MTH 210 Vector Analysis: Topics, Study Guide and Practice

MTH 210 Vector Analysis is a 1-unit, 200 level course in the National Universities Commission's Core Curriculum and Minimum Academic Standards (CCMAS). It is compulsory in the B.Sc. Mathematics programme, and it picks up the vector thread that MTH 103 started at 100 level. It is short, but it introduces the operators that much of applied mathematics and physics is written in. This guide maps every MTH 210 topic to practice.

What MTH 210 covers

The CCMAS outline covers elementary vector algebra, including vector and triple products, and the solution of vector equations. It studies plane curves and space curves, geometrical equations of lines and planes, linear independence, components of vectors, direction cosines, position vectors and scalar products. It introduces the Frenet-Serret formulas and gives a differential definition of the gradient, divergence and curl.

The learning outcomes ask students to describe vector algebra, explain the equations of lines and planes, and work problems on gradient, divergence and curl.

In a standard sequence these topics sit in Calculus 3: the opening vectors unit plus the first ideas of vector calculus.

MTH 210 topic map

MTH 210 and the vector sequence

The CCMAS also lists MTH 303 Vector and Tensor Analysis at 300 level, which builds on this course.

Example problems at the MTH 210 level

These are original practice problems.

1. A plane from a point and a normal. The plane through (1, 0, 2) with normal (2, -1, 3) is

2. A scalar triple product. For \mathbf{a} = (1, 0, 0), \mathbf{b} = (0, 2, 0), \mathbf{c} = (0, 0, 3), we have \mathbf{b}\times\mathbf{c} = (6, 0, 0), so \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = 6: the volume of the box the three vectors span.

3. Gradient, divergence and curl. For f = x^2 y + z^3, \nabla f = (2xy,\; x^2,\; 3z^2), which is (4, 1, 3) at (1, 2, 1). For \mathbf{F} = (x^2,\; yz,\; z),

and for the rotating field \mathbf{G} = (-y,\; x,\; 0), \nabla\times\mathbf{G} = (0,\; 0,\; 2).

How to do well in MTH 210

• Review MTH 103 first. Triple products and planes are built on the dot and cross products, so make those automatic with practice.
• Learn grad, div and curl as patterns. Grad turns a scalar into a vector, div turns a vector into a scalar, curl turns a vector into a vector. Keep the definitions on your formula sheet.
• Check results geometrically. A normal vector should be perpendicular to every direction in its plane, and a triple product of zero means the vectors are coplanar.
• Avoid sign slips in the cross product. The middle component is the usual culprit; our common calculus mistakes guide has more checks like this.