATH 138 and a first linear algebra course. It takes the ideas of single-variable calculus, limits, derivatives, Taylor polynomials, integrals, and rebuilds them for functions of several variables. This guide maps each MATH 237 topic to practice so you can learn by doing.

What Waterloo MATH 237 covers

The Waterloo academic calendar describes MATH 237 as calculus of functions of several variables. The listed topics are limits, continuity, differentiability and the chain rule; the gradient vector and the directional derivative; Taylor's formula; optimization problems; mappings and the Jacobian; and multiple integrals in various coordinate systems.

Compared with a standard Calculus 3, MATH 237 covers the differential and integral parts in depth, with extra attention to what differentiability really means in several variables. The calendar description does not include vector calculus topics such as line integrals or Stokes' theorem.

MATH 237 topic map

Waterloo's Calculus 3 options

All three are listed together as Calculus 3 options, so students take one. The usual route is MATH 137, then MATH 138, then MATH 237.

Example problems at the MATH 237 level

1. A limit that does not exist. For f(x, y) = \dfrac{xy}{x^2 + y^2}, approach (0, 0) along y = mx:

The value depends on m, so the limit at the origin does not exist.

2. A directional derivative. For f(x, y) = x^2 y at (1, 2) in the direction \mathbf{u} = \left(\frac{3}{5}, \frac{4}{5}\right): \nabla f = (2xy, x^2) = (4, 1), so

3. Polar coordinates. Integrate x^2 + y^2 over the unit disk:

How to do well in MATH 237

• Review linear algebra first. The derivative of a mapping is a matrix, and the Jacobian is a determinant. Comfort with MATH 136 material pays off immediately.
• Practice the computations daily. Partial derivatives, gradients and iterated integrals should be routine; short practice sets get you there.
• Test limits along several paths. Paths can prove a limit does not exist, but never that it does. Know the difference.
• Keep a reference. The calculus formula sheet and the dot and cross product guide cover the tools you use every week.

Tip: always include the Jacobian factor when you change coordinates. The extra r in polar coordinates is the most common lost mark on multiple-integral questions.