# UAlberta MATH 214 Calculus III: Syllabus, Topics and Practice

> A study guide for University of Alberta MATH 214 (Calculus III): series, Taylor series, curves, polar and spherical coordinates and multiple integrals.

Source: https://practicecalculus.com/courses/university-of-alberta-math-214
Updated: 2026-11-08

## At a glance

- **Course code:** MATH 214
- **Units:** 3 (fi 6), 3 lecture hours per week, offered in either term
- **Prerequisites:** One of MATH 101, 115, 118, 136, 146 or 156
- **Recommended:** One of MATH 102, 125 or 127 (linear algebra)
- **Credit restriction:** Not for credit if you already have MATH 209 or 217
- **Covers:** Sequences and series, convergence tests, Taylor series, curves, tangent vectors and arc length, applications of partial differentiation, polar, cylindrical and spherical coordinates, multiple integration
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 214 is the 3-unit Calculus III course at the University of Alberta.
- It starts with sequences, series, convergence tests and Taylor series, which many schools teach in Calculus 2.
- It then moves into curves, arc length, partial derivative applications and polar, cylindrical and spherical coordinates.
- It ends with multiple integration, and linear algebra (MATH 102, 125 or 127) is recommended.

**MATH 214** is Calculus III at the University of Alberta. It is an unusual mix: it opens with sequences and series, which many universities teach in Calculus 2, and then moves into curves, coordinate systems, partial derivative applications and multiple integrals. This guide maps every MATH 214 topic to free practice so you can work through both halves with instant feedback.

## What UAlberta MATH 214 covers

The UAlberta catalogue lists the topics as: sequences and series, convergence tests, and Taylor series; curves, tangent vectors, and arc length; applications of partial differentiation; polar, cylindrical, and spherical coordinates; and multiple integration. The course is worth 3 units with three lecture hours a week. You need one of MATH 101, 115, 118, 136, 146 or 156, and one of MATH 102, 125 or 127 is recommended.

In terms of the usual sequence, MATH 214 combines the series part of **Calculus 2** with the core of **Calculus 3**. Vector fields and line integrals are not in the catalogue description.

## MATH 214 topic map

| MATH 214 topic | Practice | Read |
| --- | --- | --- |
| Sequences | [Sequences](/learn/calculus-2/sequences-and-series/sequences) | [Squeeze theorem](/theorems/squeeze-theorem) |
| Series and partial sums | [Series and partial sums](/learn/calculus-2/sequences-and-series/series-and-partial-sums) | [Geometric series](/blog/geometric-series) |
| Convergence tests | [Choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) | [Convergence tests](/blog/convergence-tests), [ratio test](/rules/ratio-test) |
| Alternating series | [Alternating series](/learn/calculus-2/sequences-and-series/alternating-series) | [Alternating series test](/rules/alternating-series-test) |
| Taylor series | [Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Taylor series guide](/blog/taylor-series), [Taylor's theorem](/theorems/taylors-theorem) |
| Curves and tangent vectors | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) |
| Arc length | [Parametric arc length](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) | |
| Applications of partial differentiation | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes), [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers guide](/blog/lagrange-multipliers) |
| Extrema of several variables | [Second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Polar coordinates | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates) | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) |
| Cylindrical and spherical coordinates | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | |
| Multiple integration | [Iterated integrals](/learn/calculus-3/multiple-integrals/iterated-integrals), [triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Double integrals guide](/blog/double-integrals) |

## MATH 214 and the engineering sequence

| Course | Title | Main focus |
| --- | --- | --- |
| [MATH 146](/courses/university-of-alberta-math-146) | Calculus for the Mathematical and Physical Sciences II | Integration, ODEs, partial derivatives |
| MATH 214 | Calculus III | Series, curves, coordinates, multiple integrals |
| MATH 209 | Calculus for Engineering III | Partial derivatives, multiple integrals, vector field theory (engineering) |

MATH 214 cannot be taken for credit if you already have MATH 209 or MATH 217.

## Example problems at the MATH 214 level

**1. The ratio test.** For $\sum \frac{n}{2^n}$, $\frac{a_{n+1}}{a_n} = \frac{n+1}{2n} \to \frac{1}{2} < 1$, so the series converges.

**2. A Taylor series.** From $e^x = \sum \frac{x^n}{n!}$, substituting $-x^2$ gives $e^{-x^2} = 1 - x^2 + \frac{x^4}{2} - \cdots$.

**3. Arc length.** For $\mathbf{r}(t) = \langle 3\cos t, 3\sin t, 4t \rangle$ on $0 \le t \le 2\pi$, $|\mathbf{r}'(t)| = 5$, so the length is $10\pi$.

**4. A spherical integral.** The volume of a ball of radius $a$ is

$$\int_0^{2\pi}\int_0^{\pi}\int_0^{a} \rho^2\sin\phi\,d\rho\,d\phi\,d\theta = \tfrac{4}{3}\pi a^3$$

## How to do well in MATH 214

- **Treat series as reasoning, not computing.** Name your test, check its conditions and state the conclusion clearly.
- **Know the core Maclaurin series by heart.** $e^x$, $\sin x$, $\cos x$ and $\frac{1}{1-x}$ generate most of the others.
- **Draw regions before integrating.** Wrong bounds are the most common error in multiple integrals.
- **Do the linear algebra.** The catalogue recommends MATH 102, 125 or 127, and vectors and determinants make the second half much easier.

> **Tip:** in spherical coordinates, $\phi$ is measured down from the positive $z$-axis and runs from $0$ to $\pi$, not $0$ to $2\pi$.

## Frequently asked questions

### What is MATH 214 at the University of Alberta?

MATH 214 is Calculus III. The catalogue lists sequences and series, convergence tests and Taylor series; curves, tangent vectors and arc length; applications of partial differentiation; polar, cylindrical and spherical coordinates; and multiple integration.

### What are the prerequisites for MATH 214?

One of MATH 101, 115, 118, 136, 146 or 156. The catalogue also recommends one of MATH 102, 125 or 127.

### Does MATH 214 cover vector calculus?

The catalogue description does not list line integrals, Green's theorem or Stokes' theorem. Vector field theory is listed in MATH 209 (Calculus for Engineering III), and MATH 214 cannot be taken for credit with MATH 209 or 217.

### Why are series in Calculus III at UAlberta?

The UAlberta science sequence puts integration, differential equations and partial derivatives in MATH 146 or 136, so sequences and series move to MATH 214. Expect to spend the first part of the term on convergence tests.