(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, "Times"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Times"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, "Times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, "Times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, "Times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 14, "Times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = leftheader, inactive, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Times"; fontset = leftfooter, inactive, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] Chapter 2(2) :[font = section; inactive; Cclosed; preserveAspect; startGroup] Problem 13 :[font = text; inactive; preserveAspect] Sections of the solution surface (x,t,u) of the Example 2.3 are shown in Figure 2.8 (with was not produced with Mathematica. To produce the solution surface, we generate the characteristics emanating from the initial function: :[font = input; preserveAspect] f[x_]:=If[Abs[x]<1,1-x^2,0] :[font = input; preserveAspect] x[t_,x0_]:=x0+f[x0]t :[font = text; inactive; preserveAspect] The solution surface is then generated with ParametricPlot3D: :[font = input; preserveAspect; startGroup] ParametricPlot3D[{x[t,x0],t,f[x0]},{x0,-2,2},{t,0,2}, BoxRatios->{1,1,1},PlotPoints->30] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 276; pictureHeight = 300] %! %%Creator: Mathematica %%AspectRatio: 1.08079 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0260524 1.10184 0.0275459 1.10184 [ [(-2)] .11326 .24476 .97192 1 Msboxa [(-1)] .23087 .19508 .88712 1 Msboxa [(0)] .35479 .14275 .80233 1 Msboxa [(1)] .48553 .08758 .71753 1 Msboxa [(2)] .62368 .02932 .63273 1 Msboxa [(0)] .03657 .80845 .15454 -1 Msboxa 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.14867 .24797 L s P p .001 w .169 .23524 m .17183 .23825 L s P p .001 w .19243 .22539 m .19523 .22843 L s P p .001 w .2161 .21543 m .21887 .2185 L s P p .001 w .26417 .19522 m .26689 .19834 L s P p .001 w .28859 .18495 m .29127 .1881 L s P p .001 w .31327 .17458 m .31592 .17775 L s P p .001 w .3382 .16409 m .34082 .16729 L s P p .001 w .38888 .14279 m .39143 .14604 L s P p .001 w .41463 .13196 m .41714 .13524 L s P p .001 w .44065 .12102 m .44313 .12432 L s P p .001 w .46696 .10995 m .46941 .11328 L s P p .001 w .52045 .08746 m .52282 .09085 L s P p .001 w .54764 .07603 m .54997 .07944 L s P p .001 w .57513 .06447 m .57742 .06791 L s P p .001 w .60293 .05278 m .60518 .05625 L s P p .001 w .65948 .02901 m .66164 .03253 L s P P p p .002 w .02755 .78701 m .40586 1.05324 L s P p .002 w .03867 .79484 m .03972 .78803 L s P [(0)] .03657 .80845 .15454 -1 Mshowa p .002 w .1433 .86846 m .14411 .86163 L s P [(0.5)] .14168 .88214 .11851 -1 Mshowa p .002 w .23715 .93451 m .23775 .92765 L s P [(1)] .23595 .94823 .08756 -1 Mshowa p .002 w .32182 .99409 m .32224 .98722 L s P [(1.5)] .32098 1.00784 .0607 -1 Mshowa p .002 w .39858 1.04811 m .39884 1.04123 L s P [(2)] .39807 1.06188 .03715 -1 Mshowa p .001 w .06055 .81023 m .06115 .80614 L s P p .001 w .08193 .82528 m .0825 .82119 L s P p .001 w .10284 .83999 m .10338 .8359 L s P p .001 w .12329 .85439 m .12381 .85029 L s P p .001 w .16288 .88224 m .16334 .87814 L s P p .001 w .18204 .89573 m .18247 .89162 L s P p .001 w .2008 .90893 m .20121 .90482 L s P p .001 w .21916 .92185 m .21955 .91774 L s P p .001 w .25478 .94692 m .25511 .9428 L s P p .001 w .27204 .95907 m .27236 .95495 L s P p .001 w .28897 .97098 m .28926 .96685 L s P p .001 w .30556 .98265 m .30583 .97853 L s P p .001 w .33777 1.00532 m .338 1.00119 L s P p .001 w .35341 1.01632 m .35362 1.0122 L s P p .001 w .36875 1.02712 m .36894 1.023 L s P p .001 w .38381 1.03772 m .38398 1.03359 L s P P p p .002 w .11113 .25957 m .02755 .78701 L s P p .002 w .10947 .27006 m .11582 .2674 L s P [(0)] .09676 .27537 1 -0.41829 Mshowa p .002 w .09199 .38033 m .0984 .3778 L s P [(0.25)] .07918 .38539 1 -0.39531 Mshowa p .002 w .07298 .50028 m .07944 .49789 L s P [(0.5)] .06007 .50506 1 -0.37011 Mshowa p .002 w .05223 .63126 m .05874 .62903 L s P [(0.75)] .0392 .63572 1 -0.34234 Mshowa p .002 w .02947 .77484 m .03605 .77279 L s P [(1)] .01632 .77894 1 -0.3116 Mshowa p .001 w .10609 .2914 m .1099 .28982 L s P p .001 w .10265 .31308 m .10647 .31152 L s P p .001 w .09916 .33513 m .10299 .33358 L s P p .001 w .0956 .35754 m .09944 .356 L s P p .001 w .08832 .40351 m .09217 .402 L s P p .001 w .08458 .42708 m .08844 .42559 L s P p .001 w .08078 .45106 m .08465 .44959 L s P p .001 w .07692 .47546 m .08079 .47401 L s P p .001 w .06898 .52555 m .07286 .52413 L s P p .001 w .0649 .55127 m .06879 .54987 L s P p .001 w .06075 .57745 m .06465 .57607 L s P p .001 w .05653 .60411 m .06043 .60275 L s P p .001 w .04785 .65891 m .05176 .65759 L s P p .001 w .04338 .68708 m .0473 .68578 L s P p .001 w .03883 .71578 m .04276 .71451 L s P p .001 w .0342 .74503 m .03814 .74378 L s P P p P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] No Input Form was saved for this expression. ;[o] -Graphics3D- :[font = section; inactive; Cclosed; preserveAspect; startGroup] Problem 16 :[font = text; inactive; preserveAspect] This problem can be solved by using the Mathematica commands in Problem 12. Since we already determined xi and eta to be xi=x/y and eta=x/u, we have :[font = input; Cclosed; preserveAspect; startGroup] Clear[x,y,u,s,t,xi,eta,c1,c2,r] xi = x/y; eta = x/u; {y,u}={y,u}/. Solve[{xi==c1,eta==c2},{y,u}][[1]]; Solve[{x,y,u}=={s^2,s,s^3},{c1,c2},x] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{c1 -> s, c2 -> s^(-1)}} ;[o] 1 {{c1 -> s, c2 -> -}} s :[font = section; inactive; Cclosed; preserveAspect; startGroup] Problem 18 :[font = text; inactive; preserveAspect] The two "branches" of the solution is given by u+(x,t) and u-(x,t). The only allowed solution in the limit of small t is u+(x,t) if the initial conditions are to be satisfied. Fot sufficiently large values of t, both u+(x,t) and u-(x,t) are allowed solutions. To see the relationship between the the appearance of double-valuedness in the solution and both u+(x,t) and u-(x,t) being allowed solutions, we will plot both so9lutions as a function of time, with u-(x,t) indicated by a thicker line: :[font = input; preserveAspect] uPlus[x_,t_] := (1/(2t^2))(2x t-1+Sqrt[1-4x t+4t^2]); uMinus[x_,t_] := (1/(2t^2))(2x t-1-Sqrt[1-4x t+4t^2]); :[font = input; preserveAspect] Solution[y_]:=Plot[{uPlus[x,y],uMinus[x,y]}, {x,-1,(1/(4y))+y},AxesLabel->{x,t}, PlotRange->{-0.5,1},PlotStyle-> {Thickness[0.001],Thickness[0.01]}] :[font = input; preserveAspect; startGroup] Solution[0.4] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.494121 0.470312 0.206011 0.412023 [ [(-1)] .02381 .20601 0 2 Msboxa [(-0.5)] .25897 .20601 0 2 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.43674 m .49787 .43674 L s P p .001 w .49412 .46971 m .49787 .46971 L s P p .001 w .49412 .48619 m .49787 .48619 L s P p .001 w .49412 .50267 m .49787 .50267 L s P p .001 w .49412 .51915 m .49787 .51915 L s P p .001 w .49412 .55211 m .49787 .55211 L s P p .001 w .49412 .56859 m .49787 .56859 L s P p .001 w .49412 .58507 m .49787 .58507 L s P p .001 w .49412 .60155 m .49787 .60155 L s P p .001 w .49412 .02472 m .49787 .02472 L s P p .001 w .49412 .00824 m .49787 .00824 L s P [(t)] .49412 .61803 0 -4 Mshowa p .002 w .49412 0 m .49412 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .001 w .02381 .20601 m .06349 .24412 L .10317 .28117 L .14286 .31706 L .18254 .35172 L .22222 .38506 L .2619 .41697 L .30159 .44734 L .34127 .47601 L .38095 .50283 L .42063 .52761 L .46032 .55013 L .5 .57013 L .53968 .58727 L .57937 .60116 L .59921 .60674 L .61905 .6113 L .62897 .61318 L .63889 .61477 L .64881 .61606 L .65377 .61659 L .65873 .61704 L .66369 .61741 L .66865 .6177 L .67113 .61781 L .67361 .6179 L .67485 .61793 L .67609 .61796 L .67733 .61799 L .67857 .61801 L .67981 .61802 L .68105 .61803 L .68229 .61803 L .68353 .61803 L .68477 .61802 L .68601 .61801 L .68725 .61799 L .68849 .61796 L .69097 .61789 L .69345 .6178 L .69841 .61753 L .70337 .61717 L .70833 .61671 L .71825 .61546 L .72817 .61376 L .7381 .6116 L .75794 .60572 L .77778 .59754 L .79762 .58671 L .81746 .57277 L Mistroke .8373 .55512 L .85714 .53292 L .87698 .50498 L .89683 .46947 L .91667 .42329 L .93651 .36042 L .94643 .31877 L .95139 .29393 L .95635 .26531 L .96131 .23136 L .96627 .18906 L .96875 .1628 L .97123 .13065 L .97247 .11095 L .97371 .08709 L .97495 .05516 L Mfstroke .9758 0 m .97495 .05516 L s s P P p p .01 w s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Solution[0.5] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.47619 0.206011 0.412023 [ [(-1)] .02381 .20601 0 2 Msboxa [(-0.5)] .2619 .20601 0 2 Msboxa [(0.5)] .7381 .20601 0 2 Msboxa [(1)] .97619 .20601 0 2 Msboxa [(x)] 1.025 .20601 -1 0 Msboxa [(-0.4)] .4875 .0412 1 0 Msboxa [(-0.2)] .4875 .12361 1 0 Msboxa [(0.2)] .4875 .28842 1 0 Msboxa [(0.4)] .4875 .37082 1 0 Msboxa [(0.6)] .4875 .45322 1 0 Msboxa [(0.8)] .4875 .53563 1 0 Msboxa [(1)] .4875 .61803 1 0 Msboxa [(t)] .5 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .20601 m .02381 .21226 L s P [(-1)] .02381 .20601 0 2 Mshowa p .002 w .2619 .20601 m .2619 .21226 L s P [(-0.5)] .2619 .20601 0 2 Mshowa p .002 w .7381 .20601 m .7381 .21226 L s P [(0.5)] .7381 .20601 0 2 Mshowa p .002 w .97619 .20601 m .97619 .21226 L s P [(1)] .97619 .20601 0 2 Mshowa p .001 w .07143 .20601 m .07143 .20976 L s P p .001 w .11905 .20601 m .11905 .20976 L s P p .001 w .16667 .20601 m .16667 .20976 L s P p .001 w .21429 .20601 m .21429 .20976 L s P p .001 w .30952 .20601 m .30952 .20976 L s P p .001 w .35714 .20601 m .35714 .20976 L s P p .001 w .40476 .20601 m .40476 .20976 L s P p .001 w .45238 .20601 m .45238 .20976 L s P p .001 w .54762 .20601 m .54762 .20976 L s P p .001 w .59524 .20601 m .59524 .20976 L s P p .001 w .64286 .20601 m .64286 .20976 L s P p .001 w .69048 .20601 m .69048 .20976 L s P p .001 w .78571 .20601 m .78571 .20976 L s P p .001 w .83333 .20601 m .83333 .20976 L s P p .001 w .88095 .20601 m .88095 .20976 L s P p .001 w .92857 .20601 m .92857 .20976 L s P [(x)] 1.025 .20601 -1 0 Mshowa p .002 w 0 .20601 m 1 .20601 L s P p .002 w .5 .0412 m .50625 .0412 L s P [(-0.4)] .4875 .0412 1 0 Mshowa p .002 w .5 .12361 m .50625 .12361 L s P [(-0.2)] .4875 .12361 1 0 Mshowa p .002 w .5 .28842 m .50625 .28842 L s P [(0.2)] .4875 .28842 1 0 Mshowa p .002 w .5 .37082 m .50625 .37082 L s P [(0.4)] .4875 .37082 1 0 Mshowa p .002 w .5 .45322 m .50625 .45322 L s P [(0.6)] .4875 .45322 1 0 Mshowa p .002 w .5 .53563 m .50625 .53563 L s P [(0.8)] .4875 .53563 1 0 Mshowa p .002 w .5 .61803 m .50625 .61803 L s P [(1)] .4875 .61803 1 0 Mshowa p .001 w .5 .05768 m .50375 .05768 L s P p .001 w .5 .07416 m .50375 .07416 L s P p .001 w .5 .09064 m .50375 .09064 L s P p .001 w .5 .10713 m .50375 .10713 L s P p .001 w .5 .14009 m .50375 .14009 L s P p .001 w .5 .15657 m .50375 .15657 L s P p .001 w .5 .17305 m .50375 .17305 L s P p .001 w .5 .18953 m .50375 .18953 L s P p .001 w .5 .22249 m .50375 .22249 L s P p .001 w .5 .23897 m .50375 .23897 L s P p .001 w .5 .25545 m .50375 .25545 L s P p .001 w .5 .27193 m .50375 .27193 L s P p .001 w .5 .3049 m .50375 .3049 L s P p .001 w .5 .32138 m .50375 .32138 L s P p .001 w .5 .33786 m .50375 .33786 L s P p .001 w .5 .35434 m .50375 .35434 L s P p .001 w .5 .3873 m .50375 .3873 L s P p .001 w .5 .40378 m .50375 .40378 L s P p .001 w .5 .42026 m .50375 .42026 L s P p .001 w .5 .43674 m .50375 .43674 L s P p .001 w .5 .46971 m .50375 .46971 L s P p .001 w .5 .48619 m .50375 .48619 L s P p .001 w .5 .50267 m .50375 .50267 L s P p .001 w .5 .51915 m .50375 .51915 L s P p .001 w .5 .55211 m .50375 .55211 L s P p .001 w .5 .56859 m .50375 .56859 L s P p .001 w .5 .58507 m .50375 .58507 L s P p .001 w .5 .60155 m .50375 .60155 L s P p .001 w .5 .02472 m .50375 .02472 L s P p .001 w .5 .00824 m .50375 .00824 L s P [(t)] .5 .61803 0 -4 Mshowa p .002 w .5 0 m .5 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .001 w .02381 .20601 m .06349 .23998 L .10317 .27319 L .14286 .30558 L .18254 .3371 L .22222 .36767 L .2619 .39723 L .30159 .42569 L .34127 .45294 L .38095 .47888 L .42063 .50338 L .46032 .52626 L .5 .54734 L .53968 .5664 L .57937 .58314 L .61905 .59722 L .63889 .60312 L .65873 .60817 L .67857 .61229 L .68849 .61398 L .69841 .61539 L .70833 .61652 L .71329 .61697 L .71825 .61735 L .72321 .61764 L .72569 .61776 L .72817 .61786 L .73065 .61793 L .73189 .61797 L .73313 .61799 L .73438 .61801 L .73562 .61802 L .73686 .61803 L .7381 .61803 L .73934 .61803 L .74058 .61802 L .74182 .61801 L .74306 .61799 L .74554 .61793 L .74802 .61785 L .7505 .61775 L .75298 .61762 L .75794 .61729 L .7629 .61685 L .76786 .61632 L .77778 .61491 L .7877 .61303 L .79762 .61064 L .81746 .60416 L .8373 .59504 L Mistroke .85714 .58269 L .87698 .56625 L .89683 .54443 L .91667 .51503 L .92659 .4963 L .93651 .47376 L .94643 .44585 L .95139 .42905 L .95635 .40956 L .96131 .38627 L .96627 .35705 L .96875 .33881 L .97123 .31637 L .97247 .30258 L .97371 .28582 L .97495 .26334 L .97619 .20601 L Mfstroke P P p p .01 w s s s s s s s s s s s s s s s s s s s s s s s s s s s s .96406 0 m .96627 .02064 L s .96627 .02064 m .96875 .04746 L .97123 .07849 L .97247 .09657 L .97371 .11762 L .97495 .1444 L .97619 .20601 L s P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] Solution[0.6] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.496065 0.472255 0.206011 0.412023 [ [(-1)] .02381 .20601 0 2 Msboxa [(-0.5)] .25994 .20601 0 2 Msboxa [(0.5)] .73219 .20601 0 2 Msboxa [(1)] .96832 .20601 0 2 Msboxa [(x)] 1.025 .20601 -1 0 Msboxa [(-0.4)] .48356 .0412 1 0 Msboxa [(-0.2)] .48356 .12361 1 0 Msboxa [(0.2)] .48356 .28842 1 0 Msboxa [(0.4)] .48356 .37082 1 0 Msboxa [(0.6)] .48356 .45322 1 0 Msboxa [(0.8)] .48356 .53563 1 0 Msboxa [(1)] .48356 .61803 1 0 Msboxa [(t)] .49606 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .20601 m .02381 .21226 L s P [(-1)] .02381 .20601 0 2 Mshowa p .002 w .25994 .20601 m .25994 .21226 L s P [(-0.5)] .25994 .20601 0 2 Mshowa p .002 w .73219 .20601 m .73219 .21226 L s P [(0.5)] .73219 .20601 0 2 Mshowa p .002 w .96832 .20601 m .96832 .21226 L s P [(1)] .96832 .20601 0 2 Mshowa p .001 w .07104 .20601 m .07104 .20976 L s P p .001 w .11826 .20601 m .11826 .20976 L s P p .001 w .16549 .20601 m .16549 .20976 L s P p .001 w .21271 .20601 m .21271 .20976 L s P p .001 w .30716 .20601 m .30716 .20976 L s P p .001 w .35439 .20601 m .35439 .20976 