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Multivariable Calculus (NCAA Approved)

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Prerequisites: Qualifying math score and completion of Calculus BC

Course Format: Session Based. See calendar for session dates and application deadlines.

Course Length: 30 weeks

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

Student Expectations: Students are strongly encouraged to work at least 1 hour a day, 5 days a week.

Course Code: MVCY

Course Description


Multivariable Calculus is an online course that covers all topics in the Johns Hopkins one-semester Calculus III 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. Online interactives and assignments help to reinforce concepts taught in the class. Online course materials supplement the required textbook.

Online course materials, such as videos, notes, interactive webpages, and practice problems with solutions, are provided for the student. Students are expected to watch videos and review notes regularly. Each student is assigned to a CTY instructor to help them during their course.

In this course, participation in forums and synchronous online virtual sessions are required as part of the final grade. Discussion forums are located within the course where students will be required to respond to a given prompt and then comment on responses by other students. Online sessions, led by a CTY instructor, are held in Adobe Connect, which allows for video, voice, text, screen sharing, and whiteboard interaction. 

Students can contact their instructors via email with 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, unit exams, and a cumulative final.

The textbook must be purchased separately.

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, 7th edition, by James Stewart

  • ISBN10: 0-538-49787-4
  • ISBN13: 978-0-538-49787-9

mvc text

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.