regular smooth curve - Penn Math

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Problem 4. If �: [a, b] � R 3 is just continuous, and we attempt to define the arc length of the image �[a, b] to be the LUB of the lengths of all inscribed polygonal paths, show that this LUB may be infinite. By contrast, show that if � is of class C 1 (meaning that it has a first derivative �'(t) which is continuous), then this LUB is finite and equals �a b |�'(t)| dt . 6
Let �: I � R 3 be a regular (smooth) curve. Then the arc length along � , starting from some point �(t0) , is given by t s(t) = �to |�'(t)| dt . Note that s'(t) = |�'(t)| � 0 , so we can invert this function to obtain t = t(s) . Then �(s) = �(t(s)) is a reparametrization of our curve, and |�'(s)| = 1. We will say that � is parametrized by arc length. 7
Page 1 and 2: Math 501 - Differential Geometry Pr
Page 3 and 4: Examples. (1) The helix �(t) = (a
Page 5: Definition. A smooth curve �: I
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 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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