L s P p .001 w .40161 .20601 m .40161 .20976 L s P p .001 w .44884 .20601 m .44884 .20976 L s P p .001 w .54329 .20601 m .54329 .20976 L s P p .001 w .59052 .20601 m .59052 .20976 L s P p .001 w .63774 .20601 m .63774 .20976 L s P p .001 w .68497 .20601 m .68497 .20976 L s P p .001 w .77942 .20601 m .77942 .20976 L s P p .001 w .82664 .20601 m .82664 .20976 L s P p .001 w .87387 .20601 m .87387 .20976 L s P p .001 w .92109 .20601 m .92109 .20976 L s P [(x)] 1.025 .20601 -1 0 Mshowa p .002 w 0 .20601 m 1 .20601 L s P p .002 w .49606 .0412 m .50231 .0412 L s P [(-0.4)] .48356 .0412 1 0 Mshowa p .002 w .49606 .12361 m .50231 .12361 L s P [(-0.2)] .48356 .12361 1 0 Mshowa p .002 w .49606 .28842 m .50231 .28842 L s P [(0.2)] .48356 .28842 1 0 Mshowa p .002 w .49606 .37082 m .50231 .37082 L s P [(0.4)] .48356 .37082 1 0 Mshowa p .002 w .49606 .45322 m .50231 .45322 L s P [(0.6)] .48356 .45322 1 0 Mshowa p .002 w .49606 .53563 m .50231 .53563 L s P [(0.8)] .48356 .53563 1 0 Mshowa p .002 w .49606 .61803 m .50231 .61803 L s P [(1)] .48356 .61803 1 0 Mshowa p .001 w .49606 .05768 m .49981 .05768 L s P p .001 w .49606 .07416 m .49981 .07416 L s P p .001 w .49606 .09064 m .49981 .09064 L s P p .001 w .49606 .10713 m .49981 .10713 L s P p .001 w .49606 .14009 m .49981 .14009 L s P p .001 w .49606 .15657 m .49981 .15657 L s P p .001 w .49606 .17305 m .49981 .17305 L s P p .001 w .49606 .18953 m .49981 .18953 L s P p .001 w .49606 .22249 m .49981 .22249 L s P p .001 w .49606 .23897 m .49981 .23897 L s P p .001 w .49606 .25545 m .49981 .25545 L s P p .001 w .49606 .27193 m .49981 .27193 L s P p .001 w .49606 .3049 m .49981 .3049 L s P p .001 w .49606 .32138 m .49981 .32138 L s P p .001 w .49606 .33786 m .49981 .33786 L s P p .001 w .49606 .35434 m .49981 .35434 L s P p .001 w .49606 .3873 m .49981 .3873 L s P p .001 w .49606 .40378 m .49981 .40378 L s P p .001 w .49606 .42026 m .49981 .42026 L s P p .001 w .49606 .43674 m .49981 .43674 L s P p .001 w .49606 .46971 m .49981 .46971 L s P p .001 w .49606 .48619 m .49981 .48619 L s P p .001 w .49606 .50267 m .49981 .50267 L s P p .001 w .49606 .51915 m .49981 .51915 L s P p .001 w .49606 .55211 m .49981 .55211 L s P p .001 w .49606 .56859 m .49981 .56859 L s P p .001 w .49606 .58507 m .49981 .58507 L s P p .001 w .49606 .60155 m .49981 .60155 L s P p .001 w .49606 .02472 m .49981 .02472 L s P p .001 w .49606 .00824 m .49981 .00824 L s P [(t)] .49606 .61803 0 -4 Mshowa p .002 w .49606 0 m .49606 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .001 w .02381 .20601 m .06349 .23721 L .10317 .26782 L .14286 .29781 L .18254 .32713 L .22222 .35573 L .2619 .38356 L .30159 .41054 L .34127 .43661 L .38095 .46167 L .42063 .48562 L .46032 .50835 L .5 .5297 L .53968 .5495 L .57937 .56754 L .61905 .58354 L .65873 .59715 L .67857 .60292 L .69841 .60791 L .71825 .61203 L .72817 .61373 L .7381 .61517 L .74802 .61634 L .75298 .61682 L .75794 .61723 L .7629 .61755 L .76786 .61779 L .77034 .61789 L .77158 .61792 L .77282 .61795 L .77406 .61798 L .7753 .618 L .77654 .61802 L .77778 .61803 L .77902 .61803 L .78026 .61803 L .7815 .61803 L .78274 .61801 L .78398 .618 L .78522 .61797 L .7877 .6179 L .79018 .61781 L .79266 .6177 L .79762 .61739 L .80258 .61698 L .80754 .61646 L .81746 .61507 L .82738 .61317 L .8373 .61072 L .84722 .60766 L Mistroke .85714 .60391 L .87698 .59398 L .89683 .57993 L .90675 .57089 L .91667 .56009 L .92659 .5471 L .93651 .53119 L .94643 .51118 L .95139 .49901 L .95635 .48477 L .96131 .46764 L .96627 .44597 L .96875 .43237 L .97123 .41555 L .97247 .40518 L .97371 .39255 L .97495 .37553 L .97619 .33191 L Mfstroke P P p p .01 w s s s s s s s s s s s s s s s s s s s s s s s s .93256 0 m .93651 .01722 L s .93651 .01722 m .94643 .06608 L .95635 .12134 L .96131 .1529 L .96627 .18899 L .96875 .20981 L .97123 .23384 L .97247 .24781 L .97371 .26405 L .97495 .28468 L .97619 .33191 L s P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = text; inactive; preserveAspect; endGroup] This shows that the onset of multivaluedness in the solution coincides with both the positive and negative branches of the solution being in the "physical" region, i.e., positive values of u. :[font = section; inactive; Cclosed; preserveAspect; startGroup] Problem 20 :[font = text; inactive; preserveAspect] We proceed as in Example 2.3. The initial profile and characteristics are given by :[font = input; preserveAspect] f[x_]:=Which[x<0,1,x>1,0,True,1-x] :[font = input; preserveAspect] x[t_,x0_]:=x0+f[x0]t :[font = text; inactive; preserveAspect] The profiles of the solution are then defined with ParametricPlot: :[font = input; preserveAspect] u[y_]:=ParametricPlot[{x[y,x0],f[x0]},{x0,-2,2}, AxesLabel->{x,t}] :[font = text; inactive; preserveAspect] The solution can now be displayed at various times: :[font = input; preserveAspect; startGroup] u[0] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.238095 0.0147151 0.588604 [ [(-2)] .02381 .01472 0 2 Msboxa [(-1)] .2619 .01472 0 2 Msboxa [(1)] .7381 .01472 0 2 Msboxa [(2)] .97619 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(0.2)] .4875 .13244 1 0 Msboxa [(0.4)] .4875 .25016 1 0 Msboxa [(0.6)] .4875 .36788 1 0 Msboxa [(0.8)] .4875 .4856 1 0 Msboxa [(1)] .4875 .60332 1 0 Msboxa [(t)] .5 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .01472 m .02381 .02097 L s P [(-2)] .02381 .01472 0 2 Mshowa p .002 w .2619 .01472 m .2619 .02097 L s P [(-1)] .2619 .01472 0 2 Mshowa p .002 w .7381 .01472 m .7381 .02097 L s P [(1)] .7381 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(2)] .97619 .01472 0 2 Mshowa p .001 w .07143 .01472 m .07143 .01847 L s P p .001 w .11905 .01472 m .11905 .01847 L s P p .001 w .16667 .01472 m .16667 .01847 L s P p .001 w .21429 .01472 m .21429 .01847 L s P p .001 w .30952 .01472 m .30952 .01847 L s P p .001 w .35714 .01472 m .35714 .01847 L s P p .001 w .40476 .01472 m .40476 .01847 L s P p .001 w .45238 .01472 m .45238 .01847 L s P p .001 w .54762 .01472 m .54762 .01847 L s P p .001 w .59524 .01472 m .59524 .01847 L s P p .001 w .64286 .01472 m .64286 .01847 L s P p .001 w .69048 .01472 m .69048 .01847 L s P p .001 w .78571 .01472 m .78571 .01847 L s P p .001 w .83333 .01472 m .83333 .01847 L s P p .001 w .88095 .01472 m .88095 .01847 L s P p .001 w .92857 .01472 m .92857 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P p .002 w .5 .13244 m .50625 .13244 L s P [(0.2)] .4875 .13244 1 0 Mshowa p .002 w .5 .25016 m .50625 .25016 L s P [(0.4)] .4875 .25016 1 0 Mshowa p .002 w .5 .36788 m .50625 .36788 L s P [(0.6)] .4875 .36788 1 0 Mshowa p .002 w .5 .4856 m .50625 .4856 L s P [(0.8)] .4875 .4856 1 0 Mshowa p .002 w .5 .60332 m .50625 .60332 L s P [(1)] .4875 .60332 1 0 Mshowa p .001 w .5 .03826 m .50375 .03826 L s P p .001 w .5 .0618 m .50375 .0618 L s P p .001 w .5 .08535 m .50375 .08535 L s P p .001 w .5 .10889 m .50375 .10889 L s P p .001 w .5 .15598 m .50375 .15598 L s P p .001 w .5 .17952 m .50375 .17952 L s P p .001 w .5 .20307 m .50375 .20307 L s P p .001 w .5 .22661 m .50375 .22661 L s P p .001 w .5 .2737 m .50375 .2737 L s P p .001 w .5 .29724 m .50375 .29724 L s P p .001 w .5 .32079 m .50375 .32079 L s P p .001 w .5 .34433 m .50375 .34433 L s P p .001 w .5 .39142 m .50375 .39142 L s P p .001 w .5 .41497 m .50375 .41497 L s P p .001 w .5 .43851 m .50375 .43851 L s P p .001 w .5 .46205 m .50375 .46205 L s P p .001 w .5 .50914 m .50375 .50914 L s P p .001 w .5 .53269 m .50375 .53269 L s P p .001 w .5 .55623 m .50375 .55623 L s P p .001 w .5 .57977 m .50375 .57977 L s P [(t)] .5 .61803 0 -4 Mshowa p .002 w .5 0 m .5 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .60332 m .06349 .60332 L .10317 .60332 L .14286 .60332 L .18254 .60332 L .22222 .60332 L .2619 .60332 L .30159 .60332 L .34127 .60332 L .38095 .60332 L .42063 .60332 L .44048 .60332 L .46032 .60332 L .47024 .60332 L .48016 .60332 L .48512 .60332 L .49008 .60332 L .49256 .60332 L .49504 .60332 L .49628 .60332 L .49752 .60332 L .49876 .60332 L .5 .60332 L .50248 .59719 L .50496 .59106 L .50992 .57879 L .51984 .55427 L .53968 .50522 L .57937 .40712 L .61905 .30902 L .65873 .21092 L .69841 .11282 L .71825 .06377 L .72817 .03924 L .73313 .02698 L .73562 .02085 L .73686 .01778 L .7381 .01472 L .73934 .01472 L .74058 .01472 L .74306 .01472 L .74802 .01472 L .75794 .01472 L .77778 .01472 L .81746 .01472 L .85714 .01472 L .89683 .01472 L .93651 .01472 L .97619 .01472 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] u[0.5] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.431973 0.272109 0.0147151 0.588604 [ [(-1.5)] .02381 .01472 0 2 Msboxa [(-1)] .15986 .01472 0 2 Msboxa [(-0.5)] .29592 .01472 0 2 Msboxa [(0.5)] .56803 .01472 0 2 Msboxa [(1)] .70408 .01472 0 2 Msboxa [(1.5)] .84014 .01472 0 2 Msboxa [(2)] .97619 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(0.2)] .41947 .13244 1 0 Msboxa [(0.4)] .41947 .25016 1 0 Msboxa [(0.6)] .41947 .36788 1 0 Msboxa [(0.8)] .41947 .4856 1 0 Msboxa [(1)] .41947 .60332 1 0 Msboxa [(t)] .43197 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .01472 m .02381 .02097 L s P [(-1.5)] .02381 .01472 0 2 Mshowa p .002 w .15986 .01472 m .15986 .02097 L s P [(-1)] .15986 .01472 0 2 Mshowa p .002 w .29592 .01472 m .29592 .02097 L s P [(-0.5)] .29592 .01472 0 2 Mshowa p .002 w .56803 .01472 m .56803 .02097 L s P [(0.5)] .56803 .01472 0 2 Mshowa p .002 w .70408 .01472 m .70408 .02097 L s P [(1)] .70408 .01472 0 2 Mshowa p .002 w .84014 .01472 m .84014 .02097 L s P [(1.5)] .84014 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(2)] .97619 .01472 0 2 Mshowa p .001 w .05102 .01472 m .05102 .01847 L s P p .001 w .07823 .01472 m .07823 .01847 L s P p .001 w .10544 .01472 m .10544 .01847 L s P p .001 w .13265 .01472 m .13265 .01847 L s P p .001 w .18707 .01472 m .18707 .01847 L s P p .001 w .21429 .01472 m .21429 .01847 L s P p .001 w .2415 .01472 m .2415 .01847 L s P p .001 w .26871 .01472 m .26871 .01847 L s P p .001 w .32313 .01472 m .32313 .01847 L s P p .001 w .35034 .01472 m .35034 .01847 L s P p .001 w .37755 .01472 m .37755 .01847 L s P p .001 w .40476 .01472 m .40476 .01847 L s P p .001 w .45918 .01472 m .45918 .01847 L s P p .001 w .48639 .01472 m .48639 .01847 L s P p .001 w .51361 .01472 m .51361 .01847 L s P p .001 w .54082 .01472 m .54082 .01847 L s P p .001 w .59524 .01472 m .59524 .01847 L s P p .001 w .62245 .01472 m .62245 .01847 L s P p .001 w .64966 .01472 m .64966 .01847 L s P p .001 w .67687 .01472 m .67687 .01847 L s P p .001 w .73129 .01472 m .73129 .01847 L s P p .001 w .7585 .01472 m .7585 .01847 L s P p .001 w .78571 .01472 m .78571 .01847 L s P p .001 w .81293 .01472 m .81293 .01847 L s P p .001 w .86735 .01472 m .86735 .01847 L s P p .001 w .89456 .01472 m .89456 .01847 L s P p .001 w .92177 .01472 m .92177 .01847 L s P p .001 w .94898 .01472 m .94898 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P p .002 w .43197 .13244 m .43822 .13244 L s P [(0.2)] .41947 .13244 1 0 Mshowa p .002 w .43197 .25016 m .43822 .25016 L s P [(0.4)] .41947 .25016 1 0 Mshowa p .002 w .43197 .36788 m .43822 .36788 L s P [(0.6)] .41947 .36788 1 0 Mshowa p .002 w .43197 .4856 m .43822 .4856 L s P [(0.8)] .41947 .4856 1 0 Mshowa p .002 w .43197 .60332 m .43822 .60332 L s P [(1)] .41947 .60332 1 0 Mshowa p .001 w .43197 .03826 m .43572 .03826 L s P p .001 w .43197 .0618 m .43572 .0618 L s P p .001 w .43197 .08535 m .43572 .08535 L s P p .001 w .43197 .10889 m .43572 .10889 L s P p .001 w .43197 .15598 m .43572 .15598 L s P p .001 w .43197 .17952 m .43572 .17952 L s P p .001 w .43197 .20307 m .43572 .20307 L s P p .001 w .43197 .22661 m .43572 .22661 L s P p .001 w .43197 .2737 m .43572 .2737 L s P p .001 w .43197 .29724 m .43572 .29724 L s P p .001 w .43197 .32079 m .43572 .32079 L s P p .001 w .43197 .34433 m .43572 .34433 L s P p .001 w .43197 .39142 m .43572 .39142 L s P p .001 w .43197 .41497 m .43572 .41497 L s P p .001 w .43197 .43851 m .43572 .43851 L s P p .001 w .43197 .46205 m .43572 .46205 L s P p .001 w .43197 .50914 m .43572 .50914 L s P p .001 w .43197 .53269 m .43572 .53269 L s P p .001 w .43197 .55623 m .43572 .55623 L s P p .001 w .43197 .57977 m .43572 .57977 L s P [(t)] .43197 .61803 0 -4 Mshowa p .002 w .43197 0 m .43197 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .60332 m .06916 .60332 L .11451 .60332 L .15986 .60332 L .20522 .60332 L .25057 .60332 L .29592 .60332 L .34127 .60332 L .38662 .60332 L .43197 .60332 L .47732 .60332 L .5 .60332 L .52268 .60332 L .53401 .60332 L .54535 .60332 L .55102 .60332 L .55669 .60332 L .55952 .60332 L .56236 .60332 L .56378 .60332 L .56519 .60332 L .56661 .60332 L .56803 .60332 L .56944 .59719 L .57086 .59106 L .5737 .57879 L .57937 .55427 L .5907 .50522 L .61338 .40712 L .63605 .30902 L .65873 .21092 L .68141 .11282 L .69274 .06377 L .69841 .03924 L .70125 .02698 L .70266 .02085 L .70337 .01778 L .70408 .01472 L .7055 .01472 L .70692 .01472 L .70975 .01472 L .71542 .01472 L .72676 .01472 L .74943 .01472 L .79478 .01472 L .84014 .01472 L .88549 .01472 L .93084 .01472 L .97619 .01472 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] u[1] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.34127 0.31746 0.0147151 0.588604 [ [(-1)] .02381 .01472 0 2 Msboxa [(-0.5)] .18254 .01472 0 2 Msboxa [(0.5)] .5 .01472 0 2 Msboxa [(1)] .65873 .01472 0 2 Msboxa [(1.5)] .81746 .01472 0 2 Msboxa [(2)] .97619 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(0.2)] .32877 .13244 1 0 Msboxa [(0.4)] .32877 .25016 1 0 Msboxa [(0.6)] .32877 .36788 1 0 Msboxa [(0.8)] .32877 .4856 1 0 Msboxa [(1)] .32877 .60332 1 0 Msboxa [(t)] .34127 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .01472 m .02381 .02097 L s P [(-1)] .02381 .01472 0 2 Mshowa p .002 w .18254 .01472 m .18254 .02097 L s P [(-0.5)] .18254 .01472 0 2 Mshowa p .002 w .5 .01472 m .5 .02097 L s P [(0.5)] .5 .01472 0 2 Mshowa p .002 w .65873 .01472 m .65873 .02097 L s P [(1)] .65873 .01472 0 2 Mshowa p .002 w .81746 .01472 m .81746 .02097 L s P [(1.5)] .81746 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(2)] .97619 .01472 0 2 Mshowa p .001 w .05556 .01472 m .05556 .01847 L s P p .001 w .0873 .01472 m .0873 .01847 L s P p .001 w .11905 .01472 m .11905 .01847 L s P p .001 w .15079 .01472 m .15079 .01847 L s P p .001 w .21429 .01472 m .21429 .01847 L s P p .001 w .24603 .01472 m .24603 .01847 L s P p .001 w .27778 .01472 m .27778 .01847 L s P p .001 w .30952 .01472 m .30952 .01847 L s P p .001 w .37302 .01472 m .37302 .01847 L s P p .001 w .40476 .01472 m .40476 .01847 L s P p .001 w .43651 .01472 m .43651 .01847 L s P p .001 w .46825 .01472 m .46825 .01847 L s P p .001 w .53175 .01472 m .53175 .01847 L s P p .001 w .56349 .01472 m .56349 .01847 L s P p .001 w .59524 .01472 m .59524 .01847 L s P p .001 w .62698 .01472 m .62698 .01847 L s P p .001 w .69048 .01472 m .69048 .01847 L s P p .001 w .72222 .01472 m .72222 .01847 L s P p .001 w .75397 .01472 m .75397 .01847 L s P p .001 w .78571 .01472 m .78571 .01847 L s P p .001 w .84921 .01472 m .84921 .01847 L s P p .001 w .88095 .01472 m .88095 .01847 L s P p .001 w .9127 .01472 m .9127 .01847 L s P p .001 w .94444 .01472 m .94444 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P p .002 w .34127 .13244 m .34752 .13244 L s P [(0.2)] .32877 .13244 1 0 Mshowa p .002 w .34127 .25016 m .34752 .25016 L s P [(0.4)] .32877 .25016 1 0 Mshowa p .002 w .34127 .36788 m .34752 .36788 L s P [(0.6)] .32877 .36788 1 0 Mshowa p .002 w .34127 .4856 m .34752 .4856 L s P [(0.8)] .32877 .4856 1 0 Mshowa p .002 w .34127 .60332 m .34752 .60332 L s P [(1)] .32877 .60332 1 0 Mshowa p .001 w .34127 .03826 m .34502 .03826 L s P p .001 w .34127 .0618 m .34502 .0618 L s P p .001 w .34127 .08535 m .34502 .08535 L s P p .001 w .34127 .10889 m .34502 .10889 L s P p .001 w .34127 .15598 m .34502 .15598 L s P p .001 w .34127 .17952 m .34502 .17952 L s P p .001 w .34127 .20307 m .34502 .20307 L s P p .001 w .34127 .22661 m .34502 .22661 L s P p .001 w .34127 .2737 m .34502 .2737 L s P p .001 w .34127 .29724 m .34502 .29724 L s P p .001 w .34127 .32079 m .34502 .32079 L s P p .001 w .34127 .34433 m .34502 .34433 L s P p .001 w .34127 .39142 m .34502 .39142 L s P p .001 w .34127 .41497 m .34502 .41497 L s P p .001 w .34127 .43851 m .34502 .43851 L s P p .001 w .34127 .46205 m .34502 .46205 L s P p .001 w .34127 .50914 m .34502 .50914 L s P p .001 w .34127 .53269 m .34502 .53269 L s P p .001 w .34127 .55623 m .34502 .55623 L s P p .001 w .34127 .57977 m .34502 .57977 L s P [(t)] .34127 .61803 0 -4 Mshowa p .002 w .34127 0 m .34127 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .60332 m .07672 .60332 L .12963 .60332 L .18254 .60332 L .23545 .60332 L .28836 .60332 L .34127 .60332 L .39418 .60332 L .44709 .60332 L .5 .60332 L .55291 .60332 L .57937 .60332 L .60582 .60332 L .61905 .60332 L .63228 .60332 L .63889 .60332 L .6455 .60332 L .64881 .60332 L .65212 .60332 L .65377 .60332 L .65542 .60332 L .65708 .60332 L .65873 .60332 L .65873 .59719 L .65873 .59106 L .65873 .57879 L .65873 .55427 L .65873 .50522 L .65873 .40712 L .65873 .30902 L .65873 .21092 L .65873 .11282 L .65873 .06377 L .65873 .03924 L .65873 .02698 L .65873 .02085 L .65873 .01778 L .65873 .01472 L .66038 .01472 L .66204 .01472 L .66534 .01472 L .67196 .01472 L .68519 .01472 L .71164 .01472 L .76455 .01472 L .81746 .01472 L .87037 .01472 L .92328 .01472 L .97619 .01472 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = input; preserveAspect; startGroup] u[1.5] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.214286 0.380952 0.0147151 0.588604 [ [(-0.5)] .02381 .01472 0 2 Msboxa [(0.5)] .40476 .01472 0 2 Msboxa [(1)] .59524 .01472 0 2 Msboxa [(1.5)] .78571 .01472 0 2 Msboxa [(2)] .97619 .01472 0 2 Msboxa [(x)] 1.025 .01472 -1 0 Msboxa [(0.2)] .20179 .13244 1 0 Msboxa [(0.4)] .20179 .25016 1 0 Msboxa [(0.6)] .20179 .36788 1 0 Msboxa [(0.8)] .20179 .4856 1 0 Msboxa [(1)] .20179 .60332 1 0 Msboxa [(t)] .21429 .61803 0 -4 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .01472 m .02381 .02097 L s P [(-0.5)] .02381 .01472 0 2 Mshowa p .002 w .40476 .01472 m .40476 .02097 L s P [(0.5)] .40476 .01472 0 2 Mshowa p .002 w .59524 .01472 m .59524 .02097 L s P [(1)] .59524 .01472 0 2 Mshowa p .002 w .78571 .01472 m .78571 .02097 L s P [(1.5)] .78571 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(2)] .97619 .01472 0 2 Mshowa p .001 w .0619 .01472 m .0619 .01847 L s P p .001 w .1 .01472 m .1 .01847 L s P p .001 w .1381 .01472 m .1381 .01847 L s P p .001 w .17619 .01472 m .17619 .01847 L s P p .001 w .25238 .01472 m .25238 .01847 L s P p .001 w .29048 .01472 m .29048 .01847 L s P p .001 w .32857 .01472 m .32857 .01847 L s P p .001 w .36667 .01472 m .36667 .01847 L s P p .001 w .44286 .01472 m .44286 .01847 L s P p .001 w .48095 .01472 m .48095 .01847 L s P p .001 w .51905 .01472 m .51905 .01847 L s P p .001 w .55714 .01472 m .55714 .01847 L s P p .001 w .63333 .01472 m .63333 .01847 L s P p .001 w .67143 .01472 m .67143 .01847 L s P p .001 w .70952 .01472 m .70952 .01847 L s P p .001 w .74762 .01472 m .74762 .01847 L s P p .001 w .82381 .01472 m .82381 .01847 L s P p .001 w .8619 .01472 m .8619 .01847 L s P p .001 w .9 .01472 m .9 .01847 L s P p .001 w .9381 .01472 m .9381 .01847 L s P [(x)] 1.025 .01472 -1 0 Mshowa p .002 w 0 .01472 m 1 .01472 L s P p .002 w .21429 .13244 m .22054 .13244 L s P [(0.2)] .20179 .13244 1 0 Mshowa p .002 w .21429 .25016 m .22054 .25016 L s P [(0.4)] .20179 .25016 1 0 Mshowa p .002 w .21429 .36788 m .22054 .36788 L s P [(0.6)] .20179 .36788 1 0 Mshowa p .002 w .21429 .4856 m .22054 .4856 L s P [(0.8)] .20179 .4856 1 0 Mshowa p .002 w .21429 .60332 m .22054 .60332 L s P [(1)] .20179 .60332 1 0 Mshowa p .001 w .21429 .03826 m .21804 .03826 L s P p .001 w .21429 .0618 m .21804 .0618 L s P p .001 w .21429 .08535 m .21804 .08535 L s P p .001 w .21429 .10889 m .21804 .10889 L s P p .001 w .21429 .15598 m .21804 .15598 L s P p .001 w .21429 .17952 m .21804 .17952 L s P p .001 w .21429 .20307 m .21804 .20307 L s P p .001 w .21429 .22661 m .21804 .22661 L s P p .001 w .21429 .2737 m .21804 .2737 L s P p .001 w .21429 .29724 m .21804 .29724 L s P p .001 w .21429 .32079 m .21804 .32079 L s P p .001 w .21429 .34433 m .21804 .34433 L s P p .001 w .21429 .39142 m .21804 .39142 L s P p .001 w .21429 .41497 m .21804 .41497 L s P p .001 w .21429 .43851 m .21804 .43851 L s P p .001 w .21429 .46205 m .21804 .46205 L s P p .001 w .21429 .50914 m .21804 .50914 L s P p .001 w .21429 .53269 m .21804 .53269 L s P p .001 w .21429 .55623 m .21804 .55623 L s P p .001 w .21429 .57977 m .21804 .57977 L s P [(t)] .21429 .61803 0 -4 Mshowa p .002 w .21429 0 m .21429 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .60332 m .0873 .60332 L .15079 .60332 L .21429 .60332 L .27778 .60332 L .34127 .60332 L .40476 .60332 L .46825 .60332 L .53175 .60332 L .59524 .60332 L .65873 .60332 L .69048 .60332 L .72222 .60332 L .7381 .60332 L .75397 .60332 L .7619 .60332 L .76984 .60332 L .77381 .60332 L .77778 .60332 L .77976 .60332 L .78175 .60332 L .78373 .60332 L .78571 .60332 L .78373 .59719 L .78175 .59106 L .77778 .57879 L .76984 .55427 L .75397 .50522 L .72222 .40712 L .69048 .30902 L .65873 .21092 L .62698 .11282 L .61111 .06377 L .60317 .03924 L .59921 .02698 L .59722 .02085 L .59623 .01778 L .59524 .01472 L .59722 .01472 L .59921 .01472 L .60317 .01472 L .61111 .01472 L .62698 .01472 L .65873 .01472 L .72222 .01472 L .78571 .01472 L .84921 .01472 L .9127 .01472 L .97619 .01472 L s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] No Input Form was saved for this expression. ;[o] -Graphics- :[font = text; inactive; preserveAspect] These graphs show that the onset of multivaluedness occurs at t=1. The behavior of the characteristics can be seen by constructing the characteristic that emanates from the point x0: :[font = input; preserveAspect] X[x0_]:=ParametricPlot[{x[y,x0],y},{y,0,2},AxesLabel->{x,t}, DisplayFunction->Identity] :[font = input; preserveAspect; startGroup] Characteristics={X[-0.5],X[-0.25],X[0],X[0.25],X[0.5], X[0.75],X[1],X[1.25],X[1.5],X[1.75],X[2]} :[font = output; output; inactive; preserveAspect; endGroup] {Graphics[{{Line[{{-0.5, 0.}, {-0.4166666666666667, 0.0833333333333333}, {-0.3333333333333334, 0.1666666666666666}, {-0.25, 0.25}, {-0.1666666666666667, 0.3333333333333333}, {-0.0833333333333334, 0.4166666666666666}, {-(5.551115123125783*10^-17), 0.5}, {0.0833333333333333, 0.5833333333333333}, {0.1666666666666666, 0.6666666666666666}, {0.25, 0.75}, {0.3333333333333334, 0.833333333333333}, {0.4166666666666668, 0.916666666666667}, {0.5, 1.}, {0.5833333333333333, 1.083333333333333}, {0.6666666666666666, 1.166666666666666}, {0.7499999999999999, 1.25}, {0.833333333333333, 1.333333333333333}, {0.916666666666666, 1.416666666666666}, {1., 1.5}, {1.083333333333333, 1.583333333333333}, {1.166666666666666, 1.666666666666666}, {1.249999999999999, 1.749999999999999}, {1.333333333333333, 1.833333333333333}, {1.416666666666666, 1.916666666666666}, {1.5, 2.}}]}}, {PlotRange -> Automatic, AspectRatio -> GoldenRatio^(-1), DisplayFunction :> Identity, ColorOutput -> Automatic, Axes -> Automatic, AxesOrigin -> Automatic, PlotLabel -> None, AxesLabel -> {x, t}, Ticks -> Automatic, GridLines -> None, Prolog -> {}, Epilog -> {}, AxesStyle -> Automatic, Background -> Automatic, DefaultColor -> Automatic, DefaultFont :> $DefaultFont, RotateLabel -> True, Frame -> False, FrameStyle -> Automatic, FrameTicks -> Automatic, FrameLabel -> None, PlotRegion -> Automatic}], Graphics[{{Line[{{-0.25, 0.}, {-0.1666666666666667, 0.0833333333333333}, {-0.0833333333333334, 0.1666666666666666}, {0., 0.25}, {0.0833333333333333, 0.3333333333333333}, {0.1666666666666666, 0.4166666666666666}, {0.25, 0.5}, {0.3333333333333333, 0.5833333333333333}, {0.4166666666666666, 0.6666666666666666}, {0.5, 0.75}, {0.5833333333333334, 0.833333333333333}, {0.6666666666666668, 0.916666666666667}, {0.75, 1.}, {0.833333333333333, 1.083333333333333}, {0.916666666666666, 1.166666666666666}, {1., 1.25}, {1.083333333333333, 1.333333333333333}, {1.166666666666666, 1.416666666666666}, {1.25, 1.5}, {1.333333333333333, 1.583333333333333}, {1.416666666666666, 1.666666666666666}, {1.499999999999999, 1.749999999999999}, {1.583333333333333, 1.833333333333333}, {1.666666666666666, 1.916666666666666}, {1.75, 2.}}]}}, {PlotRange -> Automatic, AspectRatio -> GoldenRatio^(-1), DisplayFunction :> Identity, ColorOutput -> Automatic, Axes -> Automatic, AxesOrigin -> Automatic, PlotLabel -> None, AxesLabel -> {x, t}, Ticks -> Automatic, GridLines -> None, Prolog -> {}, Epilog -> {}, AxesStyle -> Automatic, Background -> Automatic, DefaultColor -> Automatic, DefaultFont :> $DefaultFont, RotateLabel -> True, Frame -> False, FrameStyle -> Automatic, FrameTicks -> Automatic, FrameLabel -> None, PlotRegion -> Automatic}], Graphics[{{Line[{{0., 0.}, {0.0833333333333333, 0.0833333333333333}, {0.1666666666666666, 0.1666666666666666}, {0.25, 0.25}, {0.3333333333333333, 0.3333333333333333}, {0.4166666666666666, 0.4166666666666666}, {0.5, 0.5}, {0.5833333333333333, 0.5833333333333333}, {0.6666666666666666, 0.6666666666666666}, {0.75, 0.75}, {0.833333333333333, 0.833333333333333}, {0.916666666666667, 0.916666666666667}, {1., 1.}, {1.083333333333333, 1.083333333333333}, {1.166666666666666, 1.166666666666666}, {1.25, 1.25}, {1.333333333333333, 1.333333333333333}, {1.416666666666666, 1.416666666666666}, {1.5, 1.5}, {1.583333333333333, 1.583333333333333}, {1.666666666666666, 1.666666666666666}, {1.749999999999999, 1.749999999999999}, {1.833333333333333, 1.833333333333333}, {1.916666666666666, 1.916666666666666}, {2., 2.}}]}}, {PlotRange -> Automatic, AspectRatio -> GoldenRatio^(-1), DisplayFunction :> Identity, ColorOutput -> Automatic, Axes -> Automatic, AxesOrigin -> Automatic, PlotLabel -> None, AxesLabel -> {x, t}, Ticks -> Automatic, GridLines -> None, Prolog -> {}, Epilog -> {}, AxesStyle -> Automatic, Background -> Automatic, DefaultColor -> Automatic, DefaultFont :> $DefaultFont, RotateLabel -> True, Frame -> False, FrameStyle -> Automatic, FrameTicks -> Automatic, FrameLabel -> None, PlotRegion -> Automatic}], Graphics[{{Line[{{0.25, 0.}, {0.3125, 0.0833333333333333}, {0.375, 0.1666666666666666}, {0.4375, 0.25}, {0.5, 0.3333333333333333}, {0.5625, 0.4166666666666666}, {0.625, 0.5}, {0.6875, 0.5833333333333333}, {0.75, 0.6666666666666666}, {0.8125, 0.75}, {0.875, 0.833333333333333}, {0.9375, 0.916666666666667}, {1., 1.}, {1.0625, 1.083333333333333}, {1.125, 1.166666666666666}, {1.1875, 1.25}, {1.25, 1.333333333333333}, {1.3125, 1.416666666666666}, {1.375, 1.5}, {1.4375, 1.583333333333333}, {1.5, 1.666666666666666}, {1.5625, 1.749999999999999}, {1.625, 1.833333333333333}, {1.687499999999999, 1.916666666666666}, {1.75, 2.}}]}}, {PlotRange -> Automatic, AspectRatio -> GoldenRatio^(-1), DisplayFunction :> Identity, ColorOutput -> Automatic, Axes -> Automatic, AxesOrigin -> Automatic, PlotLabel -> None, AxesLabel -> {x, t}, Ticks -> Automatic, GridLines -> None, Prolog -> {}, Epilog -> {}, AxesStyle -> Automatic, Background -> Automatic, DefaultColor -> Automatic, DefaultFont :> $DefaultFont, RotateLabel -> True, Frame -> False, FrameStyle -> Automatic, FrameTicks -> Automatic, FrameLabel -> None, PlotRegion -> Automatic}], Graphics[{{Line[{{0.5, 0.}, {0.5416666666666666, 0.0833333333333333}, {0.5833333333333334, 0.1666666666666666}, {0.625, 0.25}, {0.6666666666666666, 0.3333333333333333}, {0.7083333333333333, 0.4166666666666666}, {0.75, 0.5}, {0.7916666666666666, 0.5833333333333333}, {0.833333333333333, 0.6666666666666666}, {0.875, 0.75}, {0.916666666666667, 0.833333333333333}, {0.958333333333333, 0.916666666666667}, {1., 1.}, {1.041666666666666, 1.083333333333333}, {1.083333333333333, 1.166666666666666}, {1.125, 1.25}, {1.166666666666666, 1.333333333333333}, {1.208333333333333, 1.416666666666666}, {1.25, 1.5}, {1.291666666666666, 1.583333333333333}, {1.333333333333333, 1.666666666666666}, {1.375, 1.749999999999999}, {1.416666666666666, 1.833333333333333}, {1.458333333333333, 1.916666666666666}, {1.5, 2.}}]}}, {PlotRange -> Automatic, AspectRatio -> GoldenRatio^(-1), DisplayFunction :> Identity, ColorOutput -> Automatic, Axes -> Automatic, AxesOrigin -> Automatic, PlotLabel -> None, AxesLabel -> {x, t}, Ticks -> Automatic, GridLines -> None, Prolog -> {}, Epilog -> {}, AxesStyle -> Automatic, Background -> Automatic, DefaultColor -> Automatic, DefaultFont :> $DefaultFont, RotateLabel -> True, Frame -> False, FrameStyle -> Automatic, FrameTicks -> Automatic, FrameLabel -> None, PlotRegion -> Automatic}], Graphics[{{Line[{{0.75, 0.}, {0.7708333333333334, 0.0833333333333333}, {0.7916666666666666, 0.1666666666666666}, {0.8125, 0.25}, {0.833333333333333, 0.3333333333333333}, {0.854166666666667, 0.4166666666666666}, {0.875, 0.5}, {0.895833333333333, 0.5833333333333333}, {0.916666666666667, 0.6666666666666666}, {0.9375, 0.75}, {0.958333333333333, 0.833333333333333}, {0.979166666666667, 0.916666666666667}, {1., 1.