regular smooth curve - Penn Math

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Volumes of tubes...two problems. (1) Show that the area of a tube of radius � about a simple closed curve of length L in the plane is 2� L . (2) Show that the volume of a tube of radius � about a simple closed curve of length L in 3-space is � � 2 L . We will solve both of these problems, and the Frenet equations for curves will be our main tool. 40
Tubes about circles in the plane. The simplest example is that of a tube of radius � about a circle of radius r in the plane, so just an annulus between concentric circles of radii r + � and r � � , with area � (r + �) 2 � � (r � �) 2 = � 4r� = (2�r) (2�) = (circumference of circle) (width of tube) 41
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 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: Remark. If �0 and �1 : S 1 �
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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