(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 77537, 2480]*) (*NotebookOutlinePosition[ 78174, 2502]*) (* CellTagsIndexPosition[ 78130, 2498]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["The two cones problem ", "Subtitle"], Cell["After A. Oldknow and others", "Subsubtitle"], Cell[CellGroupData[{ Cell["Cardboard cones", "Section"], Cell[CellGroupData[{ Cell["Single cone", "Subsection"], Cell["\<\ We cut a sector of a certain angle from a cardboard circle, roll it \ into a cone and find the volume. We assume the radius is our unit \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(R = 1\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Closed]], Cell["\<\ The basis of the cone is called r and its height h . These \ relate by\ \>", "Text"], Cell[BoxData[ \(\(\(Clear[f, sol, volume]\)\(\n\) \)\)], "Input"], Cell[BoxData[ \(sol[angle_] := Solve[{r == \ R\ \(angle/2\)/\[Pi], r^2 + h^2 == R^2}, {r, h}]\)], "Input"], Cell[BoxData[ \(volume[angle_] := \ 1/3\ \[Pi]\ r^2\ h /. Last[sol[angle]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[volume[angle], {angle, 0, 2\ \[Pi]}, AspectRatio -> Automatic]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .06415 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.151576 0.00152738 0.151576 [ [.17539 -0.01097 -3 -9 ] [.17539 -0.01097 3 0 ] [.32696 -0.01097 -3 -9 ] [.32696 -0.01097 3 0 ] [.47854 -0.01097 -3 -9 ] [.47854 -0.01097 3 0 ] [.63011 -0.01097 -3 -9 ] [.63011 -0.01097 3 0 ] [.78169 -0.01097 -3 -9 ] [.78169 -0.01097 3 0 ] [.93327 -0.01097 -3 -9 ] [.93327 -0.01097 3 0 ] [.01131 .01668 -18 -4.5 ] [.01131 .01668 0 4.5 ] [.01131 .03184 -18 -4.5 ] [.01131 .03184 0 4.5 ] [.01131 .047 -18 -4.5 ] [.01131 .047 0 4.5 ] [.01131 .06216 -18 -4.5 ] [.01131 .06216 0 4.5 ] [ 0 0 0 0 ] [ 1 .06415 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .17539 .00153 m .17539 .00778 L s [(1)] .17539 -0.01097 0 1 Mshowa .32696 .00153 m .32696 .00778 L s [(2)] .32696 -0.01097 0 1 Mshowa .47854 .00153 m .47854 .00778 L s [(3)] .47854 -0.01097 0 1 Mshowa .63011 .00153 m .63011 .00778 L s [(4)] .63011 -0.01097 0 1 Mshowa .78169 .00153 m .78169 .00778 L s [(5)] .78169 -0.01097 0 1 Mshowa .93327 .00153 m .93327 .00778 L s [(6)] .93327 -0.01097 0 1 Mshowa .125 Mabswid .05412 .00153 m .05412 .00528 L s .08444 .00153 m .08444 .00528 L s .11476 .00153 m .11476 .00528 L s .14507 .00153 m .14507 .00528 L s .2057 .00153 m .2057 .00528 L s .23602 .00153 m .23602 .00528 L s .26633 .00153 m .26633 .00528 L s .29665 .00153 m .29665 .00528 L s .35728 .00153 m .35728 .00528 L s .38759 .00153 m .38759 .00528 L s .41791 .00153 m .41791 .00528 L s .44822 .00153 m .44822 .00528 L s .50885 .00153 m .50885 .00528 L s .53917 .00153 m .53917 .00528 L s .56948 .00153 m .56948 .00528 L s .5998 .00153 m .5998 .00528 L s .66043 .00153 m .66043 .00528 L s .69074 .00153 m .69074 .00528 L s .72106 .00153 m .72106 .00528 L s .75138 .00153 m .75138 .00528 L s .81201 .00153 m .81201 .00528 L s .84232 .00153 m .84232 .00528 L s .87264 .00153 m .87264 .00528 L s .90295 .00153 m .90295 .00528 L s .96358 .00153 m .96358 .00528 L s .9939 .00153 m .9939 .00528 L s .25 Mabswid 0 .00153 m 1 .00153 L s .02381 .01668 m .03006 .01668 L s [(0.1)] .01131 .01668 1 0 Mshowa .02381 .03184 m .03006 .03184 L s [(0.2)] .01131 .03184 1 0 Mshowa .02381 .047 m .03006 .047 L s [(0.3)] .01131 .047 1 0 Mshowa .02381 .06216 m .03006 .06216 L s [(0.4)] .01131 .06216 1 0 Mshowa .125 Mabswid .02381 .00456 m .02756 .00456 L s .02381 .00759 m .02756 .00759 L s .02381 .01062 m .02756 .01062 L s .02381 .01365 m .02756 .01365 L s .02381 .01972 m .02756 .01972 L s .02381 .02275 m .02756 .02275 L s .02381 .02578 m .02756 .02578 L s .02381 .02881 m .02756 .02881 L s .02381 .03487 m .02756 .03487 L s .02381 .03791 m .02756 .03791 L s .02381 .04094 m .02756 .04094 L s .02381 .04397 m .02756 .04397 L s .02381 .05003 m .02756 .05003 L s .02381 .05306 m .02756 .05306 L s .02381 .05609 m .02756 .05609 L s .02381 .05913 m .02756 .05913 L s .25 