}, {1.020833333333333, 1.083333333333333}, {1.041666666666666, 1.166666666666666}, {1.0625, 1.25}, {1.083333333333333, 1.333333333333333}, {1.104166666666666, 1.416666666666666}, {1.125, 1.5}, {1.145833333333333, 1.583333333333333}, {1.166666666666666, 1.666666666666666}, {1.1875, 1.749999999999999}, {1.208333333333333, 1.833333333333333}, {1.229166666666666, 1.916666666666666}, {1.25, 2.}}]}}, {PlotRange -> Automatic, AspectRatio -> GoldenRatio^(-1), DisplayFunction :> Identity, ColorOutput -> Automatic, Axes -> Automatic, AxesOrigin -> Automatic, PlotLabel -> None, AxesLabel -> {x, t}, Ticks -> Automatic, GridLines -> None, Prolog -> {}, Epilog -> {}, AxesStyle -> Automatic, Background -> Automatic, DefaultColor -> Automatic, DefaultFont :> $DefaultFont, RotateLabel -> True, Frame -> False, FrameStyle -> Automatic, FrameTicks -> Automatic, FrameLabel -> None, PlotRegion -> Automatic}], Graphics[{{Line[{{1., 0.}, {1., 0.0833333333333333}, {1., 0.1666666666666666}, {1., 0.25}, {1., 0.3333333333333333}, {1., 0.4166666666666666}, {1., 0.5}, {1., 0.5833333333333333}, {1., 0.6666666666666666}, {1., 0.75}, {1., 0.833333333333333}, {1., 0.916666666666667}, {1., 1.}, {1., 1.083333333333333}, {1., 1.166666666666666}, {1., 1.25}, {1., 1.333333333333333}, {1., 1.416666666666666}, {1., 1.5}, {1., 1.583333333333333}, {1., 1.666666666666666}, {1., 1.749999999999999}, {1., 1.833333333333333}, {1., 1.916666666666666}, {1., 2.}}]}}, {PlotRange -> Automatic, AspectRatio -> GoldenRatio^(-1), DisplayFunction :> Identity, ColorOutput -> Automatic, Axes -> Automatic, AxesOrigin -> Automatic, PlotLabel -> None, AxesLabel -> {x, t}, Ticks -> Automatic, GridLines -> None, Prolog -> {}, Epilog -> {}, AxesStyle -> Automatic, Background -> Automatic, DefaultColor -> Automatic, DefaultFont :> $DefaultFont, RotateLabel -> True, Frame -> False, FrameStyle -> Automatic, FrameTicks -> Automatic, FrameLabel -> None, PlotRegion -> Automatic}], Graphics[{{Line[{{1.25, 0.}, {1.25, 0.0833333333333333}, {1.25, 0.1666666666666666}, {1.25, 0.25}, {1.25, 0.3333333333333333}, {1.25, 0.4166666666666666}, {1.25, 0.5}, {1.25, 0.5833333333333333}, {1.25, 0.6666666666666666}, {1.25, 0.75}, {1.25, 0.833333333333333}, {1.25, 0.916666666666667}, {1.25, 1.}, {1.25, 1.083333333333333}, {1.25, 1.166666666666666}, {1.25, 1.25}, {1.25, 1.333333333333333}, {1.25, 1.416666666666666}, {1.25, 1.5}, {1.25, 1.583333333333333}, {1.25, 1.666666666666666}, {1.25, 1.749999999999999}, {1.25, 1.833333333333333}, {1.25, 1.916666666666666}, {1.25, 2.}}]}}, {PlotRange -> Automatic, AspectRatio -> GoldenRatio^(-1), DisplayFunction :> Identity, ColorOutput -> Automatic, Axes -> Automatic, AxesOrigin -> Automatic, PlotLabel -> None, AxesLabel -> {x, t}, Ticks -> Automatic, GridLines -> None, Prolog -> {}, Epilog -> {}, AxesStyle -> Automatic, Background -> Automatic, DefaultColor -> Automatic, DefaultFont :> $DefaultFont, RotateLabel -> True, Frame -> False, FrameStyle -> Automatic, FrameTicks -> Automatic, FrameLabel -> None, PlotRegion -> Automatic}], Graphics[{{Line[{{1.5, 0.}, {1.5, 0.0833333333333333}, {1.5, 0.1666666666666666}, {1.5, 0.25}, {1.5, 0.3333333333333333}, {1.5, 0.4166666666666666}, {1.5, 0.5}, {1.5, 0.5833333333333333}, {1.5, 0.6666666666666666}, {1.5, 0.75}, {1.5, 0.833333333333333}, {1.5, 0.916666666666667}, {1.5, 1.}, {1.5, 1.083333333333333}, {1.5, 1.166666666666666}, {1.5, 1.25}, {1.5, 1.333333333333333}, {1.5, 1.416666666666666}, {1.5, 1.5}, {1.5, 1.583333333333333}, {1.5, 1.666666666666666}, {1.5, 1.749999999999999}, {1.5, 1.833333333333333}, {1.5, 1.916666666666666}, {1.5, 2.}}]}}, {PlotRange -> Automatic, AspectRatio -> GoldenRatio^(-1), DisplayFunction :> Identity, ColorOutput -> Automatic, Axes -> Automatic, AxesOrigin -> Automatic, PlotLabel -> None, AxesLabel -> {x, t}, Ticks -> Automatic, GridLines -> None, Prolog -> {}, Epilog -> {}, AxesStyle -> Automatic, Background -> Automatic, DefaultColor -> Automatic, DefaultFont :> $DefaultFont, RotateLabel -> True, Frame -> False, FrameStyle -> Automatic, FrameTicks -> Automatic, FrameLabel -> None, PlotRegion -> Automatic}], Graphics[{{Line[{{1.75, 0.}, {1.75, 0.0833333333333333}, {1.75, 0.1666666666666666}, {1.75, 0.25}, {1.75, 0.3333333333333333}, {1.75, 0.4166666666666666}, {1.75, 0.5}, {1.75, 0.5833333333333333}, {1.75, 0.6666666666666666}, {1.75, 0.75}, {1.75, 0.833333333333333}, {1.75, 0.916666666666667}, {1.75, 1.}, {1.75, 1.083333333333333}, {1.75, 1.166666666666666}, {1.75, 1.25}, {1.75, 1.333333333333333}, {1.75, 1.416666666666666}, {1.75, 1.5}, {1.75, 1.583333333333333}, {1.75, 1.666666666666666}, {1.75, 1.749999999999999}, {1.75, 1.833333333333333}, {1.75, 1.916666666666666}, {1.75, 2.}}]}}, {PlotRange -> Automatic, AspectRatio -> GoldenRatio^(-1), DisplayFunction :> Identity, ColorOutput -> Automatic, Axes -> Automatic, AxesOrigin -> Automatic, PlotLabel -> None, AxesLabel -> {x, t}, Ticks -> Automatic, GridLines -> None, Prolog -> {}, Epilog -> {}, AxesStyle -> Automatic, Background -> Automatic, DefaultColor -> Automatic, DefaultFont :> $DefaultFont, RotateLabel -> True, Frame -> False, FrameStyle -> Automatic, FrameTicks -> Automatic, FrameLabel -> None, PlotRegion -> Automatic}], Graphics[{{Line[{{2., 0.}, {2., 0.0833333333333333}, {2., 0.1666666666666666}, {2., 0.25}, {2., 0.3333333333333333}, {2., 0.4166666666666666}, {2., 0.5}, {2., 0.5833333333333333}, {2., 0.6666666666666666}, {2., 0.75}, {2., 0.833333333333333}, {2., 0.916666666666667}, {2., 1.}, {2., 1.083333333333333}, {2., 1.166666666666666}, {2., 1.25}, {2., 1.333333333333333}, {2., 1.416666666666666}, {2., 1.5}, {2., 1.583333333333333}, {2., 1.666666666666666}, {2., 1.749999999999999}, {2., 1.833333333333333}, {2., 1.916666666666666}, {2., 2.}}]}}, {PlotRange -> Automatic, AspectRatio -> GoldenRatio^(-1), DisplayFunction :> Identity, ColorOutput -> Automatic, Axes -> Automatic, AxesOrigin -> Automatic, PlotLabel -> None, AxesLabel -> {x, t}, Ticks -> Automatic, GridLines -> None, Prolog -> {}, Epilog -> {}, AxesStyle -> Automatic, Background -> Automatic, DefaultColor -> Automatic, DefaultFont :> $DefaultFont, RotateLabel -> True, Frame -> False, FrameStyle -> Automatic, FrameTicks -> Automatic, FrameLabel -> None, PlotRegion -> Automatic}]} ;[o] {-Graphics-, -Graphics-, -Graphics-, -Graphics-, -Graphics-, -Graphics-, -Graphics-, -Graphics-, -Graphics-, -Graphics-, -Graphics-} :[font = input; preserveAspect; startGroup] Show[Characteristics,DisplayFunction->$DisplayFunction] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 299; pictureHeight = 185] %! %%Creator: Mathematica 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L p .672 .728 .891 r F P s P p .002 w .58626 .10381 m .60584 .09575 L .61566 .11058 L p .672 .728 .891 r F P s P p .002 w .65056 .89041 m .63087 .89534 L .61132 .8887 L p .625 .78 .956 r F P s P p .002 w .59176 .89364 m .61132 .8887 L .63087 .89534 L p .625 .78 .956 r F P s P p .002 w .51322 .89022 m .49406 .89515 L .47379 .88851 L p .625 .78 .956 r F P s P p .002 w .45476 .89345 m .47379 .88851 L .49406 .89515 L p .625 .78 .956 r F P s P p .002 w .7632 .73266 m .77692 .80662 L .75836 .79913 L p .875 .562 .453 r F P s P p .002 w .75836 .79913 m .74643 .72457 L .7632 .73266 L p .875 .562 .453 r F P s P p .002 w .68437 .10133 m .66453 .10935 L .65507 .0945 L p .672 .728 .891 r F P s P p .002 w .63529 .10257 m .65507 .0945 L .66453 .10935 L p .672 .728 .891 r F P s P p .002 w .88627 .88908 m .86564 .89402 L .84732 .88736 L p .625 .78 .956 r F P s P p .002 w .82681 .89231 m .84732 .88736 L .86564 .89402 L p .625 .78 .956 r F P s P p .002 w .74882 .88889 m .72872 .89383 L .70967 .88717 L p 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m .35245 .87149 L .33103 .86459 L p .625 .78 .956 r F P s P p .002 w .63572 .06415 m .65588 .05578 L .66551 .07119 L p .672 .728 .891 r F P s P p .002 w .50471 .26308 m .48703 .3352 L .47204 .32287 L p .929 .714 .592 r F P s P p .002 w .47204 .32287 m .49151 .25 L .50471 .26308 L p .929 .714 .592 r F P s P p .002 w .75139 .86518 m .7309 .87031 L .7115 .8634 L p .625 .78 .956 r F P s P p .002 w .69115 .86854 m .7115 .8634 L .7309 .87031 L p .625 .78 .956 r F P s P p .002 w .61131 .86498 m .59138 .87011 L .57122 .8632 L p .625 .78 .956 r F P s P p .002 w .55143 .86834 m .57122 .8632 L .59138 .87011 L p .625 .78 .956 r F P s P p .002 w .47119 .86479 m .45181 .86992 L .4309 .863 L p .625 .78 .956 r F P s P p .002 w .41166 .86814 m .4309 .863 L .45181 .86992 L p .625 .78 .956 r F P s P p .002 w .79527 .81403 m .79286 .8548 L .77356 .84772 L p .857 .609 .563 r F P s P p .002 w .77356 .84772 m .77692 .80662 L .79527 .81403 L p .857 .609 .563 r F P s P p .002 w .5308 .37122 m .5178 .44434 L 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r F P s P p .002 w .56767 .63835 m .57425 .56113 L .59122 .5709 L p .89 .628 .534 r F P s P p .002 w .59098 .84569 m .57084 .85097 L .55027 .84385 L p .625 .78 .956 r F P s P p .002 w .53026 .84915 m .55027 .84385 L .57084 .85097 L p .625 .78 .956 r F P s P p .002 w .65967 .68272 m .66239 .7604 L .64253 .75238 L p .875 .593 .507 r F P s P p .002 w .64253 .75238 m .6417 .67406 L .65967 .68272 L p .875 .593 .507 r F P s P p .002 w .44876 .84548 m .42918 .85077 L .40783 .84364 L p .625 .78 .956 r F P s P p .002 w .3884 .84894 m .40783 .84364 L .42918 .85077 L p .625 .78 .956 r F P s P p .002 w .30649 .84528 m .28749 .85057 L .26536 .84344 L p .625 .78 .956 r F P s P p .002 w .69266 .84405 m .67207 .84935 L .65203 .84221 L p .625 .78 .956 r F P s P p .002 w .63158 .84752 m .65203 .84221 L .67207 .84935 L p .625 .78 .956 r F P s P p .002 w .55027 .84385 m .53026 .84915 L .50944 .842 L p .625 .78 .956 r F P s P p .002 w .48956 .84732 m .50944 .842 L .53026 .84915 L p .625 .78 .956 r F P s P 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.002 w .17634 .75374 m .19628 .74764 L .22042 .75584 L p .625 .78 .956 r F P s P p .002 w .50207 .74811 m .48081 .75421 L .45818 .74599 L p .625 .78 .956 r F P s P p .002 w .43708 .75211 m .45818 .74599 L .48081 .75421 L p .625 .78 .956 r F P s P p .002 w .3492 .74788 m .32861 .75398 L .30507 .74575 L p .625 .78 .956 r F P s P p .002 w .28463 .75187 m .30507 .74575 L .32861 .75398 L p .625 .78 .956 r F P s P p .002 w .49686 .69359 m .47946 .73981 L .45658 .73142 L p .847 .654 .641 r F P s P p .002 w .45658 .73142 m .47504 .68479 L .49686 .69359 L p .847 .654 .641 r F P s P p .002 w .19628 .74764 m .17634 .75374 L .15191 .74551 L p .625 .78 .956 r F P s P p .002 w .13213 .75164 m .15191 .74551 L .17634 .75374 L p .625 .78 .956 r F P s P p .002 w .42911 .57152 m .40798 .65772 L .38507 .64848 L p .868 .64 .591 r F P s P p .002 w .38507 .64848 m .40831 .56149 L .42911 .57152 L p .868 .64 .591 r F P s P p .002 w .45818 .74599 m .43708 .75211 L .41415 .74386 L p .625 .78 .956 r F P s P p 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.47946 .73981 m .45818 .74599 L .43527 .73766 L p .625 .78 .956 r F P s P p .002 w .41415 .74386 m .43527 .73766 L .45818 .74599 L p .625 .78 .956 r F P s P p .002 w .32568 .73958 m .30507 .74575 L .28125 .73743 L p .625 .78 .956 r F P s P p .002 w .2608 .74362 m .28125 .73743 L .30507 .74575 L p .625 .78 .956 r F P s P p .002 w .47504 .68479 m .45658 .73142 L .43343 .72292 L p .846 .656 .646 r F P s P p .002 w .43343 .72292 m .45296 .67587 L .47504 .68479 L p .846 .656 .646 r F P s P p .002 w .43527 .73766 m .41415 .74386 L .39094 .73551 L p .625 .78 .956 r F P s P p .002 w .36998 .74172 m .39094 .73551 L .41415 .74386 L p .625 .78 .956 r F P s P p .002 w .28125 .73743 m .2608 .74362 L .23668 .73527 L p .625 .78 .956 r F P s P p .002 w .21639 .74148 m .23668 .73527 L .2608 .74362 L p .625 .78 .956 r F P s P p .002 w .40831 .56149 m .38507 .64848 L .36188 .63912 L p .867 .643 .597 r F P s P p .002 w .36188 .63912 m .38724 .55133 L .40831 .56149 L p .867 .643 .597 r F P s P p .002 w .39094 .73551 m .36998 .74172 L .34648 .73335 L p .625 .78 .956 r F P s P p .002 w .32568 .73958 m .34648 .73335 L .36998 .74172 L p .625 .78 .956 r F P s P p .002 w .23668 .73527 m .21639 .74148 L .19198 .73311 L p .625 .78 .956 r F P s P p .002 w .17185 .73934 m .19198 .73311 L .21639 .74148 L p .625 .78 .956 r F P s P p .002 w .43527 .73766 m .45658 .73142 L .47946 .73981 L p .625 .78 .956 r F P s P p .002 w .34648 .73335 m .32568 .73958 L .30188 .73118 L p .625 .78 .956 r F P s P p .002 w .28125 .73743 m .30188 .73118 L .32568 .73958 L p .625 .78 .956 r F P s P p .002 w .45658 .73142 m .43527 .73766 L .41209 .72924 L p .625 .78 .956 r F P s P p .002 w .39094 .73551 m .41209 .72924 L .43527 .73766 L p .625 .78 .956 r F P s P p .002 w .30188 .73118 m .28125 .73743 L .25715 .729 L p .625 .78 .956 r F P s P p .002 w .23668 .73527 m .25715 .729 L .28125 .73743 L p .625 .78 .956 r F P s P p .002 w .45296 .67587 m .43343 .72292 L .40999 .71432 L p .845 .659 .65 r F P s P p .002 w .40999 .71432 m .43061 .66685 L .45296 .67587 L p .845 .659 .65 r F P s P p .002 w .41209 .72924 m .39094 .73551 L .36746 .72706 L p .625 .78 .956 r F P s P p .002 w .34648 .73335 m .36746 .72706 L .39094 .73551 L p .625 .78 .956 r F P s P p .002 w .25715 .729 m .23668 .73527 L .21228 .72682 L p .625 .78 .956 r F P s P p .002 w .19198 .73311 m .21228 .72682 L .23668 .73527 L p .625 .78 .956 r F P s P p .002 w .36746 .72706 m .34648 .73335 L .3227 .72487 L p .625 .78 .956 r F P s P p .002 w .30188 .73118 m .3227 .72487 L .34648 .73335 L p .625 .78 .956 r F P s P p .002 w .41209 .72924 m .43343 .72292 L .45658 .73142 L p .625 .78 .956 r F P s P p .002 w .3227 .72487 m .30188 .73118 L .2778 .72268 L p .625 .78 .956 r F P s P p .002 w .25715 .729 m .2778 .72268 L .30188 .73118 L p .625 .78 .956 r F P s P p .002 w .43343 .72292 m .41209 .72924 L .38863 .72072 L p .625 .78 .956 r F P s P p .002 w .36746 .72706 m .38863 .72072 L .41209 .72924 L p .625 .78 .956 r F P s P p .002 w .2778 .72268 m .25715 .729 L .23276 .72048 L p .625 .78 .956 r F P s P p .002 w .21228 .72682 m .23276 .72048 L .25715 .729 L p .625 .78 .956 r F P s P p .002 w .43061 .66685 m .40999 .71432 L .38628 .70562 L p .844 .661 .654 r F P s P p .002 w .38628 .70562 m .40798 .65772 L .43061 .66685 L p .844 .661 .654 r F P s P p .002 w .38863 .72072 m .36746 .72706 L .3437 .71851 L p .625 .78 .956 r F P s P p .002 w .3227 .72487 m .3437 .71851 L .36746 .72706 L p .625 .78 .956 r F P s P p .002 w .3437 .71851 m .3227 .72487 L .29863 .7163 L p .625 .78 .956 r F P s P p .002 w .2778 .72268 m .29863 .7163 