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Multivariable Calculus

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Prerequisites: Qualifying math score and successful completion of AP Calculus BC or equivalent

Course Format: Individually Paced

Course Length: Typically 6 months

Recommended School Credit: One full year of high school credit equal to or greater than an AP class or one semester of college credit

Course Code: MVC

Course Description


Multivariable Calculus is an online and individually-paced course that covers all topics in JHU's undergraduate Calculus III: Calculus of Several Variables course. In this course, students will extend what was learned in AB & BC Calculus and learn about the subtleties, applications, and beauty of limits, continuity, differentiation, and integration in higher dimensions. Computer-based interactives, homework and quizzes help to reinforce concepts taught in the class. Online course materials supplement the required textbook.

Each student is assigned to a CTY instructor to help them during their course. Students can contact their instructor via email with any questions or concerns at any time. Live one-on-one online review sessions can be scheduled as well to prepare for the graded assessments, which include quizzes, homework, midterm exams, and a cumulative final. Instructors use virtual classroom software allowing video, voice, text, screen sharing and whiteboard interaction.

Topics include:

  • Vectors in Euclidean space
  • Vector analysis
  • Analytic geometry of three dimensions
  • Curves in space
  • Partial derivatives
  • Optimization techniques
  • Multiple integrals
  • Vector fields
  • Green's theorem
  • Divergence theorem
  • Stokes' theorem
  • Differential forms

For a detailed list of topics, click the List of Topics tab.

Image of Multivariable Calculus banner.

Materials Needed

A textbook purchase is required for this course:

Multivariable Calculus, 8th edition, by James Stewart

ISBN13: 978-1-305-26664-3

Picture of Calculus Cover 8th edition.

The material is also included in Calculus, 8th edition, by James Stewart

ISBN-13: 978-1-285-74062-1

Only one of the textbooks listed on this page is required for this course.

List of Topics

Upon successful completion of the course, students will be able to demonstrate mastery over the following topics:

Vectors and the Geometry of Space

  • Three-Dimensional Coordinate Systems
  • Vectors
  • The Dot Product
  • The Cross Product
  • Equations of Lines and Planes
  • Cylinders and Quadric Surfaces

Vector Functions

  • Vector Functions and Space Curves
  • Derivatives and Integrals of Vector Functions
  • Arc Length and Curvature
  • Motion in Space:  Velocity and Acceleration

Partial Derivatives

  • Functions of Several Variables
  • Limits and Continuity
  • Partial Derivatives
  • Tangent Planes and Linear Approximations
  • The Chain Rule
  • Directional Derivatives and the Gradient Vector
  • Maximum and Minimum Values
  • Lagrange Multipliers

Multiple Integrals

  • Double Integrals over Rectangles
  • Iterated Integrals
  • Double Integrals over General Regions
  • Double Integrals in Polar Coordinates
  • Applications of Double Integrals
  • Surface Area
  • Triple Integrals
  • Triple Integrals in Cylindrical Coordinates
  • Triple Integrals in Spherical Coordinates
  • Change of Variables in Multiple Integrals

Vector Calculus

  • Vector Fields
  • Line Integrals
  • The Fundamental Theorem for Line Integrals
  • Green's Theorem
  • Curl and Divergence
  • Parametric Surfaces and Their Areas
  • Surface Integrals
  • Stokes' Theorem
  • The Divergence Theorem
  • Differential Forms and the General Stokes' Theorem

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Sample Video Lecture

Sample Video

Vector projection video sample

Technical Requirements

This course requires a properly maintained computer with high-speed internet access and an up-to-date web browser (such as Chrome or Firefox). The student must be able to communicate with the instructor via email. Visit the Technical Requirements and Support page for more details.