regular smooth curve - Penn Math

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Problem 18. (a) Show that the total curvature of a regular closed curve in the plane is 2n� for some integer n . (b) Show that if the regular closed curve is simple, then n = +1 or —1 . (c) Suppose that a regular closed curve in the plane has curvature which is strictly positive or strictly negative, and that the above integer n equals +1 or —1 . Show that the curve is simple. 36
Let �: S 1 � R 2 be a regular closed curve in the plane. For each point � � S 1 , the unit tangent vector T(�) to the curve at the point �(�) is given by T(�) = �'(�) / |�'(�)| . Thus T: S 1 � S 1 , and then the induced map T* : �1(S 1 ) � �1(S 1 ) is a group homorphism from the integers to the integers, and hence is multiplication by some integer n , which we call the degree of the map T , or the winding number or rotation index of the curve � . 37
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 and 30: The Frenet equations T'(s) = �(s)
Page 31 and 32: Problem 15. Let �: I � R 3 be a
Page 33 and 34: Definitions. A closed plane curve i
Page 35: If �: [a, b] � R 2 is a regular
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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