Mabswid .02381 0 m .02381 .06415 L s 0 0 m 1 0 L 1 .06415 L 0 .06415 L closepath clip newpath .5 Mabswid .02381 .00153 m .02499 .00153 L .02605 .00153 L .02729 .00153 L .02846 .00153 L .03053 .00154 L .03279 .00154 L .03527 .00155 L .0379 .00156 L .04262 .00159 L .04749 .00163 L .05205 .00167 L .06244 .00179 L .07305 .00195 L .08274 .00213 L .10458 .00266 L .12357 .00326 L .14429 .00405 L .18493 .00601 L .22406 .00839 L .26565 .01143 L .30571 .01481 L .34426 .01845 L .38527 .02268 L .42475 .02705 L .46273 .03145 L .50315 .03627 L .54206 .04096 L .58342 .04587 L .62326 .05039 L .66159 .05439 L .70238 .05807 L .72263 .05959 L .74164 .06079 L .75981 .06169 L .77003 .06208 L .77939 .06235 L .78413 .06245 L .78674 .0625 L .7892 .06254 L .79149 .06256 L .79398 .06259 L .79507 .0626 L .79623 .06261 L .7973 .06261 L .79832 .06262 L .7996 .06262 L .80077 .06262 L .80203 .06262 L .80275 .06262 L Mistroke .80341 .06262 L .80462 .06262 L .80572 .06261 L .80698 .0626 L .80817 .06259 L .81089 .06257 L .81339 .06253 L .81902 .06242 L .82373 .0623 L .82878 .06213 L .83788 .06172 L .8487 .06104 L .85856 .06023 L .87768 .05804 L .88848 .05637 L .89845 .0545 L .91716 .04993 L .9278 .04653 L .93777 .04263 L .94755 .03787 L .95783 .03136 L .96207 .02795 L .9666 .02354 L .96921 .02043 L .97157 .01699 L .97271 .01498 L .97395 .01236 L .97512 .00905 L .97619 .00157 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{893.75, 57.3125}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgA000`40O003h00OogooogooogooP7oo 003oOoooOoooOon0Ool00?moo`Eoo`<005Eoo`@005Eoo`<005Moo`<005Aoo`<005Moo`800;]oo`00 ogoo1Woo00<007ooOol0EGoo00<007ooOol0F7oo00<007ooOol0EGoo00<007ooOol0EWoo00<007oo Ool0Dgoo00@007ooOol00;Yoo`00ogoo1Woo00<007ooOol0EWoo00<007ooOol0Egoo00<007ooOol0 Dgoo1000Egoo00<007ooOol0Dgoo00@007ooOol00;Yoo`00ogoo1Woo00<007ooOol0Egoo00<007oo Ool0E7oo0P00EWoo00<007oo0000EGoo0`00EWoo0`00^goo003oOol6Ool00`00Oomoo`1DOol01000 Oomoo`00F7oo00<007ooOol0Dgoo00<007oo0000EGoo00<007ooOol0EWoo00<007ooOol0^goo003o Ool5Ool2001GOol2001FOol3001GOol2001EOol4001FOol3002jOol00?mooomooomooh1oo`00ogoo ogooogooP7oo003oOoooOoooOon0Ool00?mooomooomooh1oo`00ogooogooogooP7oo003oOoooOooo Oon0Ool00?mooomooomooh1oo`00ogooogooogooP7oo002]Ool00`00Oomoo`3oOoooOoo?Ool009mo ool00?l004`009Aoo`00[Goo0`003goo00<007ooOol03Woo00<007ooOol03goo7P001Woo00<007oo Ool03Woo00<007ooOol03goo00<007ooOol03goo00<007ooOol03goo00<007ooOol03Woo00<007oo Ool03goo00<007ooOol03goo00<007ooOol03goo00<007ooOol03Woo00<007ooOol03goo00<007oo Ool03goo00<007ooOol03Woo00<007ooOol03goo00<007ooOol03goo00<007ooOol03goo00<007oo Ool03Woo00<007ooOol03goo00<007ooOol03goo00<007ooOol03goo00<007ooOol03Woo00<007oo Ool03goo00<007ooOol03goo00<007ooOol03goo00<007ooOol03Woo00<007ooOol03goo00<007oo Ool03goo00<007ooOol017oo00<007ooOol027oo00<007ooOol0UGoo002]Ool00`00Oomoo`0?Ool0 0`00Oomoo`0>Ool00`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomoo`09OolB0005Ool00`00Oomo o`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0>Ool00`00Oomoo`0?Ool00`00Oomo o`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0>Ool00`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomo o`0>Ool00`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0>Ool00`00Oomo o`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0>Ool00`00Oomoo`0?Ool00`00Oomo o`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0>Ool00`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomo o`04Ool00`00Oomoo`08Ool00`00Oomoo`2EOol00:eoo`<005Ioo`03001oogoo00Uoo`/0045oo`03 001oogoo05Ioo`03001oogoo05Ioo`03001oogoo05Eoo`03001oogoo05Ioo`03001oogoo01Ioo`03 001oogoo0:1oo`00[Goo00<007ooOol0KGoo4P00ogoo[7oo00<007ooOol0X7oo002]Ool3001oOolB 003oOonJOol00`00Oomoo`2POol009Eoo`8000Aoo`8000=oo`<000Yoo`03001oogoo095oo``00?mo oheoo`03001oogoo0:5oo`00U7oo00@007ooOol000=oo`8000Aoo`03001oogoo00Uoo`<009eoo``0 0?mooh1oo`03001oogoo0:9oo`00U7oo00@007ooOol000Uoo`03001oogoo00Uoo`@00:Qoo`/00?mo ogEoo`03001oogoo0:9oo`00U7oo00@007ooOol000Uoo`03001oogoo00Uoo`03001oogoo0;Aoo`T0 