L .3227 .72487 L p .625 .78 .956 r F P s P p .002 w .38863 .72072 m .40999 .71432 L .43343 .72292 L p .625 .78 .956 r F P s P p .002 w .29863 .7163 m .2778 .72268 L .25343 .71408 L p .625 .78 .956 r F P s P p .002 w .23276 .72048 m .25343 .71408 L .2778 .72268 L p .625 .78 .956 r F P s P p .002 w .40999 .71432 m .38863 .72072 L .36489 .7121 L p .625 .78 .956 r F P s P p .002 w .3437 .71851 m .36489 .7121 L .38863 .72072 L p .625 .78 .956 r F P s P p .002 w .40798 .65772 m .38628 .70562 L .36227 .69681 L p .843 .663 .658 r F P s P p .002 w .36227 .69681 m .38507 .64848 L .40798 .65772 L p .843 .663 .658 r F P s P p .002 w .36489 .7121 m .3437 .71851 L .31966 .70986 L p .625 .78 .956 r F P s P p .002 w .29863 .7163 m .31966 .70986 L .3437 .71851 L p .625 .78 .956 r F P s P p .002 w .31966 .70986 m .29863 .7163 L .27428 .70762 L p .625 .78 .956 r F P s P p .002 w .25343 .71408 m .27428 .70762 L .29863 .7163 L p .625 .78 .956 r F P s P p .002 w .36489 .7121 m .38628 .70562 L .40999 .71432 L p .625 .78 .956 r F P s P p .002 w .38628 .70562 m .36489 .7121 L .34087 .70337 L p .625 .78 .956 r F P s P p .002 w .31966 .70986 m .34087 .70337 L .36489 .7121 L p .625 .78 .956 r F P s P p .002 w .38507 .64848 m .36227 .69681 L .33797 .6879 L p .842 .665 .662 r F P s P p .002 w .33797 .6879 m .36188 .63912 L .38507 .64848 L p .842 .665 .662 r F P s P p .002 w .34087 .70337 m .31966 .70986 L .29532 .70111 L p .625 .78 .956 r F P s P p .002 w .27428 .70762 m .29532 .70111 L .31966 .70986 L p .625 .78 .956 r F P s P p .002 w .34087 .70337 m .36227 .69681 L .38628 .70562 L p .625 .78 .956 r F P s P p .002 w .36227 .69681 m .34087 .70337 L .31655 .69453 L p .625 .78 .956 r F P s P p .002 w .29532 .70111 m .31655 .69453 L .34087 .70337 L p .625 .78 .956 r F P s P p .002 w .31655 .69453 m .33797 .6879 L .36227 .69681 L p .625 .78 .956 r F P s P p .002 w .66295 .02755 m .90359 .41965 L s .90359 .41965 m .97245 .92345 L s .97245 .92345 m .69507 .58075 L s .69507 .58075 m .66295 .02755 L s .11113 .25957 m .02755 .78701 L s .02755 .78701 m .69507 .58075 L s .69507 .58075 m .66295 .02755 L s .66295 .02755 m .11113 .25957 L s P p p .002 w .11113 .25957 m .66295 .02755 L s P p .002 w .12286 .25464 m .12766 .25957 L s P [(-2)] .11326 .24476 .97192 1 Mshowa p .002 w .24366 .20385 m .24822 .209 L s P [(-1)] .23453 .19353 .88455 1 Mshowa p .002 w .3711 .15026 m .37539 .15565 L s P [(0)] .36251 .13949 .79718 1 Mshowa p .002 w .50575 .09364 m .50974 .09926 L s P [(1)] .49778 .08241 .70982 1 Mshowa p .002 w .64825 .03373 m .65189 .03958 L s P [(2)] .64097 .02204 .62245 1 Mshowa p .001 w .14652 .24469 m .14937 .24768 L s P p .001 w .17042 .23464 m .17324 .23766 L s P p .001 w .19457 .22448 m .19737 .22753 L s P p .001 w .21898 .21422 m .22175 .21729 L s P p .001 w .2686 .19336 m .2713 .19648 L s P p .001 w .29381 .18276 m .29648 .18591 L s P p .001 w .31929 .17204 m .32193 .17522 L s P p .001 w .34505 .16121 m .34766 .16442 L s P p .001 w .39743 .13919 m .39998 .14245 L s P p .001 w .42406 .12799 m .42657 .13128 L s P p .001 w .45099 .11667 m .45346 .11998 L s P p .001 w .47822 .10522 m .48065 .10856 L s P p .001 w .5336 .08193 m .53596 .08533 L s P p .001 w .56177 .07009 m .56408 .07351 L s P p .001 w .59027 .05811 m .59253 .06156 L s P p .001 w .61909 .04599 m .62132 .04947 L s P P p p .002 w .02755 .78701 m .40586 1.05324 L s P p .002 w .03867 .79484 m .03972 .78803 L s P [(0)] .03657 .80845 .15454 -1 Mshowa p .002 w .1433 .86846 m .14411 .86163 L s P [(0.5)] .14168 .88214 .11851 -1 Mshowa p .002 w .23715 .93451 m .23775 .92765 L s P [(1)] .23595 .94823 .08756 -1 Mshowa p .002 w .32182 .99409 m .32224 .98722 L s P [(1.5)] .32098 1.00784 .0607 -1 Mshowa p .002 w .39858 1.04811 m .39884 1.04123 L s P [(2)] .39807 1.06188 .03715 -1 Mshowa p .001 w .06055 .81023 m .06115 .80614 L s P p .001 w .08193 .82528 m .0825 .82119 L s P p .001 w .10284 .83999 m .10338 .8359 L s P p .001 w .12329 .85439 m .12381 .85029 L s P p .001 w .16288 .88224 m .16334 .87814 L s P p .001 w .18204 .89573 m .18247 .89162 L s P p .001 w .2008 .90893 m .20121 .90482 L s P p .001 w .21916 .92185 m .21955 .91774 L s P p .001 w .25478 .94692 m .25511 .9428 L s P p .001 w .27204 .95907 m .27236 .95495 L s P p .001 w .28897 .97098 m .28926 .96685 L s P p .001 w .30556 .98265 m .30583 .97853 L s P p .001 w .33777 1.00532 m .338 1.00119 L s P p .001 w .35341 1.01632 m .35362 1.0122 L s P p .001 w .36875 1.02712 m .36894 1.023 L s P p .001 w .38381 1.03772 m .38398 1.03359 L s P P p p .002 w .11113 .25957 m .02755 .78701 L s P p .002 w .10947 .27006 m .11582 .2674 L s P [(0)] .09676 .27537 1 -0.41829 Mshowa p .002 w .09208 .37978 m .09848 .37725 L s P [(0.25)] .07927 .38485 1 -0.39542 Mshowa p .002 w .07317 .49909 m .07963 .4967 L s P [(0.5)] .06026 .50388 1 -0.37036 Mshowa p .002 w .05254 .6293 m .05905 .62707 L s P [(0.75)] .03951 .63377 1 -0.34276 Mshowa p .002 w .02993 .77198 m .0365 .76993 L s P [(1)] .01678 .77609 1 -0.31222 Mshowa p .001 w .1061 .29129 m .10992 .28971 L s P p .001 w .10268 .31287 m .10651 .31131 L s P p .001 w .09921 .33481 m .10304 .33326 L s P p .001 w .09567 .35711 m .09951 .35557 L s P p .001 w .08843 .40284 m .09227 .40134 L s P p .001 w .08471 .42629 m .08856 .4248 L s P p .001 w .08093 .45014 m .08479 .44867 L s P p .001 w .07708 .4744 m .08095 .47295 L s P p .001 w .06919 .52422 m .07307 .5228 L s P p .001 w .06514 .54979 m .06903 .54839 L s P p .001 w .06101 .57582 m .06491 .57444 L s P p .001 w .05681 .60232 m .06072 .60096 L s P p .001 w .04818 .65679 m .0521 .65547 L s P p .001 w .04375 .68478 m .04767 .68348 L s P p .001 w .03923 .7133 m .04316 .71203 L s P p .001 w .03462 .74236 m .03856 .74111 L s P P p P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] No Input Form was saved for this expression. ;[o] -Graphics3D- :[font = section; inactive; Cclosed; preserveAspect; startGroup] Problem 22 :[font = text; inactive; preserveAspect] The solutions for q(t) and p(t) are given by q=q0 cos(omega t) and p(t)=(omega m q0) sin (omega t). If we now rescale these solutions by multiplying q by Sqrt[k/2] and p by Sqrt[1/2m], then these solutions become x=q0 Sqrt[k/2] cos(omega t) and y=q0 Sqrt[k/2] sin(omega t). With these rescales variables, we see x^2+y^2=(1/2)k q0^2, which is the original value of the energy and is a constant of the motion. We use ParametricPlot3D to plot these solutions. :[font = input; preserveAspect; startGroup] ParametricPlot3D[{Cos[t],Sin[t],s},{s,0,6 Pi},{t,0,2 Pi}, BoxRatios->{1,1,1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 276; pictureHeight = 300] %! %%Creator: Mathematica %%AspectRatio: 1.08079 MathPictureStart %% Graphics3D /Courier findfont 10 scalefont setfont % Scaling calculations -0.0260524 1.10184 0.0275459 1.10184 [ [(-1)] .11002 .24613 .97432 1 Msboxa [(-0.5)] .23197 .19461 .88635 1 Msboxa [(0)] .36071 .14025 .79838 1 Msboxa [(0.5)] .49683 .08282 .71042 1 Msboxa [(1)] .64097 .02204 .62245 1 Msboxa [(-1)] .03581 .80791 .15481 -1 Msboxa [(-0.5)] .14134 .8819 .11862 -1 Msboxa [(0)] .23595 .94823 .08756 -1 Msboxa [(0.5)] .32126 1.00804 .06061 -1 Msboxa [(1)] .39857 1.06223 .037 -1 Msboxa [(0)] .09676 .27537 1 -0.41829 Msboxa [(5)] .07816 .39183 1 -0.39396 Msboxa [(10)] .05782 .51912 1 -0.36713 Msboxa [(15)] 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L p .176 .077 .495 r F P s P p .002 w .8111 .54546 m .81353 .57003 L .7734 .62264 L p .176 .077 .495 r F P s P p .002 w .80636 .49753 m .80406 .47416 L .81433 .41302 L p 0 0 .302 r F P s P p .002 w .81189 .38971 m .81433 .41302 L .80406 .47416 L p 0 0 .302 r F P s P p .002 w .24147 .3589 m .24388 .33584 L .2381 .39889 L p .719 .644 .086 r F P s P p .002 w .2381 .39889 m .23572 .42166 L .24147 .3589 L p .719 .644 .086 r F P s P p .002 w .24237 .67373 m .24477 .64809 L .29801 .69458 L p .938 .685 .528 r F P s P p .002 w .29801 .69458 m .29613 .7198 L .24237 .67373 L p .938 .685 .528 r F P s P p .002 w .45205 .9022 m .45265 .87539 L .54641 .87902 L p .713 .477 .567 r F P s P p .002 w .54641 .87902 m .54658 .90578 L .45205 .9022 L p .713 .477 .567 r F P s P p .002 w .54658 .90578 m .54641 .87902 L .63842 .86785 L p .639 .417 .566 r F P s P p .002 w .63749 .84139 m .63842 .86785 L .54641 .87902 L p .639 .417 .566 r F P s P p .002 w .63842 .86785 m .63749 .84139 L .71994 .81558 L p .547 .347 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.495 r F P s P p .002 w .80871 .5213 m .80636 .49753 L .81682 .43673 L p 0 0 .302 r F P s P p .002 w .81433 .41302 m .81682 .43673 L .80636 .49753 L p 0 0 .302 r F P s P p .002 w .23902 .38235 m .24147 .3589 L .23572 .42166 L p .719 .644 .086 r F P s P p .002 w .23572 .42166 m .23331 .44481 L .23902 .38235 L p .719 .644 .086 r F P s P p .002 w .23994 .6998 m .24237 .67373 L .29613 .7198 L p .938 .685 .528 r F P s P p .002 w .29613 .7198 m .29422 .74544 L .23994 .6998 L p .938 .685 .528 r F P s P p .002 w .45144 .92945 m .45205 .9022 L .54658 .90578 L p .713 .477 .567 r F P s P p .002 w .54658 .90578 m .54677 .93298 L .45144 .92945 L p .713 .477 .567 r F P s P p .002 w .54677 .93298 m .54658 .90578 L .63937 .89475 L p .639 .417 .566 r F P s P p .002 w .63842 .86785 m .63937 .89475 L .54658 .90578 L p .639 .417 .566 r F P s P p .002 w .63937 .89475 m .63842 .86785 L .72159 .84235 L p .547 .347 .56 r F P s P p .002 w .71994 .81558 m .72159 .84235 L .63842 .86785 L p .547 .347 .56 r F P s 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.002 w .8111 .54546 m .80871 .5213 L .81934 .46085 L p 0 0 .302 r F P s P p .002 w .81682 .43673 m .81934 .46085 L .80871 .5213 L p 0 0 .302 r F P s P p .002 w .23653 .40621 m .23902 .38235 L .23331 .44481 L p .719 .644 .086 r F P s P p .002 w .23331 .44481 m .23086 .46835 L .23653 .40621 L p .719 .644 .086 r F P s P p .002 w .23746 .72633 m .23994 .6998 L .29422 .74544 L p .938 .685 .528 r F P s P p .002 w .29422 .74544 m .29228 .77151 L .23746 .72633 L p .938 .685 .528 r F P s P p .002 w .45082 .95716 m .45144 .92945 L .54677 .93298 L p .713 .477 .567 r F P s P p .002 w .54677 .93298 m .54695 .96064 L .45082 .95716 L p .713 .477 .567 r F P s P p .002 w .54695 .96064 m .54677 .93298 L .64034 .9221 L p .639 .417 .566 r F P s P p .002 w .63937 .89475 m .64034 .9221 L .54677 .93298 L p .639 .417 .566 r F P s P p .002 w .64034 .9221 m .63937 .89475 L .72327 .86957 L p .547 .347 .56 r F P s P p .002 w .72159 .84235 m .72327 .86957 L .63937 .89475 L p .547 .347 .56 r F P s P p .002 w .28625 .85243 m .28829 .825 L .36421 .85711 L p .856 .603 .555 r F P s P p .002 w .36421 .85711 m .36285 .88415 L .28625 .85243 L p .856 .603 .555 r F P s P p .002 w .81189 .38971 m .80949 .36679 L .78746 .30422 L p .916 .643 .151 r F P s P p .002 w .7852 .28143 m .78746 .30422 L .80949 .36679 L p .916 .643 .151 r F P s P p .002 w .28457 .25275 m .28665 .23026 L .24858 .29089 L p 0 0 .312 r F P s P p .002 w .24858 .29089 m .24625 .31318 L .28457 .25275 L p 0 0 .312 r F P s P p .002 w .36008 .93961 m .36148 .91164 L .45144 .92945 L p .782 .536 .564 r F P s P p .002 w .45144 .92945 m .45082 .95716 L .36008 .93961 L p .782 .536 .564 r F P s P p .002 w .72159 .84235 m .71994 .81558 L .78627 .77655 L p .415 .248 .545 r F P s P p .002 w .78403 .7498 m .78627 .77655 L .71994 .81558 L p .415 .248 .545 r F P s P p .002 w .21794 .59225 m .22061 .56661 L .24712 .62289 L p .997 .773 .431 r F P s P p .002 w .24712 .62289 m .24477 .64809 L .21794 .59225 L p .997 .773 .431 r F P s P p .002 w .78183 .72351 m 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r F P s P p .002 w .63632 .76373 m .63544 .73893 L .71465 .71216 L p .547 .347 .56 r F P s P p .002 w .71313 .68751 m .71465 .71216 L .63544 .73893 L p .547 .347 .56 r F P s P p .002 w .30119 .69352 m .30301 .6687 L .37473 .70285 L p .856 .603 .554 r F P s P p .002 w .37473 .70285 m .37352 .72737 L .30119 .69352 L p .856 .603 .554 r F P s P p .004 w .36799 .70043 m .37624 .70306 L s P p .002 w .37105 .7776 m .3723 .75228 L .45745 .77144 L p .783 .537 .564 r F P s P p .002 w .45745 .77144 m .45691 .79656 L .37105 .7776 L p .783 .537 .564 r F P s P p .002 w .71313 .68751 m .71162 .66325 L .77398 .6221 L p .414 .248 .545 r F P s P p .002 w .77195 .5979 m .77398 .6221 L .71162 .66325 L p .414 .248 .545 r F P s P p .004 w .70746 .65691 m .72083 .64747 L s P p .002 w .23659 .44387 m .23897 .42071 L .26365 .47907 L p .997 .772 .43 r F P s P p .002 w .26365 .47907 m .26154 .50191 L .23659 .44387 L p .997 .772 .43 r F P s P p .004 w .2628 .48635 m .26503 .48835 L s P p .002 w .76996 .5741 m 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.002 w .29933 .71875 m .30119 .69352 L .37352 .72737 L p .856 .603 .554 r F P s P p .002 w .37352 .72737 m .3723 .75228 L .29933 .71875 L p .856 .603 .554 r F P s P p .002 w .36979 .80334 m .37105 .7776 L .45691 .79656 L p .783 .537 .564 r F P s P p .002 w .45691 .79656 m .45636 .8221 L .36979 .80334 L p .783 .537 .564 r F P s P p .002 w .71465 .71216 m .71313 .68751 L .77604 .64669 L p .414 .248 .545 r F P s P p .002 w .77398 .6221 m .77604 .64669 L .71313 .68751 L p .414 .248 .545 r F P s P p .002 w .23416 .46741 m .23659 .44387 L .26154 .50191 L p .997 .772 .43 r F P s P p .002 w .26154 .50191 m .2594 .52512 L .23416 .46741 L p .997 .772 .43 r F P s P p .002 w .77195 .5979 m .76996 .5741 L .80893 .52093 L p .173 .075 .494 r F P s P p .002 w .80658 .49717 m .80893 .52093 L .76996 .5741 L p .173 .075 .494 r F P s P p .002 w .80199 .45079 m .79976 .42817 L .8095 .36662 L p 0 0 .3 r F P s P p .002 w .80714 .34408 m .8095 .36662 L .79976 .42817 L p 0 0 .3 r F P s P p .002 w .24934 .3124 m .25165 .29011 L .24591 .35349 L p .72 .644 .085 r F P s P p .002 w .24591 .35349 m .24363 .37553 L .24934 .3124 L p .72 .644 .085 r F P s P p .002 w .25047 .62187 m .25276 .59708 L .3048 .64428 L p .938 .685 .527 r F P s P p .002 w .3048 .64428 m .30301 .6687 L .25047 .62187 L p .938 .685 .527 r F P s P p .002 w .4558 .84804 m .45636 .8221 L .54813 .82589 L p .713 .477 .567 r F P s P p .002 w .54813 .82589 m .54832 .85179 L .4558 .84804 L p .713 .477 .567 r F P s P p .002 w .54832 .85179 m .54813 .82589 L .63814 .81455 L p .639 .417 .566 r F P s P p .002 w .63722 .78894 m .63814 .81455 L .54813 .82589 L p .639 .417 .566 r F P s P p .002 w .63814 .81455 m .63722 .78894 L .71778 .76269 L p .547 .347 .56 r F P s P p .002 w .71621 .73722 m .71778 .76269 L .63722 .78894 L p .547 .347 .56 r F P s P p .002 w .29745 .7444 m .29933 .71875 L .3723 .75228 L p .856 .603 .554 r F P s P p .002 w .3723 .75228 m .37105 .7776 