Covenant University MAT122 Vector Analysis: Study Guide

MAT122 Intro. to Vector Analysis is the vector course in the 100 level Industrial Mathematics list at Covenant University. It is a 2-unit course in the Omega semester, taken in the same semester as MAT121 Calculus. It begins with coordinates in three dimensions and ends with the gradient, divergence and curl, so it covers material that many universities leave to a third calculus course. This guide maps each MAT122 topic to free practice.

What MAT122 covers

The department's course list describes MAT122 as starting with three-dimensional Cartesian coordinates and the definition and representation of vectors. It covers elementary vector algebra: multiplying a vector by a scalar, adding vectors, scalar and vector products, and triple vector products, along with direction cosines and solving vector equations.

The second part is the calculus of vector functions: differentiating vector functions, plane curves and space curves, the Serret-Frenet formulae, a differential definition of grad, div and curl with simple applications, and the integration of vector functions.

In a standard sequence this matches the vectors unit of Calculus 3 together with the first ideas of vector calculus.

MAT122 topic map

MAT122 alongside the calculus courses

Grad, div and curl are built from partial derivatives, which MAT212 develops more fully at 200 level.

Example problems at the MAT122 level

These are original practice problems.

1. Distance and direction. From P(1, -1, 2) to Q(3, 1, 3), \overrightarrow{PQ} = (2, 2, 1) and |\overrightarrow{PQ}| = \sqrt{4 + 4 + 1} = 3. The direction cosines are \frac{2}{3}, \frac{2}{3}, \frac{1}{3}.

2. A triple product. For \mathbf{a} = (1, 1, 0), \mathbf{b} = (0, 1, 1) and \mathbf{c} = (1, 0, 1):

so \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = 1 + 1 + 0 = 2, and the three vectors are not coplanar.

3. Velocity on a helix. For \mathbf{r}(t) = (\cos t,\; \sin t,\; t), \mathbf{r}'(t) = (-\sin t,\; \cos t,\; 1), so the speed is \sqrt{\sin^2 t + \cos^2 t + 1} = \sqrt{2} at every point.

4. Divergence and curl. For \mathbf{F} = (xy,\; yz,\; zx), \nabla\cdot\mathbf{F} = y + z + x and

How to do well in MAT122

• Make the products automatic. Drill the cross product and check each answer by dotting it with the inputs.
• Keep MAT121 strong. Differentiating a vector function is just differentiating each component, so weak derivative rules show up quickly. The formula sheet helps.
• Learn grad, div and curl as a set. Write all three on one card and practise each on the same field so the patterns stick.
• Practise often. Use practice for mixed review and our common calculus mistakes list to catch sign errors.