regular smooth curve - Penn Math

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Problem 8. Show that the curvature of a circle is the reciprocal of its radius. Let �: I � R 3 be parametrized by arc length. When the curvature �(s) � 0 , the unit vector is well-defined. N(s) = �"(s) / |�"(s)| Problem 9. Show that the unit vector N(s) is normal to the curve, in the sense that N(s) � T(s) = 0 , where T(s) is the unit tangent vector to the curve. Definition. When �(s) � 0 , we call N(s) the principal normal vector to the curve. 12
Let �: I � R 3 be parametrized by arc length, and let T(s) be the unit tangent vector along � . If the curvature �(s) � 0 , then we also have the principal normal vector N(s) at �(s) . In that case, define the binormal vector B(s) to � at s by the vector cross product, B(s) = T(s) � N(s) . Problem 10. Show that B'(s) is parallel to N(s) . 13
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: Problem 7. Let �: I � R 3 be pa
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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