regular smooth curve - Penn Math

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Problem 17. Let �: I � R 3 be a regular curve, not necessarily parametrized by arc length. (a) Show that the curvature of � is given by �(t) = |�' � �"| / |�'| 3 . (b) If the curvature is nonzero, so that the torsion is well-defined, show that the torsion is given by �(t) = (�' � �") � �''' / |�' � �"| 2 . 32
Definitions. A closed plane curve is a regular curve �: [a, b] � R 2 such that � and all its derivatives agree at a and at b , that is, �(a) = �(b) , �'(a) = �'(b) , �"(a) = �"(b) , ... . Alternatively, one can use the entire real line as domain, �: R � R 2 , and require that � be periodic of some period L > 0 , that is, �(t + L) = �(t) for all t � R . Another alternative: one can use a circle (of any radius) as the domain for a closed curve. 33
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: Problem 15. Let �: I � R 3 be a
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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