regular smooth curve - Penn Math

More documents

Recommendations

Info

$Homework 7 solutions - Penn Math$

$Curriculum Vitae Alexandru A. Popa - Penn Math - University of ...$

$Slides - Penn Math$

$Free-Response - Penn Math$

$A mathematical trivium - Penn Math$

$1 Vector Spaces - Penn Math$

$Math 240 Calculus, Part III Text: Zill, Dennis and Cullen, Michael ...$

$Math 241 Calculus, Part IV (4h. 1c.u.) Text: Zill, Dennis and Cullen ...$

Problem. To get a Frenet frame along a curve in R 3 , one needs to assume that the curvature �(s) never vanishes. Without this hypothesis, one can still prove that vol �-tube = (length of curve) (area of �-disk) . (a) Let T(s) , A(s) , B(s) be an ON frame along our curve x(s) . Show that the Frenet eqns are replaced by T'(s) = �(s) A(s) + �(s) B(s) A'(s) = � �(s) T(s) + �(s) B(s) B'(s) = � �(s) T(s) � �(s) A(s) . 48
(b) Defining the �-tube about our curve x(s) by F(s, t, u) = x(s) + t A(s) + u B(s) , with T(s) = x'(s) , and hence A(s) and B(s) orthogonal to the curve, show that we get vol �-tube = �s �t 2+u 2
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: Hence the volume of the �-tube ab
Page 51 and 52: Tubes about round spheres in 3-spac
Page 53: Problem. Compute the volume of the

regular smooth curve - Penn Math

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?