Course: Calculus III
Topic: Vector-Valued Functions
Subtopic: Curvature


There is a lot we can do to analyze the shape of a space curve beyond what we have so far (eg. tangent vectors, arc length). Consider this ... as you drive a car along a winding mountain road, you can accelerate by changing the speed of the car or changing the direction of the car. The rate at which the car changes direction is related to the notion of curvature.

Curvature measures how fast a curve turns at a point which helps us analyze the shape of the curve. Curvature (denoted by kappa, κ) is a nonnegative scalar-valued function. A large κ at a point indicates a tight curve that changes direction quickly and a small κ indicates the curve is relatively flat and changes directions slowly. In particular a line has zero curvature and a circle has constant curvature.


By the end of this topic you should know and be prepared to be tested on:


Terms you should be able to define: curvature (kappa, κ), circle of curvature

Formulae to have in your notes:

Supplemental Resources (optional)

Dale Hoffman's Contemporary Calculus III: Arc Length and Curvature of Space Curves (study "Curvature" sections)

Paul's OL Notes - Calc III: Curvature

More tutorial videos if you need them are listed at James Sousa's MathIsPower4U - Calc II. Scroll down right column to "Vector Valued Functions". There are several related titles in the second half of that list.