uOttawa MAT 2322 Calculus III for Engineers: Topics and Practice

MAT 2322 is Calculus III for Engineers at the University of Ottawa, the English-language course that completes the calculus sequence for most engineering programs. It starts with optimization in several variables and then builds up the integral side of multivariable calculus, ending with the theorems of Stokes and Gauss. This guide maps every MAT 2322 topic to free practice so you can work through each idea with instant feedback.

What uOttawa MAT 2322 covers

The uOttawa catalogue lists: extrema of functions of several variables; multiple integration and applications; vector fields and their derivatives; curves; vector differential operators; line integrals; surfaces and surface integrals; and the theorems of Stokes, Gauss and related results. It is a 3-unit lecture course. The prerequisites are one of MAT 1322, MAT 1325 or MAT 1332, plus MAT 1341 or CEGEP linear algebra.

Because partial derivatives, the chain rule and directional derivatives are already in MAT 1322, MAT 2322 begins with extrema and spends most of the term on the integral half of Calculus 3.

MAT 2322 topic map

MAT 2322 in the uOttawa sequence

Example problems at the MAT 2322 level

1. A critical point. For f(x, y) = x^2 + xy + y^2 - 3x: f_x = 2x + y - 3 = 0 and f_y = x + 2y = 0 give (2, -1). With D = 4 - 1 = 3 > 0 and f_{xx} > 0, it is a minimum.

2. A cylindrical integral. The volume under z = 4 - r^2 and above z = 0 is

3. Flux by Gauss. For \mathbf{F} = \langle x, y, z \rangle through the closed cylinder r \le 1, 0 \le z \le 2: \nabla\cdot\mathbf{F} = 3, so the flux is 3 \cdot 2\pi = 6\pi.

4. Stokes' theorem. For \mathbf{F} = \langle -y, x, 0 \rangle, \nabla\times\mathbf{F} = \langle 0, 0, 2 \rangle. Over the unit disk in the plane z = 0, the circulation around its edge is 2\pi.

How to do well in MAT 2322

• Review linear algebra. The prerequisite list includes MAT 1341 for a reason: vectors, determinants and cross products are everywhere.
• Sketch every region and surface. Bounds and orientations come from the picture, and most lost marks come from wrong bounds.
• Know the operator identities. Gradient, divergence and curl each take and return different kinds of objects. Keep them straight.
• Choose the easier integral. Stokes and Gauss let you trade a hard surface or line integral for an easier one.

Tip: a closed surface uses the Divergence Theorem; a surface with a boundary curve uses Stokes' theorem. Check which one you have before you start.