Dave4Math / Courses / Vector-Valued Functions

Vector-Valued Functions

Master vector-valued functions. In this course, you'll become a pro in the calculus of single-variable vector functions, curvature, the motion along a curve. You'll also get experienced with Kepler's Laws.


January 9, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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Do you like being taught to, or do you like participating in an education that will prepare you for success? Introducing Dave’s online math course: Vector-Valued Functions. Dave prides himself as being an educator and not just a teacher. In other words, in Dave’s courses, he guides you towards uncovering what calculus is and why it’s important and valuable.

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in vector-valued functions.
  • Students who are enrolled in calculus 3 and want to improve their grade.
  • Anyone interested in getting prepared for linear algebra.
  • Anyone wanting to learn about derivatives and integrals of vector functions, arc length and curvature, and modeling 3D motion.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is required.

What You’ll Learn in Vector-Valued Functions

  • Curves Defined by Vector Functions
  • Limits and Continuity
  • The Derivative of a Vector Function
  • Higher-Order Derivatives
  • Rules of Differentiation
  • Integration of Vector Functions
  • Arc Length
  • Smooth Curves
  • Arc Length Parameter
  • Curvature
  • Radius of Curvature
  • Velocity, Acceleration, and Speed
  • Motion of a Projectile
  • The Unit Normal
  • Tangential and Normal Components of Acceleration
  • Derivation of Kepler’s First Law
  • Multiplication and Division of Complex Numbers

Course Description

In this course, we begin by studying vector-valued functions. These are functions of one variable whose output is a vector. We cover the calculus of vector-valued functions by considering limits, continuity, differentiation, and integration of vector functions.

After that introduction to vector functions, we examine the distance traveled along the graph of a vector function (arc length) and curvature. Next, we model motion along the path of a vector function in three dimensions. We study velocity, acceleration, and speed. This modeling leads us to a discussion of the tangential and normal components of acceleration. In particular, we examine the unit normal and unit tangent vectors of a vector function.

In the end, we discuss Kepler’s Laws of planetary motion.

Recommended Prerequisites for Vector-Valued Functions

I recommend the prerequisite course Vectors and the Geometry of Space. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free calculus 3 articles.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.