# Queen's MATH 221 Vector Calculus: Syllabus, Topics and Practice

> A study guide for Queen's University MATH 221 (Vector Calculus): double and triple integrals, line and surface integrals, Green, Stokes and Gauss.

Source: https://practicecalculus.com/courses/queens-university-math-221
Updated: 2026-11-08

## At a glance

- **Course code:** MATH 221
- **Units:** 3.00
- **Learning hours:** 120 (36 lecture, 84 private study)
- **Prerequisites:** MATH 120, 121, 124, 126 or 128. Recommended: some linear algebra
- **Exclusions:** Maximum of one course from MATH 221 and MATH 280
- **Covers:** Double and triple integrals including polar and spherical coordinates, parameterized curves and line integrals, gradient, divergence and curl, Green's theorem, parameterized surfaces and surface integrals, Stokes' and Gauss' theorems
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 221 is a 3.00-unit Vector Calculus course at Queen's, usually taken after MATH 120 or 121.
- It covers double and triple integrals, including polar and spherical coordinates.
- It studies line integrals, gradient, divergence and curl, and surface integrals.
- It ends with Green's, Stokes' and Gauss' theorems, and is an exclusion with MATH 280.

**MATH 221** is Vector Calculus at Queen's University, a 3.00-unit course that takes the integral side of calculus into two and three dimensions. It is the natural next step after MATH 120 or MATH 121 for students in physics, engineering-related programs, life sciences and economics who need multiple integrals and the theorems of vector calculus. This guide maps every MATH 221 topic to free practice.

## What Queen's MATH 221 covers

The Queen's Arts and Science calendar lists double and triple integrals, including polar and spherical coordinates; parameterized curves and line integrals; gradient, divergence and curl; Green's theorem; parameterized surfaces and surface integrals; and Stokes' and Gauss' theorems. Learning hours are 120 (36 lecture and 84 private study). The prerequisite is one of MATH 120, 121, 124, 126 or 128, and some linear algebra is recommended.

The calendar's learning outcomes are to work with gradients, divergence and curl and with spherical and cylindrical coordinates; to integrate vector functions and evaluate path, surface and volume integrals; to calculate with Green's and Stokes' theorems and the Divergence Theorem; and to work with multivariable functions and vectors in two and three dimensions. That is the integral half of a standard **Calculus 3** course.

## MATH 221 topic map

| MATH 221 topic | Practice | Read |
| --- | --- | --- |
| Double integrals | [Iterated integrals](/learn/calculus-3/multiple-integrals/iterated-integrals), [general regions](/learn/calculus-3/multiple-integrals/general-regions) | [Double integrals guide](/blog/double-integrals) |
| Polar coordinates | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) | |
| Triple integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | |
| Cylindrical and spherical coordinates | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Change of variables](/learn/calculus-3/multiple-integrals/change-of-variables) |
| Parameterized curves | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) |
| Line integrals | [Line integrals of functions](/learn/calculus-3/vector-calculus/line-integrals-of-functions), [of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) |
| Gradient | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Divergence and curl | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) |
| Green's theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem statement](/theorems/greens-theorem) |
| Surface integrals | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | |
| Stokes' and Gauss' theorems | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Stokes' theorem](/theorems/stokes-theorem), [divergence theorem](/theorems/divergence-theorem) |

## MATH 221 vs MATH 280

| Course | Title | Prerequisites | Style |
| --- | --- | --- | --- |
| MATH 221 | Vector Calculus | MATH 120, 121, 124, 126 or 128 | Computational, applications |
| MATH 280 | Advanced Calculus | Linear algebra and MATH 120, 121 or 124 | Adds limits, continuity, linear approximation, Jacobians |

You can count only one of them. Before MATH 221, most students take [MATH 121](/courses/queens-university-math-121) or [MATH 120](/courses/queens-university-math-120).

## Example problems at the MATH 221 level

**1. A spherical integral.** The mass of the unit ball with density $\rho$ (distance from the origin) is

$$\int_0^{2\pi}\int_0^{\pi}\int_0^{1}\rho\cdot\rho^2\sin\phi\,d\rho\,d\phi\,d\theta = \pi$$

**2. A line integral.** For $\mathbf{F} = \langle -y, x \rangle$ around the unit circle counterclockwise, $\oint \mathbf{F}\cdot d\mathbf{r} = \int_0^{2\pi} (\sin^2 t + \cos^2 t)\,dt = 2\pi$.

**3. Green's theorem check.** For the same field, $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 2$, and $2 \times \text{area} = 2\pi$, matching example 2.

**4. Curl of a gradient.** For any smooth $f$, $\nabla\times(\nabla f) = \mathbf{0}$, which is why gradient fields are conservative.

## How to do well in MATH 221

- **Draw the region and the boundary.** Bounds for multiple integrals and orientations for line integrals both come from the picture.
- **Remember the Jacobian factors.** $r$ for polar and cylindrical, $\rho^2\sin\phi$ for spherical.
- **Choose the easiest side of a theorem.** Green's, Stokes' and Gauss' theorems let you swap a hard integral for an easier one.
- **Do some linear algebra.** The calendar recommends it, and vectors and determinants appear throughout.

> **Tip:** before computing a line integral, check whether the field is conservative. If it is, you only need a potential function and the endpoints.

## Frequently asked questions

### What is MATH 221 at Queen's?

MATH 221 is Vector Calculus. The Queen's calendar lists double and triple integrals, including polar and spherical coordinates; parameterized curves and line integrals; gradient, divergence and curl; Green's theorem; parameterized surfaces and surface integrals; and Stokes' and Gauss' theorems.

### What do I need for MATH 221?

One of MATH 120, 121, 124, 126 or 128. Some linear algebra is recommended.

### MATH 221 vs MATH 280?

You can take only one of them. MATH 280, Advanced Calculus, also needs linear algebra and adds limits, continuity and linear approximation of functions of several variables and Jacobians, alongside the theorems of Green, Stokes and Gauss. MATH 221 is the more computational choice.

### How should I prepare for MATH 221?

Review integration techniques and partial derivatives from first-year calculus, and get comfortable with vectors, dot products and cross products.