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Calculus I
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Calculus II
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Calculus III

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

Overview

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.

Objectives

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

• 14.5.1 Understand the notion of curvature as the measure of how fast a curve turns at a point
• 14.5.2 Find curvature using the formula that is in terms of arc length and the formula that does not involve arc length (both below). Also understand how to derive the second formula from the first via a chain rule.
• 14.5.3 Understand why a circle with small radius (R) has large curvature and vice versa
• 14.5.4 Understand that the curvature of a curve at a point can be visualized in terms of a "circle of curvature" which is a circle of radius R that is tangent to the curve at that point and that the curvature at that point is kappa=1/R
• 14.5.5 Find the the equation of the "circle of curvature" for a curve at a given point
• 14.5.6 Find curvature using the alternate curvature formula (below) and understand the derivation of that formula
• 14.5.7 Know and understand that for straight line motion, a(t)=0 and k=0. Also that if v(t) = 0 the object is at rest and k is undefined.
• 14.5.8 Know or be able to derive the curvature for a variety of basic curves including a parabola (2D) and a helix (3D)

Terminology

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

Formulae to have in your notes:

• Curvature in terms of arc length:  kappa(s) = ||(d vecT)/(ds)|| where vecT=vec(r')/||vec(r')||

• Curvature not inv'n arc length:  kappa = 1/||vecv||*||(d vecT)/(dt)|| = ||vec(T')(t)||/||vec(r')(t)||

• Alternate curvature formula:  kappa = ||vecv xx veca|| / ||vecv||^3 = ||vec(r')(t) xx vec(r'')(t)|| / ||vec(r')(t)||^3

• Cartesian system formula for curve given by y=f(x):  kappa=|f''(x)|/(1+(f'(x))^2)^(3/2)

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.