# Arizona MATH 223 Vector Calculus: Syllabus, Topics and Practice

> A study guide for University of Arizona MATH 223 Vector Calculus: vectors, partial derivatives, multiple integrals and the theorems of Green and Stokes.

Source: https://practicecalculus.com/courses/university-of-arizona-math-223
Updated: 2026-11-08

## At a glance

- **Course code:** MATH 223
- **Units:** Not stated on the department course page
- **Prerequisites:** Not stated on the course page; the department points to its Math Course Placement Chart
- **Covers:** Vectors and vector functions, surfaces, dot and cross products, partial derivatives, gradients, critical points, parametric curves, double and triple integrals, line and surface integrals, Green's, Stokes' and the Divergence Theorem
- **Textbook:** Multivariable Calculus (Hughes-Hallett et al.) or Calculus: Concepts and Contexts (Stewart and Kokoska), depending on section
- **Previous course:** MATH 129 Calculus II
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 223 Vector Calculus is the Calculus 3 course at the University of Arizona, taken after MATH 129.
- It covers vectors, surfaces, partial derivatives and gradients, critical points, parametric curves, double and triple integrals, and line and surface integrals.
- It finishes with the Fundamental Theorem of Line Integrals, Green's Theorem, the Divergence Theorem and Stokes' Theorem.
- Sections use either a Hughes-Hallett or a Stewart textbook, and all sections share a common final exam.

**Arizona MATH 223** (Vector Calculus) is the University of Arizona's Calculus 3 course, taken after MATH 129. It extends calculus to curves, surfaces and solids in two and three dimensions and ends with the major theorems of vector calculus. This guide maps the department's course objectives to lessons you can practice with instant feedback.

## What Arizona MATH 223 covers

The Department of Mathematics lists these MATH 223 course objectives:

- recognize and sketch surfaces in three-dimensional space,
- work with vectors and vector functions in two and three dimensions,
- compute dot and cross products and interpret them geometrically,
- compute partial derivatives, directional derivatives and gradients, and explain their meaning,
- compute and classify critical points,
- parameterize curves in 2-space and 3-space,
- set up and evaluate double and triple integrals in several coordinate systems,
- evaluate integrals over scalar and vector fields,
- apply the Fundamental Theorem of Line Integrals, Green's Theorem, the Divergence Theorem and Stokes' Theorem.

The expected learning outcomes add double integrals in Cartesian and polar coordinates, triple integrals in rectangular, cylindrical and spherical coordinates, and areas and volumes by multiple integrals. The course page does not list a unit count or a printed prerequisite, so check the placement chart and your schedule for those.

## MATH 223 topic map

| MATH 223 topic | Practice | Read |
| --- | --- | --- |
| Vectors, lines and planes | [Vectors and length](/learn/calculus-3/vectors-and-space/vectors-and-length) | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) |
| Dot and cross products | [Dot product](/learn/calculus-3/vectors-and-space/dot-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Parametric curves | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) |
| Surfaces and functions of two variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Second partials](/learn/calculus-3/partial-derivatives/second-partial-derivatives) |
| Gradients and directional derivatives | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient guide](/blog/gradient-and-directional-derivatives) |
| Critical points | [Second derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integrals, Cartesian and polar | [Polar double integrals](/learn/calculus-3/multiple-integrals/polar-double-integrals) | [Double integrals](/blog/double-integrals) |
| Triple integrals | [Cylindrical and spherical](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) |
| Line integrals | [Line integrals of vector fields](/learn/calculus-3/vector-calculus/line-integrals-of-vector-fields) | [Conservative fields](/learn/calculus-3/vector-calculus/conservative-fields) |
| Green's Theorem | [Green's theorem](/learn/calculus-3/vector-calculus/greens-theorem) | [Green's theorem](/theorems/greens-theorem) |
| Surface integrals | [Surface integrals and flux](/learn/calculus-3/vector-calculus/surface-integrals-and-flux) | [Curl and divergence](/learn/calculus-3/vector-calculus/curl-and-divergence) |
| Divergence and Stokes' Theorems | [Stokes and divergence theorems](/learn/calculus-3/vector-calculus/stokes-and-divergence-theorems) | [Divergence theorem](/theorems/divergence-theorem), [Stokes' theorem](/theorems/stokes-theorem) |

## MATH 223 in the Arizona sequence

| Course | Units | Notes |
| --- | --- | --- |
| [MATH 122B First-Semester Calculus](/courses/university-of-arizona-math-122b) | 4 | Calculus 1 |
| [MATH 129 Calculus II](/courses/university-of-arizona-math-129) | 3 | Calculus 2 |
| MATH 223 Vector Calculus | See department page | Calculus 3 |

## Example problems at the MATH 223 level

**1. The angle between vectors.** For $\mathbf{a} = \langle 1, 0, 1 \rangle$ and $\mathbf{b} = \langle 0, 1, 1 \rangle$:

$$\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|} = \frac{1}{2}, \quad \theta = \frac{\pi}{3}$$

**2. A critical point.** For $f(x, y) = x^2 + y^2 - 4x$, setting $f_x = 2x - 4 = 0$ and $f_y = 2y = 0$ gives $(2, 0)$, a local minimum.

**3. A conservative line integral.** For $\mathbf{F} = \nabla(xy)$ along any path from $(0, 0)$ to $(2, 3)$:

$$\int_C \mathbf{F}\cdot d\mathbf{r} = xy\Big|_{(0,0)}^{(2,3)} = 6$$

## How to do well in MATH 223

- **Get fluent with vectors in the first weeks.** Everything later, from gradients to flux, is built on them.
- **Sketch surfaces and regions by hand,** then use a graphing tool to check.
- **Know which theorem fits which situation.** A summary like our [vector calculus summary](/learn/calculus-3/vector-calculus/vector-calculus-summary) is worth reviewing before each exam.
- **Practice [double integrals](/practice-problems/double-integrals)** and [Lagrange multipliers](/practice-problems/lagrange-multipliers) until the setup is routine.
- **Prepare for the common final** with the department study guides and timed [practice exams](/practice).

> **Tip:** if a vector field is the gradient of a function, a line integral only depends on the endpoints. Check for that before you parameterize a curve.

## Frequently asked questions

### What is MATH 223 at the University of Arizona?

MATH 223 Vector Calculus is the University of Arizona's multivariable calculus course. The department's objectives include sketching surfaces, working with vectors and vector functions, dot and cross products, partial derivatives, directional derivatives and gradients, classifying critical points, parameterizing curves, double and triple integrals in several coordinate systems, and integrals over scalar and vector fields.

### Which big theorems are in MATH 223?

The department lists the Fundamental Theorem of Line Integrals, Green's Theorem, the Divergence Theorem and Stokes' Theorem.

### What do I need before MATH 223?

MATH 223 follows MATH 129 Calculus II. The course page does not print a prerequisite, so check the department's Math Course Placement Chart for the current rule.

### What textbook does MATH 223 use?

It depends on the section. Some sections use Multivariable Calculus (6th edition, Hughes-Hallett et al.) and others use Calculus: Concepts and Contexts (5th edition, Stewart and Kokoska).

### Is there tutoring for MATH 223?

Yes. The Math Department offers free Think Tank tutoring for MATH 223.