regular smooth curve - Penn Math

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Once the point �(s0) and the initial orthonormal frame T(s0) , N(s0) , B(s0) is picked, the curve is unique. Thus any other such curve �: I � R 3 differs from � by a rigid motion of R 3 . This completes the proof of the fundamental theorem of the local theory of curves in R 3 . 30
Problem 15. Let �: I � R 3 be a regular curve with nowhere vanishing curvature. Assume that all the principal normal lines of � pass through a fixed point in R 3 . Prove that the image of � lies on a circle. Problem 16. Let r = r(�) , a � � � b , describe a plane curve in polar coordinates. (a) Show that the arc length of this curve is given by �a b [r 2 + (r') 2 ] 1/2 d� . (b) Show that the curvature is given by �(�) = [2(r') 2 — r r" + r 2 ] / [(r') 2 + r 2 ] 3/2 . 31
Page 1 and 2: Math 501 - Differential Geometry Pr
Page 3 and 4: Examples. (1) The helix �(t) = (a
Page 5 and 6: Definition. A smooth curve �: I
Page 7 and 8: Let �: I � R 3 be a regular (sm
Page 9 and 10: Problem 5. A circular disk of radiu
Page 11 and 12: Problem 7. Let �: I � R 3 be pa
Page 13 and 14: Let �: I � R 3 be parametrized
Page 15 and 16: Problem 12. Let �: I � R 3 be p
Page 17 and 18: THEOREM. Given smooth functions �
Page 19 and 20: Proof of the fundamental theorem of
Page 21 and 22: Now the fundamental existence and u
Page 23 and 24: Let's pause to check that the natur
Page 25 and 26: These six quantities satisfy a syst
Page 27 and 28: Where are we so far? We have proved
Page 29: The Frenet equations T'(s) = �(s)
Page 33 and 34: Definitions. A closed plane curve i
Page 35 and 36: If �: [a, b] � R 2 is a regular
Page 37 and 38: Let �: S 1 � R 2 be a regular c
Page 39 and 40: Remark. If �0 and �1 : S 1 �
Page 41 and 42: Tubes about circles in the plane. T
Page 43 and 44: To produce the �-tube about this
Page 45 and 46: Tubes about any curve in 3-space. P
Page 47 and 48: Hence the volume of the �-tube ab
Page 49 and 50: (b) Defining the �-tube about our
Page 51 and 52: Tubes about round spheres in 3-spac
Page 53: Problem. Compute the volume of the

regular smooth curve - Penn Math

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