# UNIBEN MTH 213 Vector Analysis: Study Guide and Practice

> A study guide for UNIBEN MTH 213 (Vector Analysis): vector and triple products, vector equations, space curves, gradient, divergence and curl, with practice.

Source: https://practicecalculus.com/courses/uniben-mth-213
Updated: 2026-11-03

## At a glance

- **Course code:** MTH 213
- **Credits:** 3
- **Semester:** 200 level, first semester (core course)
- **Prerequisites:** MTH 110 and MTH 125
- **Covers:** Vector algebra, vector and triple products, solving vector equations, plane and space curves, the Serret-Frenet formulas, differential definitions of gradient, divergence and curl, and simple applications
- **Later course:** MTH 336 Vector Field Theory (300 level)
- **Practice here:** [Calculus 3](/learn/calculus-3), from [vectors](/learn/calculus-3/vectors-and-space/vectors-and-length) to [curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence)

## Key takeaways

- MTH 213 Vector Analysis is a 3-credit core course in the first semester of 200 level at the University of Benin.
- It covers vector algebra, vector and triple products, and solving vector equations.
- It studies plane and space curves, including the Serret-Frenet formulas.
- It introduces gradient, divergence and curl through their differential definitions, which are core Calculus 3 ideas.

**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 topic | Practice | Read |
| --- | --- | --- |
| Vector algebra | [Vectors and their length](/learn/calculus-3/vectors-and-space/vectors-and-length) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Scalar product | [The dot product](/learn/calculus-3/vectors-and-space/dot-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Vector product | [The cross product](/learn/calculus-3/vectors-and-space/cross-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Triple products and vector equations | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Lines and planes | [Lines in space](/learn/calculus-3/vectors-and-space/lines-in-space) | [Linear function](/functions/linear-function) |
| Plane and space curves | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Chain rule](/rules/chain-rule) |
| Arc length along a curve | [Arc length of parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-arc-length) | [Definite integral properties](/rules/definite-integral-properties) |
| Gradient | [The gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Divergence and curl | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) | [Divergence theorem](/theorems/divergence-theorem), [Stokes' theorem](/theorems/stokes-theorem) |
| Vector fields | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) | [Partial derivatives](/blog/partial-derivatives) |

## MTH 213 in the UNIBEN sequence

| Course | Level and semester | Vector content |
| --- | --- | --- |
| MTH 123 Vectors, Coordinate Geometry and Statistics | 100, second | Vectors in 1 to 3 dimensions, scalar and vector products, lines and planes |
| MTH 125 Differential Equations and Dynamics | 100, second | Differentiating and integrating vectors, velocity and acceleration |
| MTH 213 Vector Analysis | 200, first | Triple products, space curves, grad, div and curl |
| MTH 336 Vector Field Theory | 300, first | Further vector field theory |

MTH 213 runs alongside [MTH 218 Mathematical Methods I](/courses/uniben-mth-218), which supplies the partial derivatives you need for grad, div and curl. For the single-variable foundation, see the [MTH 112 guide](/courses/uniben-mth-112).

## Example problems at the MTH 213 level

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

$$\mathbf{b} \times \mathbf{c} = (1 \cdot 1 - 0 \cdot 1,\; 0 \cdot 1 - 0 \cdot 1,\; 0 \cdot 1 - 1 \cdot 1) = (1, 0, -1)$$

$$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 1$$

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

$$\mathbf{T}(t) = \frac{1}{\sqrt{2}}(-\sin t, \cos t, 1)$$

**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 $\mathbf{F} = (x^2, xy, z)$:

$$\nabla \cdot \mathbf{F} = 2x + x + 1 = 3x + 1$$

$$\nabla \times \mathbf{F} = \left(\frac{\partial z}{\partial y} - \frac{\partial (xy)}{\partial z},\; \frac{\partial (x^2)}{\partial z} - \frac{\partial z}{\partial x},\; \frac{\partial (xy)}{\partial x} - \frac{\partial (x^2)}{\partial y}\right) = (0, 0, y)$$

## 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](/learn/calculus-3/vectors-and-space/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](/blog/partial-derivatives).
- **Keep the identities together.** The [calculus formula sheet](/formulas) collects the vector formulas you will use most, and [common calculus mistakes](/blog/common-calculus-mistakes) flags the usual slips.
- **Practice under time pressure** with [practice exams](/practice) 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.

## Frequently asked questions

### What is MTH 213 at UNIBEN?

MTH 213 Vector Analysis is a 3-credit, first-semester, 200-level core course in the University of Benin Department of Mathematics. It covers vector algebra and products, vector equations, plane and space curves with the Serret-Frenet formulas, and the gradient, divergence and curl.

### What are the prerequisites for MTH 213?

The department lists MTH 110 Algebra and Trigonometry and MTH 125 Differential Equations and Dynamics as prerequisites. Vectors from MTH 123 and differentiation from MTH 112 are also used throughout.

### Is MTH 213 hard?

The algebra of vectors is mechanical once the dot and cross products are familiar. Space curves and the Serret-Frenet formulas are the most demanding part, because they combine vector algebra with differentiation.

### What is the difference between MTH 213 and MTH 218?

Both are first-semester 200-level core courses. MTH 213 focuses on vectors, curves and the operators grad, div and curl. MTH 218 Mathematical Methods I covers integration techniques, partial differentiation, Lagrange multipliers, series and complex numbers.

### How should I prepare for MTH 213?

Revise vectors in two and three dimensions, the dot and cross products, and partial derivatives. Practice computing cross products quickly, since almost every topic in the course uses them.