UNIBEN MTH 213 Vector Analysis: Study Guide and Practice

UNIBEN MTH 213, Vector Analysis, is the 200-level vector calculus course at the University of Benin. It is a 3-credit core course in the first semester of 200 level, with MTH 110 and MTH 125 listed as prerequisites. This guide maps every MTH 213 topic to free practice, so you can learn each idea by solving problems with instant feedback.

What UNIBEN MTH 213 covers

The department's description is short but dense. It starts with elementary vector algebra, the vector product and triple products, and solving vector equations. It then studies plane curves and space curves, including the Serret-Frenet formulas that describe how a curve's tangent, normal and binormal change. Finally it gives the differential definitions of the gradient, divergence and curl, with simple applications.

In standard terms, this covers the vectors and space unit and the start of vector calculus in a Calculus 3 course. The integral theorems of vector calculus (Green, Stokes and divergence) are not in the MTH 213 description; UNIBEN offers MTH 336 Vector Field Theory at 300 level.

MTH 213 topic map

MTH 213 in the UNIBEN sequence

MTH 213 runs alongside MTH 218 Mathematical Methods I, which supplies the partial derivatives you need for grad, div and curl. For the single-variable foundation, see the MTH 112 guide.

Example problems at the MTH 213 level

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

so the parallelepiped spanned by the three vectors has volume 1.

2. Unit tangent to a helix. For \mathbf{r}(t) = (\cos t, \sin t, t), \mathbf{r}'(t) = (-\sin t, \cos t, 1) and |\mathbf{r}'(t)| = \sqrt{2}, so

3. A gradient. For \phi = x^2 y + z, \nabla\phi = (2xy,\; x^2,\; 1), so at (1, 2, 3) the gradient is (4, 1, 1).

4. Divergence and curl. For \mathbf{F} = (x^2, xy, z):

How to do well in UNIBEN MTH 213

• Make the cross product fast and accurate. Triple products, normals to planes, binormals and curl all depend on it. Drill the cross product until you never lose a sign.
• Picture the geometry. The scalar triple product is a volume, \mathbf{T} points along the curve, and the gradient points uphill. Linking each formula to a picture makes the Serret-Frenet formulas far easier to remember.
• Review partial derivatives. Gradient, divergence and curl are built from them; see partial derivatives.
• Keep the identities together. The calculus formula sheet collects the vector formulas you will use most, and common calculus mistakes flags the usual slips.
• Practice under time pressure with practice exams before each test.

Tip: a quick check for any curl calculation: the curl of a gradient is always the zero vector. If you compute \nabla \times (\nabla\phi) and get anything else, there is an arithmetic error.