ASU MAT 267 Calculus for Engineers III: Study Guide and Practice
A study guide for ASU MAT 267 Calculus for Engineers III: vector-valued functions, partial derivatives and multiple integration, with practice.
Arizona State University Updated 3 min read
At a glance
- Course code
- MAT 267
- Credit hours
- 3
- General studies
- Mathematical Studies (MATH / MA)
- Covers
- Vector-valued functions of several variables, partial derivatives, multiple integration
- Offered by
- School of Mathematical and Statistical Sciences
- 4-credit counterpart
- MAT 272 Calculus with Analytic Geometry III
- Practice here
- Calculus 3
On this page
ASU MAT 267, Calculus for Engineers III, is the multivariable calculus course that completes Arizona State University's three-course engineering calculus sequence. It takes the derivative and integral into several variables. This guide maps every MAT 267 topic to Calculus 3 practice with instant feedback.
What ASU MAT 267 covers
The ASU catalog lists three topics: vector-valued functions of several variables, partial derivatives, and multiple integration. It is a 3-credit-hour course that counts toward Mathematical Studies.
That is the core of a standard Calculus 3: vectors and curves in space, partial derivatives with gradients and optimization, and double and triple integrals. On PracticeCalculus it lines up with the first three Calculus 3 units. The 4-credit MAT 272 adds an introduction to vector analysis, which is the fourth unit, vector calculus.
A useful way to see the course is as three extensions of single-variable ideas. Vector-valued functions extend curves and motion into space, so velocity and acceleration become vectors. Partial derivatives extend the derivative to functions with several inputs, leading to gradients, tangent planes and optimization with or without constraints. Multiple integration extends the definite integral to regions in the plane and in space, often in polar, cylindrical or spherical coordinates. Each extension reuses a skill you already have.
MAT 267 topic map
| MAT 267 topic | Practice | Read |
|---|---|---|
| Vectors in space | Vectors and their length | Dot and cross product |
| Dot and cross products | The cross product | Dot and cross product |
| Lines and planes | Planes in space | Dot and cross product |
| Vector-valued functions | Curves and motion | Partial derivatives |
| Functions of several variables | Functions of several variables | Partial derivatives |
| Partial derivatives | Partial derivatives | Clairaut's theorem |
| Chain rule | The multivariable chain rule | Chain rule |
| Gradient and tangent planes | The gradient | Gradient and directional derivatives |
| Optimization | Lagrange multipliers | Lagrange multipliers |
| Double integrals | Double integrals over general regions | Double integrals |
| Triple integrals | Cylindrical and spherical | Double integrals |
MAT 267 vs MAT 272
| Course | Credit hours | Catalog topics |
|---|---|---|
| MAT 267 Calculus for Engineers III | 3 | Vector-valued functions, partial derivatives, multiple integration |
| MAT 272 Calculus with Analytic Geometry III | 4 | Vector-valued functions, multiple integration, introduction to vector analysis |
The courses before MAT 267 are MAT 265 and MAT 266.
Example problems at the MAT 267 level
1. Velocity and speed. For \mathbf{r}(t) = \langle \cos t, \sin t, t \rangle (a helix):
2. The multivariable chain rule. For z = x^2 y with x = t^2 and y = 3t:
Check: z = 3t^5 directly, and \frac{d}{dt}(3t^5) = 15t^4.
3. A triple integral in cylindrical coordinates. The volume of a cylinder of radius 1 and height 3:
How to do well in MAT 267
- Get vectors fluent first. The dot product and cross product appear in lines, planes, tangent planes and curves.
- Draw the region for every multiple integral. Then practice switching the order so you can pick the easier setup.
- Remember the Jacobian factors: r in polar and cylindrical, \rho^2 \sin\phi in spherical.
- Keep integration skills from MAT 266 sharp with the formula sheet, and rehearse with timed practice exams.
Tip: a tree diagram makes the multivariable chain rule mechanical. Draw one branch per variable and multiply along each path.
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Curves and motion
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