# SFU MATH 251 Calculus III: Syllabus, Topics and Practice

> A study guide for SFU MATH 251 (Calculus III): vectors, lines and planes, curves in space, partial derivatives, multiple integrals and Green's theorem.

Source: https://practicecalculus.com/courses/simon-fraser-university-math-251
Updated: 2026-11-08

## At a glance

- **Course code:** MATH 251
- **Units:** 3
- **Prerequisites:** MATH 152 with at least C-, or MATH 158 with at least B (life sciences: MATH 154 with C- and MATH 155 with A-)
- **Recommended:** MATH 240 or 232 (linear algebra) before or at the same time
- **Covers:** Rectangular, cylindrical and spherical coordinates, vectors, lines, planes, quadric surfaces, vector functions and motion, differential and integral calculus of several variables, vector fields, line integrals, Green's theorem
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 251 is the 3-unit Calculus III course at SFU, taken after MATH 152.
- It covers vectors and 3D geometry, curves and motion in space, and both differential and integral calculus of several variables.
- It ends with vector fields, line integrals, the fundamental theorem for line integrals and Green's theorem.
- SFU recommends taking linear algebra (MATH 240 or 232) before or alongside it.

**MATH 251** is Calculus III at Simon Fraser University. It takes the single-variable tools from MATH 151 and MATH 152 into two and three dimensions: vectors and surfaces, curves and motion in space, partial derivatives, multiple integrals and finally line integrals. This guide maps every MATH 251 topic to free practice so you can work through each idea with instant feedback.

## What SFU MATH 251 covers

The SFU calendar lists the topics as rectangular, cylindrical and spherical coordinates; vectors, lines, planes, cylinders and quadric surfaces; vector functions, curves and motion in space; differential and integral calculus of several variables; and vector fields, line integrals, the fundamental theorem for line integrals and Green's theorem.

This is a standard **Calculus 3** course. The listed description stops at Green's theorem, so surface integrals, Stokes' theorem and the Divergence Theorem are not part of the official topic list. Your instructor's outline is the place to check the exact coverage for your term.

## MATH 251 topic map

| MATH 251 topic | Practice | Read |
| --- | --- | --- |
| Coordinates and distance in space | [Points and distance](/learn/calculus-3/vectors-and-space/points-and-distance) | [Vectors, dot and cross product](/blog/vectors-dot-and-cross-product) |
| Vectors, dot and cross products | [Dot product](/learn/calculus-3/vectors-and-space/dot-product), [cross product](/learn/calculus-3/vectors-and-space/cross-product) | [Vectors guide](/blog/vectors-dot-and-cross-product) |
| Lines and planes | [Lines in space](/learn/calculus-3/vectors-and-space/lines-in-space), [planes](/learn/calculus-3/vectors-and-space/planes-in-space) | |
| Vector functions and motion | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | |
| Partial derivatives and the chain rule | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives), [chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Partial derivatives guide](/blog/partial-derivatives), [Clairaut's theorem](/theorems/clairauts-theorem) |
| Tangent planes, gradients | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Maxima, minima and Lagrange multipliers | [Second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integrals | [Iterated integrals](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals guide](/blog/double-integrals) |
| Triple integrals, cylindrical and spherical | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | |
| Vector fields and line integrals | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | |
| Fundamental theorem for line integrals | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) | |
| Green's theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem statement](/theorems/greens-theorem) |

## How MATH 251 fits the SFU sequence

| Course | Title | Main focus |
| --- | --- | --- |
| [MATH 151](/courses/simon-fraser-university-math-151) | Calculus I | Derivatives |
| [MATH 152](/courses/simon-fraser-university-math-152) | Calculus II | Integrals and series |
| MATH 251 | Calculus III | Several variables and line integrals |

## Example problems at the MATH 251 level

**1. A plane through three points.** For $P(1,0,0)$, $Q(0,1,0)$ and $R(0,0,1)$, the normal is $\vec{PQ} \times \vec{PR} = \langle 1, 1, 1 \rangle$, so the plane is $x + y + z = 1$.

**2. A gradient.** For $f(x, y) = x^2 y + y^3$, $\nabla f = \langle 2xy,\; x^2 + 3y^2 \rangle$. At $(1, 1)$ the direction of fastest increase is $\langle 2, 4 \rangle$.

**3. A polar double integral.** The area inside $x^2 + y^2 \le 4$ is

$$\int_0^{2\pi}\int_0^2 r\,dr\,d\theta = 4\pi$$

**4. A conservative field.** $\mathbf{F} = \langle 2x, 2y \rangle = \nabla(x^2 + y^2)$, so the line integral from $(0,0)$ to $(1,2)$ along any path is $5$.

## How to do well in MATH 251

- **Draw everything.** Sketch the region before setting up a multiple integral, and sketch the curve before a line integral. Most errors come from wrong limits.
- **Know your coordinate systems.** Remember the extra factors: $r$ in polar and cylindrical, $\rho^2 \sin\phi$ in spherical.
- **Check for a potential first.** If a field is conservative, the fundamental theorem for line integrals turns a hard integral into a subtraction.
- **Keep single-variable skills fresh.** Multiple integrals end in ordinary integrals, so [integration techniques](/learn/calculus-2/integration-techniques/choosing-a-strategy) still matter.

> **Tip:** with Green's theorem, check the orientation. The boundary must be traversed counterclockwise with the region on your left.

## Frequently asked questions

### What is SFU MATH 251?

MATH 251 is Calculus III at Simon Fraser University. The calendar lists rectangular, cylindrical and spherical coordinates; vectors, lines, planes, cylinders and quadric surfaces; vector functions, curves and motion in space; differential and integral calculus of several variables; and vector fields, line integrals, the fundamental theorem for line integrals and Green's theorem.

### What are the prerequisites for MATH 251?

MATH 152 with a minimum grade of C-, or MATH 158 with at least B. Life science students can enter with MATH 154 (C- or better) and MATH 155 (A- or better).

### Does MATH 251 cover Stokes' theorem?

The SFU calendar description ends with Green's theorem. It does not list Stokes' theorem or the Divergence Theorem, so check your section's outline if you need them.

### How do I prepare for MATH 251?

Refresh integration techniques from MATH 152, since multiple integrals reduce to single-variable integrals. It also helps to be comfortable with vectors and matrices from linear algebra.