0?moofaoo`03001oogoo0:9oo`00U7oo00@007ooOol000Uoo`03001oogoo00Uoo`<00;eoo`P00?mo of=oo`03001oogoo0:=oo`00UGoo0P002Goo0P002goo00<007ooOol0aGoo2000ogooFWoo00<007oo Ool0Y7oo002]Ool3003=Ool:003oOom?Ool00`00Oomoo`2UOol00:eoo`03001oogoo0=Moo``00?mo od9oo`03001oogoo0:Ioo`00[Goo0`00hgoo2P00ogoo=goo00<007ooOol0Ygoo002EOol20004Ool2 0003Ool40009Ool3003]Ool7003oOol_Ool00`00Oomoo`2XOol009Aoo`04001oogoo0003Ool20004 Ool00`00Oomoo`09Ool00`00Oomoo`3dOol7003oOolWOol00`00Oomoo`2YOol009Aoo`04001oogoo 000:Ool00`00Oomoo`08Ool4003jOol8003oOolMOol2002/Ool009Aoo`04001oogoo000;Ool00`00 Oomoo`07Ool00`00Oomoo`3oOol4Ool8003oOolDOol00`00Oomoo`2/Ool009Aoo`04001oogoo0008 Ool01000Oomoo`002Goo0`00ogoo37oo1`00ogoo2goo0P00[goo002EOol2000:Ool2000:Ool00`00 Oomoo`3oOolCOol8003oOol2Ool00`00Oomoo`2_Ool00:eoo`<00?mooa]oo`P00?Moo`800;9oo`00 [Goo00<007ooOol0ogoo8goo1`00kWoo0P00]7oo002]Ool3003oOolZOol:003ROol2002fOol009Eo o`8000Aoo`8000=oo`<000Yoo`<00?moocAoo``00=Aoo`800;Qoo`00U7oo00@007ooOol000=oo`80 00Ioo`03001oogoo00Moo`03001oogoo0?mood1oo`X00 Ool9003IOol009Aoo`04001oogoo0003Ool20005Ool00`00Oomoo`08Ool00`00Oomoo`3oOonNOol; 000eOol>003ROol009Aoo`04001oogoo0008Ool40009Ool4003oOonXOole003`Ool009Aoo`04001o ogoo0008Ool00`00Ool0000:Ool00`00Oomoo`3oOoooOoo?Ool009Aoo`04001oogoo0008Ool00`00 Ool0003oOoooOooLOol009Eoo`8000Yoo`800?mooomoomaoo`00ogooogooogooP7oo0000\ \>"], ImageRangeCache->{{{0, 892.75}, {56.3125, 0}} -> {-1.95288, -0.183876, 0.0112694, 0.0112694}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Two cones", "Subsection"], Cell["Roll another cone from the remaining cardboard and add ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[volume[\((2\ Pi\ - \ angle)\)], {angle, 0, 2\ \[Pi]}, AspectRatio -> Automatic]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .06415 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.151576 0.00152738 0.151576 [ [.17539 -0.01097 -3 -9 ] [.17539 -0.01097 3 0 ] [.32696 -0.01097 -3 -9 ] [.32696 -0.01097 3 0 ] [.47854 -0.01097 -3 -9 ] [.47854 -0.01097 3 0 ] [.63011 -0.01097 -3 -9 ] [.63011 -0.01097 3 0 ] [.78169 -0.01097 -3 -9 ] [.78169 -0.01097 3 0 ] [.93327 -0.01097 -3 -9 ] [.93327 -0.01097 3 0 ] [.01131 .01668 -18 -4.5 ] [.01131 .01668 0 4.5 ] [.01131 .03184 -18 -4.5 ] [.01131 .03184 0 4.5 ] [.01131 .047 -18 -4.5 ] [.01131 .047 0 4.5 ] [.01131 .06216 -18 -4.5 ] [.01131 .06216 0 4.5 ] [ 0 0 0 0 ] [ 1 .06415 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .17539 .00153 m .17539 .00778 L s [(1)] .17539 -0.01097 0 1 Mshowa .32696 .00153 m .32696 .00778 L s [(2)] .32696 -0.01097 0 1 Mshowa .47854 .00153 m .47854 .00778 L s [(3)] .47854 -0.01097 0 1 Mshowa .63011 .00153 m .63011 .00778 L s [(4)] .63011 -0.01097 0 1 Mshowa .78169 .00153 m .78169 .00778 L s [(5)] .78169 -0.01097 0 1 Mshowa .93327 .00153 m .93327 .00778 L s [(6)] .93327 -0.01097 0 1 Mshowa .125 Mabswid .05412 .00153 m .05412 .00528 L s .08444 .00153 m .08444 .00528 L s .11476 .00153 m .11476 .00528 L s .14507 .00153 m .14507 .00528 L s .2057 .00153 m .2057 .00528 L s .23602 .00153 m .23602 .00528 L s .26633 .00153 m .26633 .00528 L s .29665 .00153 m .29665 .00528 L s .35728 .00153 m .35728 .00528 L s .38759 .00153 m .38759 .00528 L s .41791 .00153 m .41791 .00528 L s .44822 .00153 m .44822 .00528 L s .50885 .00153 m .50885 .00528 L s .53917 .00153 m .53917 .00528 L s .56948 .00153 m .56948 .00528 L s .5998 .00153 m .5998 .00528 L s .66043 .00153 m .66043 .00528 L s .69074 .00153 m .69074 .00528 L s .72106 .00153 m .72106 .00528 L s .75138 .00153 m .75138 .00528 L s .81201 .00153 m .81201 .00528 L s .84232 .00153 m .84232 .00528 L s .87264 .00153 m .87264 .00528 L s .90295 .00153 m .90295 .00528 L s .96358 .00153 m .96358 .00528 L s .9939 .00153 m .9939 .00528 L s .25 Mabswid 0 .00153 m 1 .00153 L s .02381 .01668 m .03006 .01668 L s [(0.1)] .01131 .01668 1 0 Mshowa .02381 .03184 m .03006 .03184 L s [(0.2)] .01131 .03184 1 0 Mshowa .02381 .047 m .03006 .047 L s [(0.3)] .01131 .047 1 0 Mshowa .02381 .06216 m .03006 .06216 L s [(0.4)] .01131 .06216 1 0 Mshowa .125 Mabswid .02381 .00456 m .02756 .00456 L s .02381 .00759 m .02756 .00759 L s .02381 .01062 m .02756 .01062 L s .02381 .01365 m .02756 .01365 L s .02381 .01972 m .02756 .01972 L s .02381 .02275 m .02756 .02275 L s .02381 .02578 m .02756 .02578 L s .02381 .02881 m .02756 .02881 L s .02381 .03487 m .02756 .03487 L s .02381 .03791 m .02756 .03791 L s .02381 .04094 m .02756 .04094 L s .02381 .04397 m .02756 .04397 L s .02381 .05003 m .02756 .05003 L s .02381 .05306 m .02756 .05306 L s .02381 .05609 m .02756 .05609 L s .02381 .05913 m .02756 .05913 L s .25 Mabswid .02381 0 m .02381 .06415 L s 0 0 m 1 0 L 1 .06415 L 0 .06415 L closepath clip newpath .5 Mabswid .02381 .00157 m .02499 .00939 L .02605 .01237 L .02729 .01499 L .02846 .01704 L .03279 .02286 L .0375 .02758 L .04262 .03169 L .05293 .03813 L .06244 .04272 L .0829 .04995 L .09322 .05266 L .10458 .05511 L .12399 .05827 L .13489 .05958 L .14487 .06054 L .15471 .06128 L .16385 .06181 L .17388 .06222 L .17949 .06239 L .18192 .06245 L .18458 .0625 L .18693 .06254 L .18945 .06257 L .19084 .06259 L .19211 .0626 L .19341 .06261 L .1946 .06261 L .1957 .06262 L .19687 .06262 L .19804 .06262 L .19934 .06262 L .20045 .06262 L .20165 .06262 L .20278 .06261 L .20382 .06261 L .20633 .06259 L .20867 .06256 L .21396 .06249 L .21913 .06238 L .22464 .06224 L .23489 .0619 L .24599 .06143 L .26614 .06033 L .30706 .05728 L .34646 .05359 L .38435 .04955 L .42469 .04492 L .46351 .04029 L .50479 .03532 L Mistroke .54455 .03059 L .5828 .02619 L .62349 .02175 L .66268 .01777 L .70431 .01392 L .74443 .01064 L .78303 .00792 L .82409 .00552 L .86363 .00373 L .88353 .00302 L .90165 .0025 L .9212 .00206 L .93195 .00187 L .94213 .00173 L .95072 .00164 L .95505 .00161 L .95983 .00157 L .9639 .00155 L .96603 .00155 L .96827 .00154 L .97025 .00153 L .97137 .00153 L .97241 .00153 L .97337 .00153 L .97424 .00153 L .97524 .00153 L .97619 .00153 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{893.75, 57.3125}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgA000`40O003h00OogooogooogooP7oo 003oOoooOoooOon0Ool00?moo`Eoo`<005Eoo`@005Eoo`<005Moo`<005Aoo`<005Moo`800;]oo`00 ogoo1Woo00<007ooOol0EGoo00<007ooOol0F7oo00<007ooOol0EGoo00<007ooOol0EWoo00<007oo Ool0Dgoo00@007ooOol00;Yoo`00ogoo1Woo00<007ooOol0EWoo00<007ooOol0Egoo00<007ooOol0 Dgoo1000Egoo00<007ooOol0Dgoo00@007ooOol00;Yoo`00ogoo1Woo00<007ooOol0Egoo00<007oo Ool0E7oo0P00EWoo00<007oo0000EGoo0`00EWoo0`00^goo003oOol6Ool00`00Oomoo`1DOol01000 Oomoo`00F7oo00<007ooOol0Dgoo00<007oo0000EGoo00<007ooOol0EWoo00<007ooOol0^goo003o Ool5Ool2001GOol2001FOol3001GOol2001EOol4001FOol3002jOol00?mooomooomooh1oo`00ogoo ogooogooP7oo003oOoooOoooOon0Ool00?mooomooomooh1oo`00ogooogooogooP7oo003oOoooOooo Oon0Ool00?mooomooomooh1oo`00ogooogooogooP7oo002]Ool00`00Oomoo`3oOoooOoo?Ool009mo ool00?l004`009Aoo`00[Goo0`003goo00<007ooOol03Woo00<007ooOol03goo00<007ooOol03goo 00<007ooOol03goo00<007ooOol03Woo00<007ooOol03goo00<007ooOol03goo00<007ooOol03goo 00<007ooOol03Woo00<007ooOol03goo00<007ooOol03goo00<007ooOol03goo00<007ooOol03Woo 00<007ooOol03goo00<007ooOol03goo00<007ooOol03Woo00<007ooOol03goo00<007ooOol03goo 00<007ooOol03goo00<007ooOol03Woo00<007ooOol03goo00<007ooOol03goo00<007ooOol03goo 00<007ooOol03Woo00<007ooOol03goo00<007ooOol03goo4`0047oo00<007ooOol03goo00<007oo Ool03goo00<007ooOol03goo00<007ooOol0UGoo002]Ool00`00Oomoo`0?Ool00`00Oomoo`0>Ool0 0`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0>Ool00`00Oomoo`0?Ool0 0`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0>Ool00`00Oomoo`0?Ool00`00Oomoo`0?Ool0 0`00Oomoo`0?Ool00`00Oomoo`0>Ool00`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0>Ool0 0`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0>Ool00`00Oomoo`0?Ool0 0`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0>Ool00`00Oomoo`0:OolH000AOol00`00Oomo o`0>Ool00`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomoo`0?Ool00`00Oomoo`2EOol00:eoo`<0 05Ioo`03001oogoo05Eoo`03001oogoo05Ioo`03001oogoo05Ioo`03001oogoo055ooa4004aoo`03 