regular smooth curve - Penn Math

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The "tube" of radius � about a round sphere of radius r in 3-space is just the region between the concentric spheres of radii r + � and r � � , with volume 4/3 � (r + �) 3 � 4/3 � (r � �) 3 = 4/3 � (6 r 2 � + 2 � 3 ) = (4 � r 2 ) 2� + 8/3 � � 3 = (area of sphere) 2� + 8/3 � � 3 = 2� (area of sphere + 2�/3 �(sphere) � 2 ) , which is exactly "Weyl's tube formula" for surfaces in R 3 . 52
Problem. Compute the volume of the �-tube about a torus of revolution in 3-space, and show that it is vol �-tube = 2� (area of torus) = 2� (area of torus + 2�/3 �(torus) � 2 ) , since �(torus) = 0 , again in accord with Weyl's tube formula. 53
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 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: Tubes about round spheres in 3-spac

regular smooth curve - Penn Math

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