SFU MATH 251 Calculus III: Syllabus, Topics and Practice

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

How MATH 251 fits the SFU sequence

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

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 still matter.

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