001oogoo0;Uoo`00[Goo00<007ooOol0ogooZWoo2`00ogoo6Woo002]Ool3003oOonNOol<003oOolU Ool009Eoo`8000Aoo`8000=oo`<000Yoo`800?mooi=oo``00?mooc5oo`00U7oo00@007ooOol000=o o`8000Aoo`03001oogoo00Uoo`<00?moohIoo``00?mooceoo`00U7oo00@007ooOol000Uoo`03001o ogoo00Uoo`@00?moogUoo``00?moodUoo`00U7oo00@007ooOol000Uoo`03001oogoo00Uoo`04001o ogoo003oOom]Ool<003oOomEOol009Aoo`04001oogoo0009Ool00`00Oomoo`09Ool4003oOomROol; 003oOomQOol009Eoo`8000Uoo`8000]oo`05001oogooOol0003oOomGOol:003oOom/Ool00:eoo`<0 009oo`03001oogoo0?moodaoo`P00?moogIoo`00[Goo00<007ooOol00goo00<007ooOol0ogoo@goo 2000ogooOWoo002]Ool30004Ool00`00Oomoo`3oOoljOol8003oOon6Ool009Eoo`8000Aoo`8000=o o`@000Uoo`<000Eoo`800?mooc=oo`L00?moohioo`00U7oo00@007ooOol000=oo`8000Aoo`03001o ogoo00Uoo`03001oogoo00Moo`03001oogoo0?moobUoo`L00?mooiEoo`00U7oo00@007ooOol000Yo o`03001oogoo00Qoo`@000Moo`800?mooamoo`X00?mooiaoo`00U7oo00@007ooOol000]oo`03001o ogoo00Moo`03001oogoo00Yoo`03001oogoo0?mooa1oo``00?moojIoo`00U7oo00@007ooOol000Qo o`04001oogoo0009Ool3000;Ool2003oOol6Ool:003oOonbOol009Eoo`8000Yoo`8000Yoo`03001o ogoo00eoo`03001oogoo0?Yoo`P00?mookaoo`00[Goo0`003Woo0P00lWoo2000ogooa7oo002]Ool0 0`00Oomoo`0@Ool2003XOol8003oOoo"], ImageRangeCache->{{{0, 892.75}, {56.3125, 0}} -> {-1.95288, -0.183876, 0.0112694, 0.0112694}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell["Summing:", "Text"], Cell[BoxData[ \(f[angle_] := volume[angle] + volume[\((2\ \[Pi]\ - \ angle)\)]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[f[angle], {angle, 0, 2\ \[Pi]}\n, AspectRatio -> Automatic]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .05161 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.151576 -0.018839 0.151576 [ [.17539 -0.00102 -3 -9 ] [.17539 -0.00102 3 0 ] [.32696 -0.00102 -3 -9 ] [.32696 -0.00102 3 0 ] [.47854 -0.00102 -3 -9 ] [.47854 -0.00102 3 0 ] [.63011 -0.00102 -3 -9 ] [.63011 -0.00102 3 0 ] [.78169 -0.00102 -3 -9 ] [.78169 -0.00102 3 0 ] [.93327 -0.00102 -3 -9 ] [.93327 -0.00102 3 0 ] [.01131 .0039 -24 -4.5 ] [.01131 .0039 0 4.5 ] [.01131 .01906 -24 -4.5 ] [.01131 .01906 0 4.5 ] [.01131 .02663 -18 -4.5 ] [.01131 .02663 0 4.5 ] [.01131 .03421 -24 -4.5 ] [.01131 .03421 0 4.5 ] [.01131 .04179 -18 -4.5 ] [.01131 .04179 0 4.5 ] [.01131 .04937 -24 -4.5 ] [.01131 .04937 0 4.5 ] [ 0 0 0 0 ] [ 1 .05161 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .17539 .01148 m .17539 .01773 L s [(1)] .17539 -0.00102 0 1 Mshowa .32696 .01148 m .32696 .01773 L s [(2)] .32696 -0.00102 0 1 Mshowa .47854 .01148 m .47854 .01773 L s [(3)] .47854 -0.00102 0 1 Mshowa .63011 .01148 m .63011 .01773 L s [(4)] .63011 -0.00102 0 1 Mshowa .78169 .01148 m .78169 .01773 L s [(5)] .78169 -0.00102 0 1 Mshowa .93327 .01148 m .93327 .01773 L s [(6)] .93327 -0.00102 0 1 Mshowa .125 Mabswid .05412 .01148 m .05412 .01523 L s .08444 .01148 m .08444 .01523 L s .11476 .01148 m .11476 .01523 L s .14507 .01148 m .14507 .01523 L s .2057 .01148 m .2057 .01523 L s .23602 .01148 m .23602 .01523 L s .26633 .01148 m .26633 .01523 L s .29665 .01148 m .29665 .01523 L s .35728 .01148 m .35728 .01523 L s .38759 .01148 m .38759 .01523 L s .41791 .01148 m .41791 .01523 L s .44822 .01148 m .44822 .01523 L s .50885 .01148 m .50885 .01523 L s .53917 .01148 m .53917 .01523 L s .56948 .01148 m .56948 .01523 L s .5998 .01148 m .5998 .01523 L s .66043 .01148 m .66043 .01523 L s .69074 .01148 m .69074 .01523 L s .72106 .01148 m .72106 .01523 L s .75138 .01148 m .75138 .01523 L s .81201 .01148 m .81201 .01523 L s .84232 .01148 m .84232 .01523 L s .87264 .01148 m .87264 .01523 L s .90295 .01148 m .90295 .01523 L s .96358 .01148 m .96358 .01523 L s .9939 .01148 m .9939 .01523 L s .25 Mabswid 0 .01148 m 1 .01148 L s .02381 .0039 m .03006 .0039 L s [(0.15)] .01131 .0039 1 0 Mshowa .02381 .01906 m .03006 .01906 L s [(0.25)] .01131 .01906 1 0 Mshowa .02381 .02663 m .03006 .02663 L s [(0.3)] .01131 .02663 1 0 Mshowa .02381 .03421 m .03006 .03421 L s [(0.35)] .01131 .03421 1 0 Mshowa .02381 .04179 m .03006 .04179 L s [(0.4)] .01131 .04179 1 0 Mshowa .02381 .04937 m .03006 .04937 L s [(0.45)] .01131 .04937 1 0 Mshowa .125 Mabswid .02381 .00541 m .02756 .00541 L s .02381 .00693 m .02756 .00693 L s .02381 .00844 m .02756 .00844 L s .02381 .00996 m .02756 .00996 L s .02381 .01299 m .02756 .01299 L s .02381 .01451 m .02756 .01451 L s .02381 .01602 m .02756 .01602 L s .02381 .01754 m .02756 .01754 L s .02381 .02057 m .02756 .02057 L s .02381 .02209 m .02756 .02209 L s .02381 .0236 m .02756 .0236 L s .02381 .02512 m .02756 .02512 L s .02381 .02815 m .02756 .02815 L s .02381 .02967 m .02756 .02967 L s .02381 .03118 m .02756 .03118 L s .02381 .0327 m .02756 .0327 L s .02381 .03573 m .02756 .03573 L s .02381 .03724 m .02756 .03724 L s .02381 .03876 m .02756 .03876 L s .02381 .04028 m .02756 .04028 L s .02381 .04331 m .02756 .04331 L s .02381 .04482 m .02756 .04482 L s .02381 .04634 m .02756 .04634 L s .02381 .04785 m .02756 .04785 L s .02381 .00238 m .02756 .00238 L s .02381 .00087 m .02756 .00087 L s .02381 .05089 m .02756 .05089 L s .25 Mabswid .02381 0 m .02381 .05161 L s 0 0 m 1 0 L 1 .05161 L 0 .05161 L closepath clip newpath .5 Mabswid .03092 0 m .03279 .00251 L .0375 .00725 L .04262 .01138 L .05293 .01791 L .06244 .02262 L .0829 .03019 L .10458 .03588 L .12411 .03967 L .14241 .0424 L .16266 .04472 L .18119 .04634 L .20017 .0476 L .2209 .04862 L .24057 .04931 L .25157 .0496 L .26154 .04982 L .2709 .04998 L .28079 .05011 L .29163 .05022 L .29685 .05026 L .30161 .05029 L .30615 .05032 L .31109 .05034 L .3138 .05035 L .3163 .05036 L .31885 .05036 L .32125 .05037 L .32345 .05037 L .32547 .05037 L .32654 .05037 L .32772 .05038 L .32889 .05038 L .33013 .05038 L .33137 .05038 L .3325 .05038 L .3338 .05038 L .33503 .05038 L .33616 .05038 L .33721 .05038 L .33958 .05037 L .34202 .05037 L .34425 .05037 L .34927 .05036 L .35834 .05034 L .36893 .0503 L .37898 .05026 L .41723 .0501 L .43859 .05001 L .45792 .04995 L Mistroke .46763 .04993 L .47273 .04992 L .47811 .04991 L .48284 .0499 L .48534 .0499 L .48801 .0499 L .49048 .0499 L .49272 .04989 L .4938 .04989 L .49497 .04989 L .49601 .04989 L .49711 .04989 L .49833 .04989 L .49945 .04989 L .50065 .04989 L .50197 .04989 L .50311 .04989 L .50433 .04989 L .50549 .04989 L .50653 .04989 L .5091 .0499 L .51188 .0499 L .51692 .0499 L .52224 .04991 L .52786 .04992 L .53794 .04994 L .55755 .05 L .57588 .05007 L .61476 .05024 L .63454 .05031 L .63996 .05033 L .64573 .05035 L .65071 .05036 L .65333 .05036 L .65609 .05037 L .65851 .05037 L .6598 .05037 L .66117 .05037 L .66244 .05037 L .66359 .05038 L .66475 .05038 L .66584 .05038 L .66695 .05038 L .66812 .05038 L .66913 .05038 L .67023 .05038 L .67142 .05038 L .6727 .05037 L .6739 .05037 L .675 .05037 L Mistroke .67766 .05037 L .68013 .05036 L .68572 .05035 L .69065 .05033 L .6959 .05031 L .70619 .05024 L .71199 .05019 L .71733 .05013 L .72776 .05 L .7375 .04983 L .75634 .0494 L .76632 .0491 L .77701 .04871 L .79826 .04769 L .81839 .04637 L .83831 .04462 L .85674 .04251 L .87542 .03975 L .89603 .03574 L .91539 .03072 L .92535 .02746 L .93625 .02319 L .94566 .01868 L .95571 .01258 L .96127 .00832 L .96632 .0035 L .96876 .00065 L Mfstroke .96876 .00065 m .96922 0 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{3111.69, 160.562}, ImageMargins->{{40, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgOol00`00Oomoo`3oOoooOoooOoooOoooOoooOoooOoooOoooOoooOomiOol00`00 Oomoo`3HOol00<5oo`03001oogoo01Moo`03001oogoo0?mooomooomooomooomooomooomooomooomo oomoogMoo`03001oogoo0=Uoo`00Qgoo0P0017oo00<007ooOol00Woo0`000goo0`009Woo00<007oo Ool067oo00<007ooOol0ogooogooogooogooogooogooogooogooogooogooMGoo00<007ooOol0fWoo 0026Ool01000Oomoo`002Goo00D007ooOomoo`0000=oo`03001oogoo02=oo`03001oogoo01Uoo`03 001oogoo0?mooomooomooomooomooomooomooomooomooomoog9oo`800=eoo`00QWoo00@007ooOol0 00Uoo`03001oogoo00Eoo`03001oogoo02=ooa8000]oo`03001oogoo0?mooomooomooomooomooomo oomooomooomooomoog1oo`03001oogoo0=eoo`00QWoo00@007ooOol000Uoo`03001oogoo00Eoo`03 001oogoo02=oo`03001oogoo01]oo`03001oogoo0?mooomooomooomooomooomooomooomooomooomo