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.
Queen's University Updated 3 min read
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
Try it yourself
On this page
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, general regions | Double integrals guide |
| Polar coordinates | Polar double integrals | |
| Triple integrals | Triple integrals | |
| Cylindrical and spherical coordinates | Cylindrical and spherical | Change of variables |
| Parameterized curves | Curves and motion | Parametric curves |
| Line integrals | Line integrals of functions, of vector fields | Conservative fields |
| Gradient | Gradient and directional derivatives | Gradient guide |
| Divergence and curl | Curl and divergence | Vector fields |
| Green's theorem | Green's theorem | Green's theorem statement |
| Surface integrals | Surface integrals and flux | |
| Stokes' and Gauss' theorems | Stokes and divergence theorems | Stokes' theorem, 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 or 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
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.
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Double integrals over rectangles
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