Arizona MATH 223 Vector Calculus: Syllabus, Topics and Practice

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 in the Arizona sequence

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:

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):

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 is worth reviewing before each exam.
• Practice double integrals and Lagrange multipliers until the setup is routine.
• Prepare for the common final with the department study guides and timed practice exams.

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.