lect11_4

lect11_4 - Tangent Vector and Curvature - (11.4) 1. Tangent...

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Tangent Vector and Curvature - (11.4) 1. Tangent Vector Let C be the curve traced out by the vector-valued function r ! ! t " "# f ! t " , g ! t " , h ! t " $ . The vector T ! ! t " " 1 r ! % ! t " r ! % ! t " is the unit tangent vector to the curve C . Observe that r ! % ! t " " r ! % ! t " # r ! % ! t " " f % ! t " 2 & g % ! t " 2 & h % ! t " 2 . We know the arc length of C for a ! t ! b is L " " a b f % ! t " 2 & g % ! t " 2 & h % ! t " 2 dt " " a b r ! % ! t " dt Example Let r ! ! t " "# 4 cos ! t " , sin ! t " , t $ . Find the unit vector to the curve traced out by r ! ! t " . Sketch the unit tangent vectors at the points respectively when t " 0, t " " 6 and t " " 2 . Evaluate the length of arc of the curve for 0 ! t ! " 2 . r ! % ! t " "# # 4sin ! t " , cos ! t " ,1 $ . || r ! ! t " || " 16sin 2 ! t " & cos 2 ! t " & 1 T ! ! t " " 1 r ! % ! t " r ! % ! t " " 1 2 ! t " & cos 2 ! t " & 1 # # ! t " , cos ! t " $ T ! ! 0 " " 1 1 & 1 # 0,1,1 $"# 0, 1 2 , 1 2 $ T " 6 " 1 4 & 3 4 & 1 # # 2, 3 2 $"# # 4 23 , 3 23 , 2 23 $ T " 2 " 1 16 & 1 # # 4,0,1 $"# # 4 17 ,0 , 1 17 $ 0 2 4 -1 -0.5 0.5 1 y -4 -2 2 4 t 1
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s " " 0 2 " ! # 4sin ! t "" 2 & ! cos ! t 2 & 1 dt " 18. 45342 the perimeter of the ellipse x 2 4 & y 2 " 1: " 0 2 " ! # ! t 2 & ! cos ! t 2 & 0 dt " 17. 15684 2. Curvature Let s ! t " be the length of arc of the curve C traced out by r ! ! t " "# f ! t " , g ! t " , h ! t " $ for t be between a and b .
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This note was uploaded on 02/05/2012 for the course MATH 2142 taught by Professor Lerna during the Fall '10 term at FIU.

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lect11_4 - Tangent Vector and Curvature - (11.4) 1. Tangent...

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