regular smooth curve - Penn Math

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Problem 6. Let �: [a, b] � R 3 be a parametrized <strong>curve</strong>, and set �(a) = p and �(b) = q . (1) Show that for any constant vector V with |V| = 1 , b b (q — p) � V = �a �'(t) � V dt � �a |�'(t)| dt . (2) Set V = (q — p) / |q — p| and conclude that b |�(b) — �(a)| � �a |�'(t)| dt . This shows that the <strong>curve</strong> of shortest length from �(a) to �(b) is the straight line segment joining these points. 10
Problem 7. Let �: I � R 3 be parametrized by arc length. Thus the tangent vector �'(s) has unit length. Show that the norm |�"(s)| of the second derivative measures the rate of change of the angle which neighboring tangents make with the tangent at s . Definition. If �: I � R 3 is parametrized by arc length, then the number �(s) = |�"(s)| is called the curvature of � at s . 11
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: Problem 5. A circular disk of radiu
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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