# McGill MATH 222 Calculus 3: Topics, Study Guide and Practice

> A study guide for McGill MATH 222 (Calculus 3): Taylor series, partial derivatives, directional derivatives, extrema and multiple integrals, with free practice.

Source: https://practicecalculus.com/courses/mcgill-math-222
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 222
- **Credits:** 3
- **Offered:** Summer 2026, Fall 2026 and Winter 2027
- **Prerequisites:** MATH 141, plus familiarity with vector geometry or MATH 133 as a corequisite
- **Restrictions:** Not open to students who have taken or are taking MATH 150, MATH 151, MATH 206 or MATH 262
- **Covers:** Taylor series and Taylor's theorem in one and several variables, vector geometry, partial and directional derivatives, extrema, parametric curves and arc length, polar and spherical coordinates, multiple integrals
- **Next course:** MATH 248 (Honours Vector Calculus), with MATH 133
- **Practice here:** [Calculus 3](/learn/calculus-3) and the series units of [Calculus 2](/learn/calculus-2)

## Key takeaways

- McGill MATH 222 is a 3-credit Calculus 3 course that follows MATH 141.
- It starts with Taylor series and Taylor's theorem in one and several variables, then moves into multivariable calculus.
- It covers partial and directional derivatives, extrema, parametric curves, polar and spherical coordinates and multiple integrals.
- Line and surface integrals and the theorems of Green, Gauss and Stokes come later, in MATH 248.

**MATH 222** is Calculus 3 at McGill University, the third course in the main calculus sequence after [MATH 141](/courses/mcgill-math-141). It is a bridge course: it finishes the series story from Calculus 2 with Taylor's theorem, then extends derivatives and integrals to functions of several variables. This guide maps every MATH 222 topic to practice so you can learn each idea by working problems.

## What McGill MATH 222 covers

McGill's Department of Mathematics and Statistics lists the topics as Taylor series and Taylor's theorem in one and several variables, a review of vector geometry, partial differentiation, directional derivatives, extrema of functions of two or three variables, parametric curves and arc length, polar and spherical coordinates, and multiple integrals. The course is worth 3 credits and runs in summer, fall and winter.

Compared with a standard sequence, MATH 222 contains the Taylor series and parametric and polar topics that many universities put in **Calculus 2**, plus the differential and integral core of **Calculus 3**. Vector calculus is left for MATH 248.

## MATH 222 topic map

| MATH 222 topic | Practice | Read |
| --- | --- | --- |
| Taylor polynomials and series | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor series](/blog/taylor-series), [power series](/blog/power-series) |
| Taylor's theorem | [Estimating with Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/estimating-with-taylor) | [Taylor's theorem](/theorems/taylors-theorem) |
| Vector geometry | [The dot product](/learn/calculus-3/vectors-and-space/dot-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Parametric curves and arc length | [Arc length of parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) |
| Partial differentiation | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives](/blog/partial-derivatives), [Clairaut's theorem](/theorems/clairauts-theorem) |
| Directional derivatives | [The gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Extrema of functions of two or three variables | [Critical points and the second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Polar coordinates | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates) | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) |
| Multiple integrals | [Double integrals over general regions](/learn/calculus-3/multiple-integrals/general-regions) | [Double integrals](/blog/double-integrals) |
| Spherical coordinates | [Cylindrical and spherical coordinates](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Double integrals](/blog/double-integrals) |

## MATH 222 in McGill's calculus sequence

| Course | Title | Credits | Prerequisite |
| --- | --- | --- | --- |
| [MATH 140](/courses/mcgill-math-140) | Calculus 1 | 3 | High school calculus |
| [MATH 141](/courses/mcgill-math-141) | Calculus 2 | 4 | MATH 139, 140 or 150 |
| MATH 222 | Calculus 3 | 3 | MATH 141 and vector geometry (or MATH 133 as corequisite) |
| MATH 248 | Honours Vector Calculus | 3 | MATH 133 and MATH 222 |

Students on the MATH 150 and 151 track skip MATH 222, since that pair covers its material.

## Example problems at the MATH 222 level

**1. A Taylor series.** Starting from $e^t = \sum_{n=0}^{\infty} \frac{t^n}{n!}$ and substituting $t = -x^2$:

$$e^{-x^2} = 1 - x^2 + \frac{x^4}{2} - \frac{x^6}{6} + \cdots$$

**2. Extrema of a function of two variables.** For $f(x, y) = x^2 - y^2$, the only critical point is $(0, 0)$. There $D = f_{xx}f_{yy} - f_{xy}^2 = (2)(-2) - 0 = -4 < 0$, so the origin is a saddle point.

**3. A polar double integral.** Over the disk $x^2 + y^2 \le 4$:

$$\iint_D e^{-(x^2 + y^2)}\,dA = \int_0^{2\pi}\!\int_0^2 e^{-r^2} r\,dr\,d\theta = 2\pi \cdot \frac{1 - e^{-4}}{2} = \pi(1 - e^{-4})$$

## How to do well in McGill MATH 222

- **Refresh series first.** The course opens with Taylor series, so review the common Maclaurin series and the [ratio test](/rules/ratio-test) before the term starts.
- **Get comfortable with vectors.** If you have not taken MATH 133, work through the [vectors and space](/learn/calculus-3/vectors-and-space/vectors-and-length) lessons early.
- **Remember the Jacobian factor.** Changing to polar or spherical coordinates adds $r$ or $\rho^2\sin\phi$; leaving it out is the most common lost mark.
- **Practice and review.** Short daily [practice](/practice) sets and the [calculus formula sheet](/formulas) keep both the series and multivariable halves fresh.

> **Tip:** Taylor's theorem in several variables is the same idea as in one variable: match the value, the first derivatives and the second derivatives at a point. Seeing it that way makes the second derivative test easy to remember.

## Frequently asked questions

### What is McGill MATH 222?

MATH 222 is Calculus 3 at McGill. It covers Taylor series and Taylor's theorem in one and several variables, a review of vector geometry, partial differentiation, directional derivatives, extrema, parametric curves and arc length, polar and spherical coordinates, and multiple integrals.

### What do I need before MATH 222?

MATH 141 and familiarity with vector geometry. If you have not studied vector geometry, you can take MATH 133 (linear algebra and geometry) at the same time.

### Is McGill MATH 222 hard?

It combines series from Calculus 2 with new multivariable ideas, so there is a lot of variety. Students who are comfortable with Taylor series and vectors before the term starts usually find the multivariable material easier.

### What comes after MATH 222?

MATH 248, Honours Vector Calculus, requires MATH 133 and MATH 222. It covers Jacobians, implicit functions, line and surface integrals, and the theorems of Green, Gauss and Stokes.

### Can I take MATH 222 after MATH 150 and 151?

No. MATH 222 is not open to students who have taken or are taking MATH 150 or 151, because that pair already covers the material of MATH 140, 141 and 222.