Vector Calculus

MATH 223

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Course Objectives

  • Recognize and sketch surfaces in three-dimensional space;

  • Recognize and apply the algebraic and geometric properties of vectors and vector functions in two and three dimensions;

  • Compute dot products and cross products and interpret their geometric meaning;

  • Compute partial derivatives of functions of several variables and explain their meaning;

  • Compute directional derivatives and gradients of scalar functions and explain their meaning;

  • Compute and classify the critical points;

  • Parameterize curves in 2- and 3-space;

  • Set up and evaluate double and triple integrals using a variety of coordinate systems;

  • Evaluate integrals through scalar or vector fields and explain some physical interpretation of these integrals;

  • Recognize and apply Fundamental theorem of line integrals, Green’s theorem, Divergence Theorem, and Stokes’ theorem correctly.

Expected Learning Outcomes

Upon completion of this course, students should be able to:

  • Perform vector operations, determine equations of lines and planes, parametrize 2D & 3D curves.

  • Graphically and analytically synthesize and apply multivariable and vector-valued functions and their derivatives, using correct notation and mathematical precision.

  • Synthesize the key concepts differential, integral and multivariate calculus.

  • Evaluate double integrals in Cartesian and polar coordinates; evaluate triple integrals in rectangular, cylindrical, and spherical coordinates; and calculate areas and volumes using multiple integrals.

  • Use double, triple and line integrals in applications, including Green's Theorem, Stokes' Theorem, Divergence Theorem and Fundamental theorem of line integrals.


Section information and links to instructor class webpages are found here.

MATH 223 (4 credits)

MATH 196V (1 unit)


Students must meet eligibility requirements.

See the current Math Course Placement Chart here.


Math 223 covers chapters 12 - 20 of Multivariable Calculus; 6th edition; McCallum, Hughes-Hallett, Gleason, et al.; Wiley.

The Textbook and WebAssign access for homework are being delivered digitally via D2L through the Inclusive Access program.

Inclusive Access FAQs


A graphing calculator is required for Math 223. We recommend any model in the TI-83 or TI-84 series. Students are not allowed to share calculators during exams and quizzes.


Math 223 uses a computer grading program called WebAssign for textbook assignments. Logging into WebAssign can only be done through the WebAssign Login link in your Math 223 D2L site.

Additional homework and/or quizzes may be assigned by instructors. Be sure to check with your instructor.

Things to Know About WebAssign at the UA

Written Homework


Individual course sections may deviate from the suggested calendar and policies. Be sure to check with your instructor.

Math 223 General Course Policies and Calendar (Spring 2023)


A common comprehensive final exam will be given for all sections of Math 223 on Tuesday, May 9th, 2023, from 1:00pm - 3:00pm.

See instructor or syllabus for more details regarding your final exam.

Final Exam Info - TBA

Final Exam Study Guide, Answers

Final Exam Room Locations - TBA

Archived Exams


On-line Calculator

On-line Graphing Tool (browser)

Calculator Programs

Math 223 demonstrations (Wolfram Project)

Dot Product (Paul Falstad)

Cross Product (Syracuse University)

Plotting surfaces and contours (Kaskosz & Ensley)

Plotting (MIT Open Courseware)

Surfaces (JavaView)

Vector Field Analyzer (Matthias Kawski - ASU)

3-D Vector Field Applet (Paul Falstad)

3-D Vector Fields (OR State University)

Trig Formulas

Derivative Rules

Basic Integral Rules

Integral Table

Geometry Formulas

Area Formulas