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Calculus III

Course: Calculus III
Topic: Vector-Valued Functions
Subtopic: Normal and Binormal Vectors

Overview

Consider an object moving along a curve in space. Recall that two ways to change the velocity of an object (to accelerate) are to change its speed and change its direction. Curvature tells how fast the curve turns. At a point P on the curve, the normal vector describes the direction in which the curve turns. The binormal vector describes the twisting or torsion (denoted by tau, τ) of the curve.

The tangent vector T and the normal vector N may provide insight into how moving objects accelerate. Changing the speed produces acceleration in the direction of T and changing the direction produces acceleration in the direction of N. The binormal vector B is orthogonal to T and N. Investigating these three vectors and the three planes each pair spans as an object moves along a curve in space will provide much information about the direction, acceleration, and orientation of the object.

There is a lot going on here, so plan to spend quality time with the concepts making connections between the visuals of points moving and accelerating on curves, the vectors associated with them, and the physical applications these motions can help analyze.

Objectives

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

• 14.6.1 Find the principal unit normal vector at a point on a curve:
N = (dT/ds) / ||dT/ds|| = 1/κ · dT/ds = (dT/dt) / ||dT/dt||
evaluated at the value of t corresponding to P
• 14.6.2 Understand visually the two "Properties of the Principal Unit Normal Vector" which state that (1) T and N are orthogonal at all points of the curve, and (2) the principal unit normal vector points to the inside of the curve i.e. in the direction the curve is turning
• 14.6.3 Understand visually and algebraically how changing the speed or direction produces acceleration in the direction of T and N, respectively
• 14.6.4 Be able to derive from a = dv/dt that a = κ N ||v||2 + T d2s/dt2
• 14.6.5 Know what the tangential component aT and normal component aN are and how they relate to the acceleration vector a algebraically and visually
• 14.6.6 Know the "Tangential and Normal Components of Acceleration" theorem: a = aNN + aTT where aN=κ||v||2=||v×a||/||v|| and aT=d2s/dt2
• 14.6.7 Understand graphical relationships between the vectors a, aNN, and aTT, physical properties of motion described by each, and their connection to the statement, "higher speeds on tighter curves produce greater normal accelerations"
• 14.6.8 Find the tangential and normal components of acceleration on basic trajectories including circular and parabolic
• 14.6.9 Know that the unit binormal vector B = T × N thus B is orthogonal to T and N and that these three vectors form a right-handed coordinate system (TNB frame) that changes its orientation as we move along the curve
• 14.6.10 Understand visually the relationship of the osculating plane, normal plane, rectifying plane, and the TNB frame as we move along the curve
• 14.6.11 Understand graphically the statement, "the rate at which the curve twists out from the plane spanned by T and N is the rate at which B changes as we move along the curve" and how it algebraically relates to: dB/ds = d/dt (T × N) = T × dN/ds
• 14.6.12 Understand what torsion (τ) is including that it is the rate at which the curve twists out from the TN-plane
• 14.6.13 Understand graphically and physically that torsion (||τ|| = ||dB/ds||) gives the rate at which the the curve moves out from the osculating plane, and complementarily, curvature (κ = ||dT/ds||) gives the rate at which the curve turns within the osculating plane
• 14.6.14 Find torsion using all three of the formulae given in the last line of the chart below

Terminology

Terms you should be able to define: principal unit normal vector, tangential and normal components of the acceleration vector, unit binormal vector, torsion (tau, τ), TNB frame, mutually orthogonal vectors, osculating plane, normal plane, rectifying plane

Formulae to have in your notes: The following list is a screenshot from Briggs' Calculus: Early Transcendentals (3rd ed., ch. 14.6). Supplemental Resources (optional)

Paul's OL Notes - Calc III: Tangent, Normal, and Binormal Vectors (study "Normal Vectors" and "Binormal Vectors" sections)

James Sousa's MathIsPower4U - Calc II: Determining the Binormal Vector