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7. Let 7: R R" be a path parameterized by arclength - i.e. the speed is a constant one. a) Prove that 7'(s) IF"(s) for all s. T"(s) |F"()" b) Let T(s) = F'(s) and N Define a new vector field B(s) = T(s) × N (s). Prove that ||B(s)|| = 1. Note: T, N, and B form a mutually perpendicular trihedron of unit vectors along r called the Frenet frame. This frame is crucial to the study of the geometry of curves and is used in the study of objects moving in space.

7. Let 7: R R" be a path parameterized by arclength - i.e. the speed is a constant one. a) Prove that 7'(s) IF"(s) for all s. T"(s) |F"()" b) Let T(s) = F'(s) and N Define a new vector field B(s) = T(s) × N (s). Prove that ||B(s)|| = 1. Note: T, N, and B form a mutually perpendicular trihedron of unit vectors along r called the Frenet frame. This frame is crucial to the study of the geometry of curves and is used in the study of objects moving in space.

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Vectors In Two And Three Dimensions
Section9.FOM: Focus On Modeling: Vectors Fields
Problem 12P
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7. Let r: R → R" be a path parameterized by arclength - i.e. the speed is a constant one. a) Prove that r (s) I7"(s) for all s. b) Let T(s) = r'(s) and N T"(s) |F"(s)|| Define a new vector field B(s) = T(s) × N(s). Prove %3D that ||B(s)|| = 1. Note: T, N, and B form a mutually perpendicular trihedron of unit vectors along r called the Frenet frame. This frame is crucial to the study of the geometry of curves and is used in the study of objects moving in space.
Transcribed Image Text:7. Let r: R → R" be a path parameterized by arclength - i.e. the speed is a constant one. a) Prove that r (s) I7"(s) for all s. b) Let T(s) = r'(s) and N T"(s) |F"(s)|| Define a new vector field B(s) = T(s) × N(s). Prove %3D that ||B(s)|| = 1. Note: T, N, and B form a mutually perpendicular trihedron of unit vectors along r called the Frenet frame. This frame is crucial to the study of the geometry of curves and is used in the study of objects moving in space.
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