{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 9 "Math 4530" }}{PARA 257 "" 0 "" {TEXT -1 30 "Homework for Friday February 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Useful procedures are posted on our lecture page, at " }{URLLINK 17 "http://www.math.utah.edu/~koreva ar/4530spring05/4530lectures.html" 4 "http://www.math.utah.edu/~koreva ar/4530spring05/4530lectures.html" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 495 "1) Rework exercise 1.4.7 using Maple; this was a messy hand calculation to compute the curvature, torsion, \+ T,N,B for a specific curve which was not parameterized by arclength. I n \"curtor\" are already written procedures to compute curvature and t orsion. You can modify these to write procedures which will compute T , N, B. Note, that since it is easy to compute T, and also B, as the \+ normalized cross product of the first and second derivatives of the cu rve, you can get N by taking B cross T." }}{PARA 0 "" 0 "" {TEXT -1 239 " After all those computations, use Maple to draw a nice pictur e of the curve, say for t between zero and one. You can decide whethe r you prefer the \"plot3d\" or \"tubeplot\" commands. (See below for a n example of what \"tubeplot\" creates.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 173 "2) Exercise 1.7.8, for kap3 and k ap4: Use the plane-curve recreation template which I've posted, or cre ate your own procedure, to recreate nice pictures of these two curves. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "3) \+ Exercises 1.7.7 and 1.5.15: Again use the plane-curve recreation temp late which I've posted or your own procedure, to recreate a curve with curvature " }{XPPEDIT 18 0 "kappa(s) = 1/(1+s^2);" "6#/-%&kappaG6#% \"sG*&\"\"\"F),&F)F)*$F'\"\"#F)!\"\"" }{TEXT -1 466 " , so that at s=0 the curve is horizontal and at the origin. Now, the real fun begins. \+ Using the plane- Frenet system which we talked about in class (see al so exercise 1.5.13 on page 41), you can find an explict formula for th e curve by antidifferentiation (this is the exercise 1.5.15 part). Pr ove that the curve you've created is actually a catenary (graph of y=c osh(x) or Euclidean motion of such)! (We saw the catenary before as t he shape of a hanging cable.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 188 "4) Use the procedure \"recreate3dview\" , which is one of the files posted on our web page, to create a 3-d cu rve which is beautiful, with your own choice of curvature and torsion \+ functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "5) Here is a picture of part of an a=2, b=1 helix, sitting on \+ a radius 2 cylinder:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with (plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "cylinder:=\npl ot3d([2*cos(theta),2*sin(theta),z],theta=0..2*Pi,z=0..2*Pi,\nstyle=wir eframe,color=black):\n#this is a parametric way of drawing the cylinde r" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "helix2:=tubeplot([2*c os(t),2*sin(t),t],\nt=0..2*Pi,radius=0.1,color=black):\n#this is a cor responding piece of the helix" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 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" }}{PARA 0 " " 0 "" {TEXT -1 91 " (Recall exercise 1.3.28 from last week.) Add thi s osculating circle to the display above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "11 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }