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

> A study guide for Waterloo MATH 237: limits, differentiability, gradients, Taylor's formula, Jacobians and multiple integrals, with practice.

Source: https://practicecalculus.com/courses/university-of-waterloo-math-237
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 237
- **Units:** 0.50
- **Prerequisites:** A linear algebra course (MATH 106, 114, 115, 136 or 146), plus MATH 148, or 60% in MATH 138, or 70% in MATH 128; enrolment in Honours Mathematics or Mathematical Physics
- **Antirequisites:** MATH 207, 247, 227, 217, 212 and ECE 206
- **Covers:** Functions of several variables, limits, continuity, differentiability, chain rule, gradient and directional derivative, Taylor's formula, optimization, mappings and the Jacobian, multiple integrals
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 237 is Calculus 3 for Honours Mathematics at Waterloo, worth 0.50 units.
- It covers differential calculus of several variables: limits, differentiability, the chain rule, gradients and Taylor's formula.
- It also covers optimization, mappings and the Jacobian, and multiple integrals in several coordinate systems.
- It needs a linear algebra course as well as MATH 138, MATH 148 or MATH 128.

**MATH 237** is Calculus 3 for Honours Mathematics at the University of Waterloo, the multivariable calculus course that follows [MATH 138](/courses/university-of-waterloo-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

| MATH 237 topic | Practice | Read |
| --- | --- | --- |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Limits, continuity and partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives](/blog/partial-derivatives) |
| Differentiability and linear approximation | [Tangent planes and linear approximation](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| The chain rule | [The multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Chain rule](/rules/chain-rule) |
| Gradient and directional derivative | [The gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Taylor's formula and second derivatives | [Second partial derivatives](/learn/calculus-3/partial-derivatives/second-partial-derivatives) | [Taylor's theorem](/theorems/taylors-theorem), [Clairaut's theorem](/theorems/clairauts-theorem) |
| Optimization | [Critical points and the second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Mappings and the Jacobian | [Change of variables and the Jacobian](/learn/calculus-3/multiple-integrals/change-of-variables) | [Double integrals](/blog/double-integrals) |
| Double integrals | [Double integrals over general regions](/learn/calculus-3/multiple-integrals/general-regions) | [Double integrals](/blog/double-integrals) |
| Polar, cylindrical and spherical coordinates | [Cylindrical and spherical coordinates](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Double integrals](/blog/double-integrals) |

## Waterloo's Calculus 3 options

| Course | Title | Notes |
| --- | --- | --- |
| MATH 237 | Calculus 3 for Honours Mathematics | Standard course for Honours Mathematics |
| MATH 247 | Calculus 3 (Advanced Level) | Advanced version, antirequisite of MATH 237 |
| MATH 207 | Calculus 3 (Non-Specialist Level) | Non-specialist version, antirequisite of MATH 237 |

All three are listed together as Calculus 3 options, so students take one. The usual route is [MATH 137](/courses/university-of-waterloo-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 $y = mx$:

$$f(x, mx) = \frac{mx^2}{x^2 + m^2x^2} = \frac{m}{1 + m^2}$$

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

$$D_{\mathbf{u}} f(1, 2) = 4 \cdot \frac{3}{5} + 1 \cdot \frac{4}{5} = \frac{16}{5}$$

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

$$\int_0^{2\pi}\!\int_0^1 r^2 \cdot r\,dr\,d\theta = 2\pi \cdot \frac{1}{4} = \frac{\pi}{2}$$

## 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](/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](/formulas) and the [dot and cross product guide](/blog/vectors-dot-and-cross-product) 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.

## Frequently asked questions

### What is MATH 237 at Waterloo?

MATH 237 is Calculus 3 for Honours Mathematics, the multivariable calculus course for Honours Mathematics students. It covers limits, continuity and differentiability in several variables, gradients, Taylor's formula, optimization, Jacobians and multiple integrals.

### What do I need before MATH 237?

A first linear algebra course (such as MATH 136) and a second calculus course: MATH 148, at least 60% in MATH 138, or at least 70% in MATH 128. You must also be enrolled in an Honours Mathematics program or Mathematical Physics.

### Is MATH 237 hard?

It asks you to think carefully about limits and differentiability in several variables, which is less intuitive than in one variable. Students who are comfortable with matrices and with partial derivatives find the course far more manageable.

### MATH 237 vs MATH 247 vs MATH 207?

All three are Calculus 3 courses and are antirequisites of each other. MATH 247 is the advanced-level version and MATH 207 is the non-specialist version; MATH 237 is the standard course for Honours Mathematics.

### Does MATH 237 cover vector calculus?

The calendar description ends with multiple integrals in various coordinate systems. It does not list line integrals or Green's and Stokes' theorems.