ofioo`03001oogoo0=ioo`00QWoo00@007ooOol000Uoo`04001oogooOol4000VOol00`00Oomoo`0L Ool00`00Oomoo`3oOoooOoooOoooOoooOoooOoooOoooOoooOoooOom/Ool00`00Oomoo`3OOol008Io o`04001oogoo0008Ool20003Ool00`00Oomoo`0WOol00`00Oomoo`0MOol00`00Oomoo`3oOoooOooo OoooOoooOoooOoooOoooOoooOoooOomZOol00`00Oomoo`3POol008Moo`8000Yoo`04001oogooOol5 000UOol;000FOol00`00Oomoo`3oOoooOoooOoooOoooOoooOoooOoooOoooOoooOomXOol00`00Oomo o`3QOol00<5oo`03001oogoo01moo`03001oogoo0?mooomooomooomooomooomooomooomooomooomo ofIoo`03001oogoo0>9oo`00`Goo00<007ooOol087oo00<007ooOol0ogooogooogooogooogooogoo ogooogooogooogooI7oo00<007ooOol0hgoo0031Ool00`00Oomoo`0QOol00`00Oomoo`3oOoooOooo OoooOoooOoooOoooOoooOoooOoooOomROol00`00Oomoo`3TOol00<5oo`03001oogoo029oo`03001o ogoo0?mooomooomooomooomooomooomooomooomooomoof1oo`03001oogoo0>Eoo`00`Goo2`006goo 00<007ooOol0ogooogooogooogooogooogooogooogooogooogooGWoo00<007ooOol0iWoo0031Ool0 0`00Oomoo`0TOol2003oOoooOoooOoooOoooOoooOoooOoooOoooOoooOomMOol00`00Oomoo`3WOol0 0<5oo`03001oogoo02Ioo`03001oogoo0?mooomooomooomooomooomooomooomooomooomooeUoo`03 001oogoo0>Qoo`00`Goo00<007ooOol09goo00<007ooOol0ogooogooogooogooogooogooogooogoo ogooogooEgoo00<007ooOol0jGoo0031Ool;000POol00`00Oomoo`3oOoooOoooOoooOoooOoooOooo OoooOoooOoooOomDOol2003/Ool00<5oo`03001oogoo02Uoo`800?mooomooomooomooomooomooomo oomooomooomooe=oo`03001oogoo0>aoo`00`Goo00<007ooOol0:goo00<007ooOol0ogooogooogoo ogooogooogooogooogooogooogooCgoo00<007ooOol0kGoo0031Ool00`00Oomoo`0/Ool00`00Oomo o`3oOoooOoooOoooOoooOoooOoooOoooOoooOoooOom=Ool00`00Oomoo`3^Ool00<5oo`/002Eoo`80 0?mooomooomooomooomooomooomooomooomooomood]oo`800?5oo`00`Goo00<007ooOol0;goo00<0 07ooOol0ogooogooogooogooogooogooogooogooogooogooAgoo00<007ooOol0lGoo0031Ool00`00 Oomoo`0`Ool00`00Oomoo`3oOoooOoooOoooOoooOoooOoooOoooOoooOoooOom5Ool00`00Oomoo`3b Ool00<5oo`03001oogoo035oo`03001oogoo0?mooomooomooomooomooomooomooomooomooomood=o o`03001oogoo0?=oo`00`Goo00<007ooOol0Woo00<007ooOol05goo00<007ooOol0E7oo00<007ooOol0S7oo0031Ool00`00Oomoo`0m Ool2000DOol00`00Oomoo`1DOol00`00Oomoo`1DOol00`00Oomoo`1COol00`00Oomoo`1DOol00`00 Oomoo`1DOol00`00Oomoo`1DOol00`00Oomoo`1COol00`00Oomoo`1DOol00`00Oomoo`1DOol00`00 Oomoo`1DOol00`00Oomoo`1COol00`00Oomoo`1DOol00`00Oomoo`1DOol00`00Oomoo`1COol00`00 Oomoo`1DOol00`00Oomoo`1DOol00`00Oomoo`1DOol00`00Oomoo`1COol00`00Oomoo`1DOol00`00 Oomoo`1DOol00`00Oomoo`1DOol00`00Oomoo`1COol00`00Oomoo`1DOol00`00Oomoo`1DOol00`00 Oomoo`1COol00`00Oomoo`1DOol00`00Oomoo`1DOol00`00Oomoo`1DOol00`00Oomoo`1COol00`00 Oomoo`0hOol2000JOol00`00Oomoo`1DOol00`00Oomoo`27oo0P004Goo00<007oo Ool0E7oo00<007ooOol0E7oo00<007ooOol0Dgoo00<007ooOol0E7oo00<007ooOol0E7oo00<007oo Ool0E7oo00<007ooOol0Dgoo00<007ooOol0E7oo00<007ooOol0E7oo00<007ooOol0E7oo00<007oo Ool0Dgoo00<007ooOol0E7oo00<007ooOol0E7oo00<007ooOol0Dgoo00<007ooOol0E7oo00<007oo Ool0E7oo00<007ooOol0E7oo00<007ooOol0Dgoo00<007ooOol0E7oo00<007ooOol0E7oo00<007oo Ool0E7oo00<007ooOol0Dgoo00<007ooOol0E7oo00<007ooOol0E7oo00<007ooOol0Dgoo00<007oo Ool0E7oo00<007ooOol0E7oo00<007ooOol0E7oo00<007ooOol0Dgoo00<007ooOol0=Goo00<007oo Ool077oo00<007ooOol0E7oo00<007ooOol0S7oo0031Ool00`00Oomoo`12Ool2000?Ool00`00Oomo o`1DOol00`00Oomoo`1DOol00`00Oomoo`1COol00`00Oomoo`1DOol00`00Oomoo`1DOol00`00Oomo o`1DOol00`00Oomoo`1COol00`00Oomoo`1DOol00`00Oomoo`1DOol00`00Oomoo`1DOol00`00Oomo o`1COol00`00Oomoo`1DOol00`00Oomoo`1DOol00`00Oomoo`1COol00`00Oomoo`1DOol00`00Oomo o`1DOol00`00Oomoo`1DOol00`00Oomoo`1COol00`00Oomoo`1DOol00`00Oomoo`1DOol00`00Oomo o`1DOol00`00Oomoo`1COol00`00Oomoo`1DOol00`00Oomoo`1DOol00`00Oomoo`1COol00`00Oomo o`1DOol00`00Oomoo`1DOol00`00Oomoo`1DOol00`00Oomoo`1COol00`00Oomoo`0cOol2000OOol0 0`00Oomoo`1DOol00`00Oomoo`2Goo0031Ool00`00Oomoo`1fOol3003oOoooOoooOoooOoooOoooOoooOooo OoooOonfOol3003oOolkOol00<5oo`03001oogoo07Uoo`800?mooomooomooomooomooomooomooomo oomook5oo`<00?moocioo`00`Goo00<007ooOol0Ngoo0`00ogooogooogooogooogooogooogooogoo ogoo[7oo0P00ogoo@Goo0031Ool;001fOol3003oOoooOoooOoooOoooOoooOoooOoooOoooOonVOol3 003oOom3Ool00<5oo`03001oogoo085oo`<00?mooomooomooomooomooomooomooomooomooj5oo`80 