L .29745 .7444 L p .856 .603 .554 r F P s P p .002 w .3685 .8295 m .36979 .80334 L .45636 .8221 L p .783 .537 .564 r F P s P p .002 w .45636 .8221 m .4558 .84804 L .3685 .8295 L p .783 .537 .564 r F P s P p .002 w .71621 .73722 m .71465 .71216 L .77814 .6717 L p .414 .248 .545 r F P s P p .002 w .77604 .64669 m .77814 .6717 L .71465 .71216 L p .414 .248 .545 r F P s P p .002 w .23169 .49136 m .23416 .46741 L .2594 .52512 L p .997 .772 .43 r F P s P p .002 w .2594 .52512 m .25722 .54871 L .23169 .49136 L p .997 .772 .43 r F P s P p .002 w .77398 .6221 m .77195 .5979 L .81132 .5451 L p .173 .075 .494 r F P s P p .002 w .80893 .52093 m .81132 .5451 L .77195 .5979 L p .173 .075 .494 r F P s P p .004 w .76417 .61269 m .77875 .59631 L s P p .002 w .80427 .47379 m .80199 .45079 L .8119 .38954 L p 0 0 .3 r F P s P p .002 w .8095 .36662 m .8119 .38954 L .80199 .45079 L p 0 0 .3 r F P s P p .002 w .247 .33507 m .24934 .3124 L .24363 .37553 L p .72 .644 .085 r F P s P p .002 w .24363 .37553 m .24132 .39794 L .247 .33507 L p .72 .644 .085 r F P s P p .002 w .24814 .64708 m .25047 .62187 L .30301 .6687 L p .938 .685 .527 r F P s P p .002 w .30301 .6687 m .30119 .69352 L .24814 .64708 L p .938 .685 .527 r F P s P p .002 w .45523 .87442 m .4558 .84804 L .54832 .85179 L p .713 .477 .567 r F P s P p .002 w .54832 .85179 m .54851 .87812 L .45523 .87442 L p .713 .477 .567 r F P s P p .002 w .54851 .87812 m .54832 .85179 L .63907 .84059 L p .639 .417 .566 r F P s P p .002 w .63814 .81455 m .63907 .84059 L .54832 .85179 L p .639 .417 .566 r F P s P p .002 w .63907 .84059 m .63814 .81455 L .71939 .78858 L p .547 .347 .56 r F P s P p .002 w .71778 .76269 m .71939 .78858 L .63814 .81455 L p .547 .347 .56 r F P s P p .002 w .29553 .77048 m .29745 .7444 L .37105 .7776 L p .856 .603 .554 r F P s P p .002 w .37105 .7776 m .36979 .80334 L .29553 .77048 L p .856 .603 .554 r F P s P p .002 w .36719 .8561 m .3685 .8295 L .4558 .84804 L p .783 .537 .564 r F P s P p .002 w .4558 .84804 m .45523 .87442 L .36719 .8561 L p .783 .537 .564 r F P s P p .002 w .71778 .76269 m .71621 .73722 L .78027 .69713 L p .414 .248 .545 r F P s P p .002 w .77814 .6717 m .78027 .69713 L .71621 .73722 L p .414 .248 .545 r F P s P p .004 w .53424 .87007 m .55215 .8685 L s P p .002 w .22918 .51571 m .23169 .49136 L .25722 .54871 L p .997 .772 .43 r F P s P p .002 w .25722 .54871 m .25501 .57269 L .22918 .51571 L p .997 .772 .43 r F P s P p .002 w .77604 .64669 m .77398 .6221 L .81375 .56968 L p .173 .075 .494 r F P s P p .002 w .81132 .5451 m .81375 .56968 L .77398 .6221 L p .173 .075 .494 r F P s P p .002 w .80658 .49717 m .80427 .47379 L .81434 .41285 L p 0 0 .3 r F P s P p .002 w .8119 .38954 m .81434 .41285 L .80427 .47379 L p 0 0 .3 r F P s P p .002 w .24461 .35813 m .247 .33507 L .24132 .39794 L p .72 .644 .085 r F P s P p .002 w .24132 .39794 m .23897 .42071 L .24461 .35813 L p .72 .644 .085 r F P s P p .002 w .24578 .67272 m .24814 .64708 L .30119 .69352 L p .938 .685 .527 r F P s P p .002 w .30119 .69352 m .29933 .71875 L .24578 .67272 L p .938 .685 .527 r F P s P p .004 w .29323 .67034 m .31062 .67989 L s P p .002 w .45465 .90124 m .45523 .87442 L .54851 .87812 L p .713 .477 .567 r F P s P p .002 w .54851 .87812 m .54871 .90489 L .45465 .90124 L p .713 .477 .567 r F P s P p .002 w .54871 .90489 m .54851 .87812 L .64002 .86705 L p .639 .417 .566 r F P s P p .002 w .63907 .84059 m .64002 .86705 L .54851 .87812 L p .639 .417 .566 r F P s P p .002 w .64002 .86705 m .63907 .84059 L .72102 .81491 L p .547 .347 .56 r F P s P p .002 w .71939 .78858 m .72102 .81491 L .63907 .84059 L p .547 .347 .56 r F P s P p .002 w .80714 .34408 m .80482 .32191 L .78299 .25903 L p .917 .643 .152 r F P s P p .004 w .80371 .33201 m .80396 .34213 L s P p .002 w .29358 .79701 m .29553 .77048 L .36979 .80334 L p .856 .603 .554 r F P s P p .002 w .36979 .80334 m .3685 .8295 L .29358 .79701 L p .856 .603 .554 r F P s P p .004 w .63409 .85051 m .65326 .84352 L s P p .002 w .36586 .88315 m .36719 .8561 L .45523 .87442 L p .783 .537 .564 r F P s P p .004 w .44077 .87287 m .46741 .87377 L s P p .002 w .45523 .87442 m .45465 .90124 L .36586 .88315 L p .783 .537 .564 r F P s P p .002 w .71939 .78858 m .71778 .76269 L .78243 .72298 L p .414 .248 .545 r F P s P p .002 w .78027 .69713 m .78243 .72298 L .71778 .76269 L p .414 .248 .545 r F P s P p .002 w .22662 .54049 m .22918 .51571 L .25501 .57269 L p .997 .772 .43 r F P s P p .002 w .25501 .57269 m .25276 .59708 L .22662 .54049 L p .997 .772 .43 r F P s P p .002 w .77814 .6717 m .77604 .64669 L .81622 .59467 L p .173 .075 .494 r F P s P p .002 w .81375 .56968 m .81622 .59467 L .77604 .64669 L p .173 .075 .494 r F P s P p .002 w .80893 .52093 m .80658 .49717 L .81683 .43656 L p 0 0 .3 r F P s P p .002 w .81434 .41285 m .81683 .43656 L .80658 .49717 L p 0 0 .3 r F P s P p .002 w .24219 .38159 m .24461 .35813 L .23897 .42071 L p .72 .644 .085 r F P s P p .002 w .23897 .42071 m .23659 .44387 L .24219 .38159 L p .72 .644 .085 r F P s P p .002 w .24337 .69881 m .24578 .67272 L .29933 .71875 L p .938 .685 .527 r F 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.45465 .90124 L p .783 .537 .564 r F P s P p .002 w .45465 .90124 m .45406 .9285 L .36451 .91066 L p .783 .537 .564 r F P s P p .002 w .72102 .81491 m .71939 .78858 L .78464 .74929 L p .414 .248 .545 r F P s P p .002 w .78243 .72298 m .78464 .74929 L .71939 .78858 L p .414 .248 .545 r F P s P p .002 w .22402 .5657 m .22662 .54049 L .25276 .59708 L p .997 .772 .43 r F P s P p .002 w .25276 .59708 m .25047 .62187 L .22402 .5657 L p .997 .772 .43 r F P s P p .002 w .78027 .69713 m .77814 .6717 L .81874 .6201 L p .173 .075 .494 r F P s P p .002 w .81622 .59467 m .81874 .6201 L .77814 .6717 L p .173 .075 .494 r F P s P p .002 w .81132 .5451 m .80893 .52093 L .81935 .46068 L p 0 0 .3 r F P s P p .002 w .81683 .43656 m .81935 .46068 L .80893 .52093 L p 0 0 .3 r F P s P p .002 w .23972 .40545 m .24219 .38159 L .23659 .44387 L p .72 .644 .085 r F P s P p .002 w .23659 .44387 m .23416 .46741 L .23972 .40545 L p .72 .644 .085 r F P s P p .004 w .23823 .42649 m .23978 .44085 L s P p .002 w .24092 .72535 m .24337 .69881 L .29745 .7444 L p .938 .685 .527 r F P s P p .002 w .29745 .7444 m .29553 .77048 L .24092 .72535 L p .938 .685 .527 r F P s P p .002 w .45346 .95623 m .45406 .9285 L .5489 .93211 L p .713 .477 .567 r F P s P p .002 w .5489 .93211 m .54911 .95978 L .45346 .95623 L p .713 .477 .567 r F P s P p .002 w .54911 .95978 m .5489 .93211 L .64196 .92133 L p .639 .417 .566 r F P s P p .002 w .64098 .89396 m .64196 .92133 L .5489 .93211 L p .639 .417 .566 r F P s P p .002 w .64196 .92133 m .64098 .89396 L .72437 .86891 L p .547 .347 .56 r F P s P p .002 w .72268 .84168 m .72437 .86891 L .64098 .89396 L p .547 .347 .56 r F P s P p .002 w .8119 .38954 m .8095 .36662 L .78746 .30422 L p .917 .643 .152 r F P s P p .002 w .7852 .28143 m .78746 .30422 L .8095 .36662 L p .917 .643 .152 r F P s P p .002 w .28958 .85144 m .2916 .82399 L .36719 .8561 L p .856 .603 .554 r F P s P p .002 w .36719 .8561 m .36586 .88315 L .28958 .85144 L p .856 .603 .554 r F P s P p .004 w .36 .86303 m .38306 .86681 L s P p .002 w .28738 .2522 m .28943 .22971 L .25165 .29011 L p 0 0 .312 r F P s P p .002 w .25165 .29011 m .24934 .3124 L .28738 .2522 L p 0 0 .312 r F P s P p .002 w .36313 .93864 m .36451 .91066 L .45406 .9285 L p .783 .537 .564 r F P s P p .002 w .45406 .9285 m .45346 .95623 L .36313 .93864 L p .783 .537 .564 r F P s P p .002 w .72268 .84168 m .72102 .81491 L .78688 .77604 L p .414 .248 .545 r F P s P p .002 w .78464 .74929 m .78688 .77604 L .72102 .81491 L p .414 .248 .545 r F P s P p .004 w .70915 .8199 m .73091 .80766 L s P p .002 w .22137 .59135 m .22402 .5657 L .25047 .62187 L p .997 .772 .43 r F P s P p .002 w .25047 .62187 m .24814 .64708 L .22137 .59135 L p .997 .772 .43 r F P s P p .002 w .78243 .72298 m .78027 .69713 L .8213 .64597 L p .173 .075 .494 r F P s P p .002 w .81874 .6201 m .8213 .64597 L .78027 .69713 L p .173 .075 .494 r F P s P p .002 w .81375 .56968 m .81132 .5451 L .82192 .48522 L p 0 0 .3 r F P s P p .002 w .81935 .46068 m .82192 .48522 L .81132 .5451 L p 0 0 .3 r F P s P p .004 w .8041 .55644 m .81213 .53949 L s P p .002 w .23721 .42974 m .23972 .40545 L .23416 .46741 L p .72 .644 .085 r F P s P p .004 w .23603 .40598 m .23788 .42321 L s P p .002 w .23416 .46741 m .23169 .49136 L .23721 .42974 L p .72 .644 .085 r F P s P p .002 w .23843 .75235 m .24092 .72535 L .29553 .77048 L p .938 .685 .527 r F P s P p .002 w .29553 .77048 m .29358 .79701 L .23843 .75235 L p .938 .685 .527 r F P s P p .002 w .45285 .98443 m .45346 .95623 L .54911 .95978 L p .713 .477 .567 r F P s P p .002 w .54911 .95978 m .54931 .98793 L .45285 .98443 L p .713 .477 .567 r F P s P p .002 w .54931 .98793 m .54911 .95978 L .64296 .94916 L p .639 .417 .566 r F P s P p .002 w .64196 .92133 m .64296 .94916 L .54911 .95978 L p .639 .417 .566 r F P s P p .002 w .64296 .94916 m .64196 .92133 L .72608 .89661 L p .547 .347 .56 r F P s P p .002 w .72437 .86891 m .72608 .89661 L .64196 .92133 L p .547 .347 .56 r F P s P p .002 w .81434 .41285 m .8119 .38954 L 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L s P p .002 w .8239 .6723 m .82655 .69909 L .78464 .74929 L p .173 .075 .494 r F P s P p .002 w .81874 .6201 m .81622 .59467 L .8272 .53561 L p 0 0 .3 r F P s P p .002 w .82454 .5102 m .8272 .53561 L .81622 .59467 L p 0 0 .3 r F P s P p .002 w .23205 .47962 m .23465 .45446 L .22918 .51571 L p .72 .644 .085 r F P s P p .002 w .22918 .51571 m .22662 .54049 L .23205 .47962 L p .72 .644 .085 r F P s P p .002 w .23331 .80781 m .23589 .77983 L .2916 .82399 L p .938 .685 .527 r F P s P p .002 w .2916 .82399 m .28958 .85144 L .23331 .80781 L p .938 .685 .527 r F P s P p .002 w .72783 .92479 m .7296 .95346 L .64397 .97746 L p .547 .347 .56 r F P s P p .002 w .81935 .46068 m .81683 .43656 L .79446 .37501 L p .917 .643 .152 r F P s P p .002 w .79208 .351 m .79446 .37501 L .81683 .43656 L p .917 .643 .152 r F P s P p .002 w .28332 .9367 m .28544 .90778 L .36313 .93864 L p .856 .603 .554 r F P s P p .002 w .36313 .93864 m .36174 .9671 L .28332 .9367 L p .856 .603 .554 r F P s P p .002 w .78746 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.16638 L p .756 .472 .501 r F P s P p .002 w .55036 .14365 m .55059 .16638 L .64973 .18515 L p .756 .472 .501 r F P s P p .002 w .34801 .32261 m .34965 .29782 L .2788 .34623 L p .35 .141 .449 r F P s P p .002 w .2788 .34623 m .27656 .37081 L .34801 .32261 L p .35 .141 .449 r F P s P p .002 w .21539 .64062 m .2183 .61253 L .21315 .67108 L p .72 .644 .085 r F P s P p .002 w .21315 .67108 m .21031 .69863 L .21539 .64062 L p .72 .644 .085 r F P s P p .002 w .83548 .61465 m .83267 .58783 L .80964 .52854 L p .917 .643 .152 r F P s P p .002 w .80699 .50176 m .80964 .52854 L .83267 .58783 L p .917 .643 .152 r F P s P p .002 w .80184 .44967 m .79933 .42433 L .74279 .37002 L p .949 .634 .401 r F P s P p .002 w .74077 .34486 m .74279 .37002 L .79933 .42433 L p .949 .634 .401 r F P s P p .002 w .26717 .47378 m .26958 .44731 L .2294 .50524 L p 0 0 .312 r F P s P p .002 w .2294 .50524 m .2267 .53133 L .26717 .47378 L p 0 0 .312 r F P s P p .002 w .7388 .32016 m .73686 .29591 L .6532 .25572 L p .852 .55 .477 r F P s P p .002 w .65202 .23176 m .6532 .25572 L .73686 .29591 L p .852 .55 .477 r F P s P p .002 w .55082 .18952 m .55059 .16638 L .44771 .17273 L p .659 .392 .505 r F P s P p .002 w .44771 .17273 m .44701 .19586 L .55082 .18952 L p .659 .392 .505 r F P s P p .004 w .22229 .54983 m .22533 .54455 L s P p .002 w .19529 .84426 m .19841 .81403 L .22801 .86528 L p .997 .772 .43 r F P s P p .002 w .22801 .86528 m .22528 .89481 L .19529 .84426 L p .997 .772 .43 r F P s P p .002 w .44555 .24341 m .44629 .21941 L .35284 .24959 L p .538 .293 .492 r F P s P p .002 w .35284 .24959 m .35126 .27348 L .44555 .24341 L p .538 .293 .492 r F P s P p .002 w .83764 .81119 m .83479 .7824 L .84724 .72702 L p 0 0 .3 r F P s P p .002 w .84422 .69813 m .84724 .72702 L .83479 .7824 L p 0 0 .3 r F P s P p .002 w .65202 .23176 m .65087 .20825 L .55082 .18952 L p .756 .472 .501 r F P s P p .002 w .55059 .16638 m .55082 .18952 L .65087 .20825 L p .756 .472 .501 r F P s P p .002 w .34634 .34787 m .34801 .32261 L .27656 .37081 L p .35 .141 .449 r F P s P p .004 w .28559 .32567 m .33908 .29418 L s P p .002 w .27656 .37081 m .27428 .39584 L .34634 .34787 L p .35 .141 .449 r F P s P p .002 w .21243 .66925 m .21539 .64062 L .21031 .69863 L p .72 .644 .085 r F P s P p .002 w .21031 .69863 m .20742 .72668 L .21243 .66925 L p .72 .644 .085 r F P s P p .002 w .83834 .64197 m .83548 .61465 L .81234 .55583 L p .917 .643 .152 r F P s P p .002 w .80964 .52854 m .81234 .55583 L .83548 .61465 L p .917 .643 .152 r F P s P p .002 w .80439 .47548 m .80184 .44967 L .74484 .39565 L p .949 .634 .401 r F P s P p .002 w .74279 .37002 m .74484 .39565 L .80184 .44967 L p .949 .634 .401 r F P s P p .004 w .77248 .3741 m .78777 .39675 L s P p .004 w .75092 .35604 m .77248 .3741 L s P p .002 w .26471 .50075 m .26717 .47378 L .2267 .53133 L p 0 0 .312 r F P s P p .002 w .2267 .53133 m .22395 .55789 L .26471 .50075 L p 0 0 .312 r F P s P p .002 w .74077 .34486 m .7388 .32016 L .65441 .28012 L p .852 .55 .477 r F P 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.58364 m .81789 .61198 L .84125 .66979 L p .917 .643 .152 r F P s P p .004 w .83112 .60881 m .83425 .63053 L s P p .002 w .80964 .52854 m .80699 .50176 L .74907 .44839 L p .949 .634 .401 r F P s P p .002 w .74693 .42177 m .74907 .44839 L .80699 .50176 L p .949 .634 .401 r F P s P p .002 w .25964 .55627 m .2622 .52824 L .22115 .58496 L p 0 0 .312 r F P s P p .004 w .23278 .53159 m .24454 .51116 L s P p .002 w .22115 .58496 m .2183 .61253 L .25964 .55627 L p 0 0 .312 r F P s P p .002 w .74484 .39565 m .74279 .37002 L .65688 .33032 L p .852 .55 .477 r F P s P p .002 w .65563 .30498 m .65688 .33032 L .74279 .37002 L p .852 .55 .477 r F P s P p .004 w .66521 .30227 m .71905 .32935 L s P p .002 w .55156 .26157 m .55131 .23711 L .44555 .24341 L p .659 .392 .505 r F P s P p .002 w .44555 .24341 m .44481 .26785 L .55156 .26157 L p .659 .392 .505 r F P s P p .002 w .44327 .31815 m .44404 .29277 L .34801 .32261 L p .538 .293 .492 r F P s P p .002 w .34801 .32261 m .34634 .34787 L .44327 .31815 L p .538 .293 .492 r F P s P p .002 w .85347 .78645 m .85667 .81704 L .84349 .87038 L p 0 0 .3 r F P s P p .002 w .65563 .30498 m .65441 .28012 L .55156 .26157 L p .756 .472 .501 r F P s P p .004 w .64996 .2946 m .65279 .29602 L s P p .002 w .55131 .23711 m .55156 .26157 L .65441 .28012 L p .756 .472 .501 r F P s P p .002 w .34113 .4266 m .3429 .39985 L .26958 .44731 L p .35 .141 .449 r F P s P p .002 w .26958 .44731 m .26717 .47378 L .34113 .4266 L p .35 .141 .449 r F P s P p .004 w .40811 .27325 m .43102 .27105 L s P p .004 w .8186 .58671 m .83112 .60881 L s P p .002 w .2032 .75852 m .20634 .72818 L .20147 .78437 L p .72 .644 .085 r F P s P p .004 w .20086 .75116 m .20328 .7577 L s P p .002 w .20147 .78437 m .19841 .81403 L .2032 .75852 L p .72 .644 .085 r F P s P p .004 w .24454 .51116 m .25901 .49967 L s P p .002 w .84724 .72702 m .84422 .69813 L .82075 .64088 L p .917 .643 .152 r F P s P p .002 w .81789 .61198 m .82075 .64088 L .84422 .69813 L p .917 .643 .152 r F P s P p .002 w 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.28651 L p .756 .472 .501 r F P s P p .002 w .55156 .26157 m .55181 .28651 L .65563 .30498 L p .756 .472 .501 r F P s P p .004 w .57053 .27279 m .64996 .2946 L s P p .002 w .33933 .45387 m .34113 .4266 L .26717 .47378 L p .35 .141 .449 r F P s P p .002 w .26717 .47378 m .26471 .50075 L .33933 .45387 L p .35 .141 .449 r F P s P p .002 w .2 .78946 m .2032 .75852 L .19841 .81403 L p .72 .644 .085 r F P s P p .002 w .19841 .81403 m .19529 .84426 L .2 .78946 L p .72 .644 .085 r F P s P p .004 w .48749 .26563 m .53225 .26949 L s P p .004 w .55941 .27183 m .57053 .27279 L s P p .002 w .85033 .75645 m .84724 .72702 L .82366 .67035 L p .917 .643 .152 r F P s P p .002 w .82075 .64088 m .82366 .67035 L .84724 .72702 L p .917 .643 .152 r F P s P p .002 w .81509 .58364 m .81234 .55583 L .75345 .50319 L p .949 .634 .401 r F P s P p .002 w .75124 .47552 m .75345 .50319 L .81234 .55583 L p .949 .634 .401 r F P s P p .002 w .25438 .61399 m .25704 .58484 L .21539 .64062 L p 0 0 .312 r F P s P p .002 w 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L p .852 .55 .477 r F P s P p .002 w .66077 .40932 m .66211 .4367 L .75124 .47552 L p .852 .55 .477 r F P s P p .002 w .55261 .36425 m .55234 .33783 L .44248 .34404 L p .659 .392 .505 r F P s P p .002 w .44248 .34404 m .44167 .37043 L .55261 .36425 L p .659 .392 .505 r F P s P p .002 w .44001 .42478 m .44085 .39734 L .34113 .4266 L p .538 .293 .492 r F P s P p .002 w .34113 .4266 m .33933 .45387 L .44001 .42478 L p .538 .293 .492 r F P s P p .002 w .66077 .40932 m .65944 .38248 L .55261 .36425 L p .756 .472 .501 r F P s P p .002 w .55234 .33783 m .55261 .36425 L .65944 .38248 L p .756 .472 .501 r F P s P p .002 w .33369 .539 m .33561 .51005 L .25964 .55627 L p .35 .141 .449 r F P s P p .002 w .25964 .55627 m .25704 .58484 L .33369 .539 L p .35 .141 .449 r F P s P p .002 w .82966 .73106 m .83276 .76234 L .85667 .81704 L p .917 .643 .152 r F P s P p .002 w .82366 .67035 m .82075 .64088 L .76037 .58954 L p .949 .634 .401 r F P s P p .002 w .75802 .56018 m .76037 .58954 L .82075 .64088 L p .949 .634 .401 r F P s P p .002 w .24608 .70499 m .2489 .67404 L .20634 .72818 L p 0 0 .312 r F P s P p .002 w .20634 .72818 m .2032 .75852 L .24608 .70499 L p 0 0 .312 r F P s P p .002 w .75571 .53141 m .75345 .50319 L .66349 .46462 L p .852 .55 .477 r F P s P p .004 w .75212 .51102 m .75315 .51198 L s P p .002 w .66211 .4367 m .66349 .46462 L .75345 .50319 L p .852 .55 .477 r F P s P p .002 w .55288 .3912 m .55261 .36425 L .44167 .37043 L p .659 .392 .505 r F P s P p .002 w .44167 .37043 m .44085 .39734 L .55288 .3912 L p .659 .392 .505 r F P s P p .002 w .43915 .45279 m .44001 .42478 L .33933 .45387 L p .538 .293 .492 r F P s P p .002 w .33933 .45387 m .33749 .48168 L .43915 .45279 L p .538 .293 .492 r F P s P p .004 w .34448 .45284 m .35033 .45037 L s P p .002 w .66211 .4367 m .66077 .40932 L .55288 .3912 L p .756 .472 .501 r F P s P p .002 w .55261 .36425 m .55288 .3912 L .66077 .40932 L p .756 .472 .501 r F P s P p .002 w .33174 .56854 m .33369 .539 L .25704 .58484 L p .35 .141 .449 r F P s P p .002 w .25704 .58484 m .25438 .61399 L .33174 .56854 L p .35 .141 .449 r F P s P p .002 w .82663 .7004 m .82366 .67035 L .76277 .6195 L p .949 .634 .401 r F P s P p .002 w .76037 .58954 m .76277 .6195 L .82366 .67035 L p .949 .634 .401 r F P s P p .004 w .35033 .45037 m .42795 .43515 L s P p .002 w .24319 .73658 m .24608 .70499 L .2032 .75852 L p 0 0 .312 r F P s P p .004 w .20897 .71506 m .23958 .68073 L s P p .002 w .2032 .75852 m .2 .78946 L .24319 .73658 L p 0 0 .312 r F P s P p .002 w .75802 .56018 m .75571 .53141 L .66489 .49312 L p .852 .55 .477 r F P s P p .002 w .66349 .46462 m .66489 .49312 L .75571 .53141 L p .852 .55 .477 r F P s P p .002 w .55317 .41868 m .55288 .3912 L .44085 .39734 L p .659 .392 .505 r F P s P p .002 w .44085 .39734 m .44001 .42478 L .55317 .41868 L p .659 .392 .505 r F P s P p .002 w .43828 .48136 m .43915 .45279 L .33749 .48168 L p .538 .293 .492 r F P s P p .002 w .33749 .48168 m .33561 .51005 L .43828 .48136 L p .538 .293 .492 r F P 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.66295 .02755 L s P p .002 w .12286 .25464 m .12766 .25957 L s P [(-1)] .11326 .24476 .97192 1 Mshowa p .002 w .24366 .20385 m .24822 .209 L s P [(-0.5)] .23453 .19353 .88455 1 Mshowa p .002 w .3711 .15026 m .37539 .15565 L s P [(0)] .36251 .13949 .79718 1 Mshowa p .002 w .50575 .09364 m .50974 .09926 L s P [(0.5)] .49778 .08241 .70982 1 Mshowa p .002 w .64825 .03373 m .65189 .03958 L s P [(1)] .64097 .02204 .62245 1 Mshowa p .001 w .14652 .24469 m .14937 .24768 L s P p .001 w .17042 .23464 m .17324 .23766 L s P p .001 w .19457 .22448 m .19737 .22753 L s P p .001 w .21898 .21422 m .22175 .21729 L s P p .001 w .2686 .19336 m .2713 .19648 L s P p .001 w .29381 .18276 m .29648 .18591 L s P p .001 w .31929 .17204 m .32193 .17522 L s P p .001 w .34505 .16121 m .34766 .16442 L s P p .001 w .39743 .13919 m .39998 .14245 L s P p .001 w .42406 .12799 m .42657 .13128 L s P p .001 w .45099 .11667 m .45346 .11998 L s P p .001 w .47822 .10522 m .48065 .10856 L s P p .001 w .5336 .08193 m .53596 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.25513 .94281 L s P p .001 w .27208 .95909 m .27239 .95497 L s P p .001 w .28901 .97101 m .28931 .96689 L s P p .001 w .30561 .98269 m .30589 .97857 L s P p .001 w .33785 1.00538 m .33808 1.00125 L s P p .001 w .35351 1.01639 m .35372 1.01227 L s P p .001 w .36886 1.0272 m .36905 1.02307 L s P p .001 w .38393 1.0378 m .3841 1.03367 L s P P p P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup] Graphics3D["<<>>"] ;[o] -Graphics3D- :[font = text; inactive; preserveAspect; endGroup] The cylinder in this diagram has a radius equal to the energy of the oscillator. The trajectory of this oscillator is clearly seen to lie on this surface. :[font = section; inactive; Cclosed; preserveAspect; startGroup] Problem 23 :[font = subsection; inactive; preserveAspect; startGroup] Problem 23(a) :[font = input; preserveAspect; startGroup] Clear[x,y,u,a,b,p,q] Equation0 = u[x,y]-a x-b y-a^2 b^2 == 0; Equations = {Equation0,D[Equation0,x],D[Equation0,y]}/. {u[x,y]->u, Derivative[1,0][u][x,y]->p, Derivative[0,1][u][x,y]->q}; Eliminate[Equations,{a,b}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] p^2*q^2 + p*x == u - q*y ;[o] 2 2 p q + p x == u - q y :[font = subsection; inactive; preserveAspect; startGroup] Problem 23(b) :[font = input; preserveAspect; startGroup] Clear[x,y,u,a,b,p,q] Equation0 = (u[x,y])^2-a x^2-y+b == 0; Equations = {Equation0,D[Equation0,x],D[Equation0,y]}/. {u[x,y]->u, Derivative[1,0][u][x,y]->p, Derivative[0,1][u][x,y]->q}; Eliminate[Equations,{a,b}][[1]] :[font = output; output; inactive; preserveAspect; endGroup] x != 0 && 2*q*u == 1 ;[o] x != 0 && 2 q u == 1 :[font = text; inactive; preserveAspect; endGroup] The differential`equation is 2uq=1. :[font = subsection; inactive; preserveAspect; startGroup] Problem 23(c) :[font = input; preserveAspect; startGroup] Clear[x,y,u,a,b,p,q] Equation0 = (x u[x,y])^2-a x-(y+b)^2 == 0; Equations = {Equation0,D[Equation0,x],D[Equation0,y]}/. {u[x,y]->u, Derivative[1,0][u][x,y]->p, Derivative[0,1][u][x,y]->q}; Eliminate[Equations,{a,b}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 2*p*u*x^3 + u^2*x^2*(1 + q^2*x^2) == 0 ;[o] 3 2 2 2 2 2 p u x + u x (1 + q x ) == 0 :[font = subsection; inactive; preserveAspect; startGroup] Problem 23(d) :[font = input; preserveAspect; startGroup] Clear[x,y,u,a,b,p,q] Equation0 = (x+u[x,y])^2-a x-(y+b)^2 == 0; Equations = {Equation0,D[Equation0,x],D[Equation0,y]}/. {u[x,y]->u, Derivative[1,0][u][x,y]->p, Derivative[0,1][u][x,y]->q}; Eliminate[Equations,{a,b}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] (-1 + q^2)*u^2 + (2*p + 2*q^2)*u*x == (-1 - 2*p - q^2)*x^2 ;[o] 2 2 2 2 2 (-1 + q ) u + (2 p + 2 q ) u x == (-1 - 2 p - q ) x :[font = subsection; inactive; preserveAspect; startGroup] Problem 23(e) :[font = input; preserveAspect; startGroup] Clear[x,y,u,a,b,p,q] Equation0 = a x+u[x,y]-Sqrt[a x+(y+b)^2] == 0; Equations = {Equation0,D[Equation0,x],D[Equation0,y]}/. {u[x,y]->u, Derivative[1,0][u][x,y]->p, Derivative[0,1][u][x,y]->q}; Simplify[Eliminate[Equations,{a,b}]][[1]] :[font = output; output; inactive; preserveAspect; endGroup] u - p*x != 0 && x != 0 && p != 0 && 1 + q^2 != 0 && (1 - q^2)*u*(1 + q^2 - 4*u + 8*p*x) == p*x*(2 + 2*q^2 + 4*p*x - 4*p*q^2*x) && ((-1 + q^2)*u*(1 + q^2 - 4*u + 8*p*x))/(1 + q^2) == p*x*(-2 + 4*p*x - (8*p*x)/(1 + q^2)) ;[o] 2 u - p x != 0 && x != 0 && p != 0 && 1 + q != 0 && 2 2 (1 - q ) u (1 + q - 4 u + 8 p x) == 2 2 p x (2 + 2 q + 4 p x - 4 p q x) && 2 2 (-1 + q ) u (1 + q - 4 u + 8 p x) ---------------------------------- == 2 1 + q 8 p x p x (-2 + 4 p x - ------) 2 1 + q :[font = text; inactive; preserveAspect; endGroup] The differential equation is u(1-q^2)(1+q^2-4u+8xp)=xp(2+2q^2+4xp- 4xpq^2). :[font = subsection; inactive; preserveAspect; startGroup] Problem 23(f) :[font = input; preserveAspect; startGroup] Clear[x,y,u,a,b,p,q] Equation0 = u[x,y]-(a x+b y)^2 == 0; Equations = {Equation0,D[Equation0,x],D[Equation0,y]}/. {u[x,y]->u, Derivative[1,0][u][x,y]->p, Derivative[0,1][u][x,y]->q}; Eliminate[Equations,{a,b}][[1]] :[font = output; output; inactive; preserveAspect; endGroup] q != 0 && u != 0 && p*x == 2*u - q*y ;[o] q != 0 && u != 0 && p x == 2 u - q y :[font = text; inactive; preserveAspect; endGroup] The differential equation is xp+yq=2u. :[font = subsection; inactive; preserveAspect; startGroup] Problem 23(g) :[font = input; preserveAspect; startGroup] Clear[x,y,u,a,b,p,q] Equation0 = (u[x,y])^2-(a x+y+b)^(1/2) == 0; Equations = {Equation0,D[Equation0,x],D[Equation0,y]}/. {u[x,y]->u, Derivative[1,0][u][x,y]->p, Derivative[0,1][u][x,y]->q}; Eliminate[Equations,{a,b}] :[font = output; output; inactive; preserveAspect; endGroup] u != 0 && 4*q*u^3 == 1 ;[o] 3 u != 0 && 4 q u == 1 :[font = text; inactive; preserveAspect] The differential equation is 4u^3q=1. An alternative way of obtaining this result is :[font = input; preserveAspect; startGroup] Clear[x,y,u,a,b,p,q] Equation0 = (u[x,y])^4-(a x+y+b) == 0; Equations = {Equation0,D[Equation0,x],D[Equation0,y]}/. {u[x,y]->u, Derivative[1,0][u][x,y]->p, Derivative[0,1][u][x,y]->q}; Eliminate[Equations,{a,b}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] 4*q*u^3 == 1 ;[o] 3 4 q u == 1 :[font = section; inactive; Cclosed; preserveAspect; startGroup] Problem 28 :[font = input; preserveAspect; startGroup] Plot3D[-y Sqrt[x^2+1],{x,-1,1},{y,-1,1}, BoxRatios->{1,1,1},ViewPoint->{1,1,1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 100; pictureWidth = 300; pictureHeight = 297] %! %%Creator: Mathematica %%AspectRatio: .9903 MathPictureStart %% SurfaceGraphics /Courier findfont 10 scalefont setfont % Scaling calculations -0.0458364 1.09167 0.0272918 1.09167 [ [(-1)] .5 .97666 0 -1 Msboxa [(-0.5)] .40838 .95415 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.89392 .57313 m .88766 .57584 L s P [(0)] .90645 .56771 -1 .43301 Mshowa p .002 w .94373 .74565 m .93722 .74769 L s P [(1)] .95675 .74157 -1 .31361 Mshowa p .001 w .86147 .4607 m .85783 .46257 L s P p .001 w .86907 .48703 m .8654 .48885 L s P p .001 w .877 .5145 m .8733 .51625 L s P p .001 w .88527 .54317 m .88155 .54486 L s P p .001 w .90297 .60446 m .89918 .60602 L s P p .001 w .91244 .63727 m .90862 .63875 L s P p .001 w .92237 .67166 m .91852 .67306 L s P p .001 w .93278 .70774 m .92891 .70906 L s P p .001 w .84716 .41115 m .84358 .41313 L s P p .001 w .84043 .38782 m .83687 .38985 L s P p .001 w .95524 .78553 m .9513 .78666 L s P p .001 w .96736 .82753 m .9634 .82856 L s P P 0 0 m 1 0 L 1 .9903 L 0 .9903 L closepath clip newpath p .002 w .5 .57313 m .5 .96301 L s .5 .96301 m .97271 .84605 L s .97271 .84605 m .83765 .37819 L s .83765 .37819 m .5 .57313 L s .16235 .37819 m .5 .02729 L s .5 .02729 m .5 .57313 L s .5 .57313 m .02729 .84605 L s .02729 .84605 m .16235 .37819 L s .5 .57313 m .5 .96301 L s .5 .96301 m .02729 .84605 L s .02729 .84605 m .16235 .37819 L s .16235 .37819 m .5 .57313 L s .83765 .37819 m .5 .02729 L s .5 .02729 m .5 .57313 L s .5 .57313 m .97271 .84605 L s .97271 .84605 m .83765 .37819 L s P p .791 .698 .762 r .0015 w .5 .95116 .52398 .91131 .5 .89051 .47578 .92857 Metetra .79 .71 .776 r .47578 .92857 .5 .89051 .4754 .87044 .45097 .90696 Metetra .791 .727 .792 r .45097 .90696 .4754 .87044 .45008 .85136 .42547 .88664 Metetra .796 .748 .809 r .42547 .88664 .45008 .85136 .42393 .83361 .39915 .86801 Metetra .803 .774 .826 r .39915 .86801 .42393 .83361 .39678 .81752 .37182 .85147 Metetra .814 .804 .842 r .37182 .85147 .39678 .81752 .36845 .80345 .34326 .83743 Metetra .827 .836 .855 r .34326 .83743 .36845 .80345 .3387 .79168 .31321 .82623 Metetra .842 .868 .862 r .31321 .82623 .3387 .79168 .30725 .78239 .28135 .81809 Metetra .856 .895 .863 r .28135 .81809 .30725 .78239 .27381 .77562 .24735 .81309 Metetra .87 .915 .857 r .24735 .81309 .27381 .77562 .23806 .77128 .21086 .81114 Metetra .883 .928 .847 r .21086 .81114 .23806 .77128 .19965 .76921 .17149 .81209 Metetra .895 .934 .834 r .17149 .81209 .19965 .76921 .15822 .76922 .12883 .81575 Metetra .906 .936 .82 r .12883 .81575 .15822 .76922 .11336 .77112 .08243 .82196 Metetra .916 .934 .806 r .08243 .82196 .11336 .77112 .06461 .77478 .03175 .8306 Metetra .803 .716 .768 r .52398 .91131 .548 .87139 .52433 .85228 .5 .89051 Metetra .802 .727 .782 r .5 .89051 .52433 .85228 .5 .83366 .4754 .87044 Metetra .802 .742 .798 r .4754 .87044 .5 .83366 .47492 .81575 .45008 .85136 Metetra .804 .761 .813 r .45008 .85136 .47492 .81575 .44899 .79881 .42393 .83361 Metetra .81 .784 .829 r .42393 .83361 .44899 .79881 .42208 .78313 .39678 .81752 Metetra .818 .811 .844 r .39678 .81752 .42208 .78313 .39401 .76899 .36845 .80345 Metetra .829 .839 .855 r .36845 .80345 .39401 .76899 .36458 .75662 .3387 .79168 Metetra .841 .866 .862 r .3387 .79168 .36458 .75662 .33355 .74615 .30725 .78239 Metetra .854 .889 .862 r .30725 .78239 .33355 .74615 .30066 .73761 .27381 .77562 Metetra .867 .907 .858 r .27381 .77562 .30066 .73761 .26562 .73089 .23806 .77128 Metetra .879 .918 .848 r .23806 .77128 .26562 .73089 .22813 .72584 .19965 .76921 Metetra .891 .924 .836 r .19965 .76921 .22813 .72584 .18787 .72226 .15822 .76922 Metetra .902 .925 .823 r .15822 .76922 .18787 .72226 .14448 .71997 .11336 .77112 Metetra .912 .923 .809 r .11336 .77112 .14448 .71997 .09756 .71881 .06461 .77478 Metetra .816 .734 .774 r .548 .87139 .57207 .83138 .54878 .81387 .52433 .85228 Metetra .813 .745 .788 r .52433 .85228 .54878 .81387 .52478 .79661 .5 .83366 Metetra .812 .759 .803 r .5 .83366 .52478 .79661 .5 .77979 .47492 .81575 Metetra .813 .775 .818 r .47492 .81575 .5 .77979 .47435 .76361 .44899 .79881 Metetra .816 .795 .833 r .44899 .79881 .47435 .76361 .44771 .74828 .42208 .78313 Metetra .822 .817 .846 r .42208 .78313 .44771 .74828 .41993 .73403 .39401 .76899 Metetra .83 .841 .856 r .39401 .76899 .41993 .73403 .39085 .72103 .36458 .75662 Metetra .84 .864 .861 r .36458 .75662 .39085 .72103 .36024 .70937 .33355 .74615 Metetra .851 .883 .862 r .33355 .74615 .36024 .70937 .32789 .69904 .30066 .73761 Metetra .863 .898 .858 r .30066 .73761 .32789 .69904 .29355 .68996 .26562 .73089 Metetra .875 .908 .849 r .26562 .73089 .29355 .68996 .25695 .68197 .22813 .72584 Metetra .886 .913 .838 r .22813 .72584 .25695 .68197 .2178 .67488 .18787 .72226 Metetra .898 .914 .825 r .18787 .72226 .2178 .67488 .1758 .66851 .14448 .71997 Metetra .908 .911 .812 r .14448 .71997 .1758 .66851 .1306 .66269 .09756 .71881 Metetra .829 .754 .78 r .57207 .83138 .59619 .7913 .57334 .77528 .54878 .81387 Metetra .825 .763 .794 r .54878 .81387 .57334 .77528 .54974 .7593 .52478 .79661 Metetra .823 .775 .808 r .52478 .79661 .54974 .7593 .52532 .74349 .5 .77979 Metetra .821 .79 .822 r .5 .77979 .52532 .74349 .5 .72799 .47435 .76361 Metetra .822 .806 .836 r .47435 .76361 .5 .72799 .47368 .71297 .44771 .74828 Metetra .825 .824 .847 r .44771 .74828 .47368 .71297 .44623 .69856 .41993 .73403 Metetra .831 .843 .856 r .41993 .73403 .44623 .69856 .41751 .6849 .39085 .72103 Metetra .839 .861 .861 r .39085 .72103 .41751 .6849 .38734 .67202 .36024 .70937 Metetra .848 .877 .861 r .36024 .70937 .38734 .67202 .35553 .65992 .32789 .69904 Metetra .859 .889 .857 r .32789 .69904 .35553 .65992 .32185 .64849 .29355 .68996 Metetra .87 .897 .85 r .29355 .68996 .32185 .64849 .28609 .63759 .25695 .68197 Metetra .881 .901 .839 r .25695 .68197 .28609 .63759 .248 .62706 .2178 .67488 Metetra .892 .901 .827 r .2178 .67488 .248 .62706 .20731 .61672 .1758 .66851 Metetra .902 .899 .813 r .1758 .66851 .20731 .61672 .16372 .60641 .1306 .66269 Metetra .842 .774 .785 r .59619 .7913 .62035 .75115 .59802 .73651 .57334 .77528 Metetra .837 .782 .799 r .57334 .77528 .59802 .73651 .57488 .72171 .54974 .7593 Metetra .833 .792 .813 r .54974 .7593 .57488 .72171 .55088 .70684 .52532 .74349 Metetra .83 .804 .826 r .52532 .74349 .55088 .70684 .52595 .69196 .5 .72799 Metetra .828 .817 .839 r .5 .72799 .52595 .69196 .5 .67718 .47368 .71297 Metetra .829 .831 .849 r .47368 .71297 .5 .67718 .47292 .66258 .44623 .69856 Metetra .832 .846 .857 r .44623 .69856 .47292 .66258 .44459 .64821 .41751 .6849 Metetra .838 .859 .861 r .41751 .6849 .44459 .64821 .41486 .6341 .38734 .67202 Metetra .845 .871 .861 r .38734 .67202 .41486 .6341 .38357 .62021 .35553 .65992 Metetra .854 .88 .857 r .35553 .65992 .38357 .62021 .35054 .60645 .32185 .64849 Metetra .865 .886 .85 r .32185 .64849 .35054 .60645 .31558 .59269 .28609 .63759 Metetra .875 .888 .839 r .28609 .63759 .31558 .59269 .27849 .5788 .248 .62706 Metetra .886 .888 .827 r .248 .62706 .27849 .5788 .23902 .56461 .20731 .61672 Metetra .896 .885 .814 r .20731 .61672 .23902 .56461 .19694 .54998 .16372 .60641 Metetra .855 .794 .79 r .62035 .75115 .64457 .71091 .62281 .69755 .59802 .73651 Metetra .849 .801 .804 r .59802 .73651 .62281 .69755 .60021 .68385 .57488 .72171 Metetra .843 .809 .817 r .57488 .72171 .60021 .68385 .5767 .66983 .55088 .70684 Metetra .838 .818 .83 r .55088 .70684 .5767 .66983 .55221 .65551 .52595 .69196 Metetra .834 .828 .841 r .52595 .69196 .55221 .65551 .52668 .64091 .5 .67718 Metetra .833 .838 .851 r .5 .67718 .52668 .64091 .5 .62606 .47292 .66258 Metetra .833 .848 .857 r .47292 .66258 .5 .62606 .47208 .61096 .44459 .64821 Metetra .836 .857 .86 r .44459 .64821 .47208 .61096 .4428 .59559 .41486 .6341 Metetra .842 .865 .86 r .41486 .6341 .4428 .59559 .41203 .57991 .38357 .62021 Metetra .85 .871 .856 r .38357 .62021 .41203 .57991 .37962 .56384 .35054 .60645 Metetra .859 .874 .849 r .35054 .60645 .37962 .56384 .34542 .54727 .31558 .59269 Metetra .869 .875 .839 r .31558 .59269 .34542 .54727 .30925 .53009 .27849 .5788 Metetra .879 .874 .827 r .27849 .5788 .30925 .53009 .27093 .51218 .23902 .56461 Metetra .889 .871 .814 r .23902 .56461 .27093 .51218 .23025 .4934 .19694 .54998 Metetra .868 .815 .795 r .64457 .71091 .66882 .6706 .64772 .65842 .62281 .69755 Metetra .86 .821 .808 r .62281 .69755 .64772 .65842 .62572 .64571 .60021 .68385 Metetra .853 .827 .821 r .60021 .68385 .62572 .64571 .60276 .63246 .5767 .66983 Metetra .846 .833 .833 r .5767 .66983 .60276 .63246 .57878 .61861 .55221 .65551 Metetra .84 .839 .844 r .55221 .65551 .57878 .61861 .55372 .60414 .52668 .64091 Metetra .836 .845 .852 r .52668 .64091 .55372 .60414 .52748 .589 .5 .62606 Metetra .834 .85 .858 r .5 .62606 .52748 .589 .5 .57313 .47208 .61096 Metetra .835 .855 .86 r .47208 .61096 .5 .57313 .47118 .55649 .4428 .59559 Metetra .839 .859 .859 r .4428 .59559 .47118 .55649 .44091 .53901 .41203 .57991 Metetra .845 .861 .855 r .41203 .57991 .44091 .53901 .40909 .52064 .37962 .56384 Metetra .853 .862 .848 r .37962 .56384 .40909 .52064 .3756 .50131 .34542 .54727 Metetra .862 .861 .838 r .34542 .54727 .3756 .50131 .3403 .48093 .30925 .53009 Metetra .871 .859 .826 r .30925 .53009 .3403 .48093 .30304 .45941 .27093 .51218 Metetra .88 .856 .813 r .27093 .51218 .30304 .45941 .26365 .43667 .23025 .4934 Metetra .88 .836 .798 r .66882 .6706 .69313 .63021 .67275 .61909 .64772 .65842 Metetra .872 .84 .812 r .64772 .65842 .67275 .61909 .65142 .60729 .62572 .64571 Metetra .863 .844 .825 r .62572 .64571 .65142 .60729 .62908 .59472 .60276 .63246 Metetra .854 .847 .836 r .60276 .63246 .62908 .59472 .60567 .58128 .57878 .61861 Metetra .846 .85 .846 r .57878 .61861 .60567 .58128 .58113 .56687 .55372 .60414 Metetra .84 .851 .853 r .55372 .60414 .58113 .56687 .55538 .55138 .52748 .589 Metetra .836 .852 .858 r .52748 .589 .55538 .55138 .52836 .5347 .5 .57313 Metetra .834 .853 .86 r .5 .57313 .52836 .5347 .5 .51676 .47118 .55649 Metetra .836 .852 .858 r .47118 .55649 .5 .51676 .47023 .4975 .44091 .53901 Metetra .84 .851 .853 r .44091 .53901 .47023 .4975 .43898 .47685 .40909 .52064 Metetra .846 .85 .846 r .40909 .52064 .43898 .47685 .40615 .4548 .3756 .50131 Metetra .854 .847 .836 r .3756 .50131 .40615 .4548 .37164 .4313 .3403 .48093 Metetra .863 .844 .825 r .3403 .48093 .37164 .4313 .33535 .40632 .30304 .45941 Metetra .872 .84 .812 r .30304 .45941 .33535 .40632 .29713 .37978 .26365 .43667 Metetra .892 .857 .8 r .69313 .63021 .71748 .58974 .69789 .57959 .67275 .61909 Metetra .882 .86 .814 r .67275 .61909 .69789 .57959 .67731 .56859 .65142 .60729 Metetra .872 .861 .827 r .65142 .60729 .67731 .56859 .65566 .55661 .62908 .59472 Metetra .861 .862 .838 r .62908 .59472 .65566 .55661 .63289 .5435 .60567 .58128 Metetra .852 .86 .848 r .60567 .58128 .63289 .5435 .60892 .52909 .58113 .56687 Metetra .843 .858 .854 r .58113 .56687 .60892 .52909 .58369 .51319 .55538 .55138 Metetra .837 .855 .858 r .55538 .55138 .58369 .51319 .55716 .49567 .52836 .5347 Metetra .833 .85 .859 r .52836 .5347 .55716 .49567 .52928 .47641 .5 .51676 Metetra .832 .846 .857 r .5 .51676 .52928 .47641 .5 .45535 .47023 .4975 Metetra .834 .842 .851 r .47023 .4975 .5 .45535 .46928 .43245 .43898 .47685 Metetra .839 .837 .844 r .43898 .47685 .46928 .43245 .43706 .40774 .40615 .4548 Metetra .846 .833 .834 r .40615 .4548 .43706 .40774 .40328 .38121 .37164 .4313 Metetra .854 .828 .822 r .37164 .4313 .40328 .38121 .36786 .35288 .33535 .40632 Metetra .862 .824 .809 r .33535 .40632 .36786 .35288 .33072 .32274 .29713 .37978 Metetra .903 .878 .801 r .71748 .58974 .74188 .54919 .72316 .53989 .69789 .57959 Metetra .892 .879 .815 r .69789 .57959 .72316 .53989 .70339 .5296 .67731 .56859 Metetra .88 .878 .829 r .67731 .56859 .70339 .5296 .68251 .51813 .65566 .55661 Metetra .868 .876 .84 r .65566 .55661 .68251 .51813 .66043 .50527 .63289 .5435 Metetra .857 .871 .849 r .63289 .5435 .66043 .50527 .63709 .49078 .60892 .52909 Metetra .846 .865 .856 r .60892 .52909 .63709 .49078 .61244 .47443 .58369 .51319 Metetra .838 .857 .859 r .58369 .51319 .61244 .47443 .58643 .45602 .55716 .49567 Metetra .832 .848 .859 r .55716 .49567 .58643 .45602 .55903 .43542 .52928 .47641 Metetra .829 .84 .855 r .52928 .47641 .55903 .43542 .53022 .41255 .5 .45535 Metetra .829 .832 .849 r .5 .45535 .53022 .41255 .5 .38743 .46928 .43245 Metetra .832 .824 .841 r .46928 .43245 .5 .38743 .46834 .36011 .43706 .40774 Metetra .837 .818 .831 r .43706 .40774 .46834 .36011 .43521 .33065 .40328 .38121 Metetra .844 .812 .819 r .40328 .38121 .43521 .33065 .40058 .29911 .36786 .35288 Metetra .852 .807 .806 r .36786 .35288 .40058 .29911 .36439 .26554 .33072 .32274 Metetra .912 .898 .8 r .74188 .54919 .76633 .50856 .74854 .50001 .72316 .53989 Metetra .9 .898 .815 r .72316 .53989 .74854 .50001 .72967 .49032 .70339 .5296 Metetra .888 .895 .829 r .70339 .5296 .72967 .49032 .70961 .47926 .68251 .51813 Metetra .875 .89 .841 r .68251 .51813 .70961 .47926 .68831 .46656 .66043 .50527 Metetra .862 .882 .85 r .66043 .50527 .68831 .46656 .66566 .45193 .63709 .49078 Metetra .85 .871 .856 r .63709 .49078 .66566 .45193 .64163 .43507 .61244 .47443 Metetra .839 .859 .859 r .61244 .47443 .64163 .43507 .61616 .41573 .58643 .45602 Metetra .831 .846 .858 r .58643 .45602 .61616 .41573 .58926 .39376 .55903 .43542 Metetra .825 .833 .854 r .55903 .43542 .58926 .39376 .56091 .3691 .53022 .41255 Metetra .823 .822 .847 r .53022 .41255 .56091 .3691 .53116 .34178 .5 .38743 Metetra .824 .811 .838 r .5 .38743 .53116 .34178 .5 .31191 .46834 .36011 Metetra .828 .803 .827 r .46834 .36011 .5 .31191 .46745 .27961 .43521 .33065 Metetra .834 .796 .815 r .43521 .33065 .46745 .27961 .43351 .245 .40058 .29911 Metetra .841 .79 .802 r .40058 .29911 .43351 .245 .39815 .20818 .36439 .26554 Metetra .92 .917 .797 r .76633 .50856 .79083 .46785 .77405 .45993 .74854 .50001 Metetra .908 .915 .813 r .74854 .50001 .77405 .45993 .75614 .45074 .72967 .49032 Metetra .894 .911 .828 r .72967 .49032 .75614 .45074 .737 .44 .70961 .47926 Metetra .881 .903 .841 r .70961 .47926 .737 .44 .71652 .42739 .68831 .46656 Metetra .867 .892 .851 r .68831 .46656 .71652 .42739 .69464 .41253 .66566 .45193 Metetra .853 .878 .857 r .66566 .45193 .69464 .41253 .67127 .3951 .64163 .43507 Metetra .84 .861 .86 r .64163 .43507 .67127 .3951 .64638 .37479 .61616 .41573 Metetra .83 .844 .858 r .61616 .41573 .64638 .37479 .61997 .35143 .58926 .39376 Metetra .822 .827 .853 r .58926 .39376 .61997 .35143 .59208 .32497 .56091 .3691 Metetra .817 .811 .845 r .56091 .3691 .59208 .32497 .56275 .29547 .53116 .34178 Metetra .816 .798 .835 r .53116 .34178 .56275 .29547 .53205 .26311 .5 .31191 Metetra .819 .788 .823 r .5 .31191 .53205 .26311 .5 .22807 .46745 .27961 Metetra .823 .779 .811 r .46745 .27961 .5 .22807 .46665 .19054 .43351 .245 Metetra .83 .772 .798 r .43351 .245 .46665 .19054 .43201 .15067 .39815 .20818 Metetra .925 .935 .791 r .79083 .46785 .81537 .42706 .79968 .41967 .77405 .45993 Metetra .913 .932 .809 r .77405 .45993 .79968 .41967 .78281 .41087 .75614 .45074 Metetra .9 .926 .826 r .75614 .45074 .78281 .41087 .76465 .40035 .737 .44 Metetra .886 .916 .84 r .737 .44 .76465 .40035 .74509 .38773 .71652 .42739 Metetra .871 .902 .851 r .71652 .42739 .74509 .38773 .72402 .37258 .69464 .41253 Metetra .856 .884 .858 r .69464 .41253 .72402 .37258 .70136 .35451 .67127 .3951 Metetra .841 .864 .86 r .67127 .3951 .70136 .35451 .67709 .33318 .64638 .37479 Metetra .828 .842 .857 r .64638 .37479 .67709 .33318 .65119 .3084 .61997 .35143 Metetra .818 .82 .851 r .61997 .35143 .65119 .3084 .62374 .28014 .59208 .32497 Metetra .811 .801 .842 r .59208 .32497 .62374 .28014 .5948 .24851 .56275 .29547 Metetra .808 .785 .831 r .56275 .29547 .5948 .24851 .56448 .21373 .53205 .26311 Metetra .809 .772 .819 r .53205 .26311 .56448 .21373 .53286 .17605 .5 .22807 Metetra .812 .762 .806 r .5 .22807 .53286 .17605 .5 .13573 .46665 .19054 Metetra .818 .754 .793 r .46665 .19054 .5 .13573 .46596 .09299 .43201 .15067 Metetra .928 .95 .783 r .81537 .42706 .83996 .3862 .82543 .37921 .79968 .41967 Metetra .917 .947 .803 r .79968 .41967 .82543 .37921 .80968 .3707 .78281 .41087 Metetra .904 .94 .822 r .78281 .41087 .80968 .3707 .79259 .3603 .76465 .40035 Metetra .89 .929 .839 r .76465 .40035 .79259 .3603 .77401 .34758 .74509 .38773 Metetra .875 .912 .851 r .74509 .38773 .77401 .34758 .75382 .33205 .72402 .37258 Metetra 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m .71881 .38097 L .70544 .40637 L p .863 .774 .762 r F P s P p .002 w .70544 .40637 m .72795 .41021 L .742 .38404 L p .863 .774 .762 r F P s P p .002 w .51085 .85273 m .49872 .83714 L .48411 .86435 L p .923 .883 .776 r F P s P p .002 w .48411 .86435 m .49555 .8806 L .51085 .85273 L p .923 .883 .776 r F P s P p .002 w .72795 .41021 m .70544 .40637 L .69203 .43183 L p .867 .78 .763 r F P s P p .002 w .69203 .43183 m .71387 .43643 L .72795 .41021 L p .867 .78 .763 r F P s P p .002 w .49555 .8806 m .48411 .86435 L .46947 .89163 L p .926 .889 .775 r F P s P p .002 w .46947 .89163 m .48022 .90851 L .49555 .8806 L p .926 .889 .775 r F P s P p .002 w .71387 .43643 m .69203 .43183 L .67858 .45737 L p .871 .786 .764 r F P s P p .002 w .67858 .45737 m .69976 .46271 L .71387 .43643 L p .871 .786 .764 r F P s P p .002 w .69976 .46271 m .67858 .45737 L .66509 .48299 L p .875 .792 .765 r F P s P p .002 w .66509 .48299 m .68561 .48906 L .69976 .46271 L p .875 .792 .765 r F P s P p .002 w .68561 .48906 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