regular smooth curve - Penn Math

More documents

Recommendations

Info

$Homework 7 solutions - Penn Math$

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

$Slides - Penn Math$

$Free-Response - Penn Math$

$A mathematical trivium - Penn Math$

$1 Vector Spaces - Penn Math$

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

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

In this chapter, we will be dealing with smooth curves �: I � R 3 , where I = (a, b) is an open interval in the real line R 3 , allowing a = — � or b = + � . Do Carmo calls these "parametrized differentiable curves", to emphasize that the specific function � is part of the definition. Thus �(t) = (cos t, sin t) and �(t) = (cos 2t, sin 2t) are considered to be different curves in the plane, even though their images are the same circle. 2
Examples. (1) The helix �(t) = (a cos t, a sint t, bt) , t � R (2) �(t) = (t 3 , t 2 ) . Problem 1. Let �(t) be a smooth curve which does not pass through the origin. If �(t0) is the point of its image which is closest to the origin (assuming such a point exists), and if �'(t0) � 0 , show that the position vector �(t0) is orthogonal to the velocity vector �'(t0) . 3
Page 1: Math 501 - Differential Geometry Pr
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 and 52: Tubes about round spheres in 3-spac
Page 53:
Problem. Compute the volume of the
show all

regular smooth curve - Penn Math

Create successful ePaper yourself

Delete template?

Save as template?