0?moodIoo`00SGoo0P0017oo00<007ooOol00Woo0`009Woo00<007ooOol0Q7oo0`00ogooogooogoo ogooogooogooogooogooogooVgoo0`00ogooB7oo002Woo 1000ogooMgoo0031Ool00`00Oomoo`2hOol4003oOoooOoooOoooOoooOoooOoooOoooOoooOolbOol4 003oOomkOol00<5oo`03001oogoo0;aoo`@00?mooomooomooomooomooomooomooomooomoobYoo`@0 0?moogmoo`00`Goo2`00^7oo1000ogooogooogooogooogooogooogooogooogoo8Woo1000ogooPgoo 0031Ool00`00Oomoo`34Ool4003oOoooOoooOoooOoooOoooOoooOoooOoooOolJOol4003oOon7Ool0 08Moo`8000Aoo`03001oogoo009oo`<000=oo`<002Ioo`03001oogoo0eoo`D00?mooomooomooomooomooomooomooomoolIoo`D00?moojmoo`00 `Goo00<007ooOol0lWoo1@00ogooogooogooogooogooogooogooogoo_7oo1@00ogoo]7oo0031Ool; 003_Ool5003oOoooOoooOoooOoooOoooOoooOoooOonbOol5003oOoniOol00<5oo`03001oogoo0?ao o`D00?mooomooomooomooomooomooomooomoojQoo`D00?mookioo`00`Goo00<007ooOol0ogoo0Woo 1@00ogooogooogooogooogooogooogooogooWWoo1@00ogoo`goo0031Ool00`00Oomoo`3oOol7Ool5 003oOoooOoooOoooOoooOoooOoooOoooOonDOol5003oOoo8Ool00<5oo`03001oogoo0?moo`aoo`D0 0?mooomooomooomooomooomooomooomoohYoo`D00?mooleoo`00`Goo2`00ogoo2Goo1@00ogooogoo ogooogooogooogooogooogooP7oo1@00ogoodWoo0031Ool00`00Oomoo`3oOolFOol5003oOoooOooo OoooOoooOoooOoooOoooOomfOol5003oOooGOol00<5oo`03001oogoo0?mooa]oo`H00?mooomooomo oomooomooomooomooomoof]oo`D00?moomaoo`00`Goo00<007ooOol0ogoo8Goo1`00ogooogooogoo ogooogooogooogooogooGWoo1P00ogoohGoo0031Ool;003oOolPOol6003oOoooOoooOoooOoooOooo OoooOoooOom@Ool8003oOooWOol00<5oo`03001oogoo0?moobioo`L00?mooomooomooomooomooomo oomooomood9oo`L00?moonmoo`00SGoo0P0017oo00<007ooOol00goo0`009Goo00<007ooOol0ogoo =Goo1P00ogooogooogooogooogooogooogooogoo=7oo2000ogoomWoo002goo1`00ogooogooogooogooogooogooogooogoo9Goo2000 ogoooWoo002goo0031Ool00`00Oomoo`3oOon0Ool8 003oOoooOoooOoooOoooOoooOoooOonKOol7003oOoooOom3Ool00<5oo`/00?mooh1oo`/00?mooomo oomooomooomooomooomoohIoo`X00?mooomoodYoo`00`Goo00<007ooOol0ogooTgoo3@00ogooogoo ogooogooogooogooogooKWoo2`00ogooogooE7oo0031Ool00`00Oomoo`3oOonPOol=003oOoooOooo OoooOoooOoooOoooOomEOol<003oOoooOomOOol00<5oo`03001oogoo0?moojeoo`d00?mooomooomo oomooomooomooomooceoo`/00?mooomoof]oo`00`Goo2`00ogoo/Woo3P00ogooogooogooogooogoo ogooogoo97oo2`00ogooogooMWoo0031Ool00`00Oomoo`3oOoo8Ool=003oOoooOoooOoooOoooOooo OoooOol7Ool@003oOoooOon1Ool00<5oo`03001oogoo0?moomEoo`h00?mooomooomooomooomooomo onEooa<00?mooomooi5oo`00`Goo00<007ooOol0ogoohgoo3@00ogooogooogooogooogooogooaGoo 4`00ogooogooY7oo0031Ool00`00Oomoo`3oOoo`OolA003oOoooOoooOoooOoooOoooOonPOolD003o OoooOongOol00<5oo`/00?moooUooa@00?mooomooomooomooomooomoogMooaD00?mooomool]oo`00 `Goo00<007ooOol0ogooogoo5Woo4`00ogooogooogooogooogooogooD7oo5000ogooogooh7oo0027 Ool20004Ool00`00Oomoo`03Ool30002Ool3000VOol00`00Oomoo`3oOoooOolYOolH003oOoooOooo OoooOoooOoooOolOOolI003oOoooOoodOol008Ioo`04001oogoo000:Ool01000Oomoo`000goo00<0 07ooOol08goo00<007ooOol0ogooogoo@Goo7@00ogooogooogooogooogooi7oo7@00ogooogooogoo 3Woo0026Ool01000Oomoo`001goo1@001Goo00<007ooOol08goo4P00ogooogooCgoo7P00ogooogoo ogooogooogooW7oo:P00ogooogooogoo:goo0026Ool01000Oomoo`001goo00@007ooOol000Ioo`03 001oogoo02=oo`03001oogoo0?mooomoogaoocT00?mooomooomooomooomoocYoobT00?mooomooomo oeEoo`00QWoo00@007ooOol000Qoo`03001oo`00009oo`@002Ioo`03001oogoo0?mooomookEoocX0 0?mood1oool00?l003L00?moodEood<00?mooomooomoogioo`00QWoo00@007ooOol000Uoo`80009o o`03001oogoo02Moo`03001oogoo0?mooomoonmoool004000?mooomoocMoool004D00?mooomooomo ol5oo`00Qgoo0P002goo00<007ooOol01@009Goo2`00ogooogooogooogooogooogooogooogooogoo ogooogooIWoo0031Ool00`00Oomoo`3oOoooOoooOoooOoooOoooOoooOoooOoooOoooOoooOom^Ool0 0<5oo`03001oogoo0?mooomooomooomooomooomooomooomooomooomooomoofioo`00\ \>"], ImageRangeCache->{{{0, 3110.69}, {159.562, 0}} -> {-0.445284, 0.096801, \ 0.00230617, 0.00230617}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell["It seems this function has a single hump:", "Text"], Cell[BoxData[ \(NSolve[D[f[angle], angle] == 0]\)], "Input"], Cell["Computing the derivative reveals a double hump:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f[angle] /. First[%]\)], "Input"], Cell[BoxData[ \(0.456640590999588535`\)], "Output"] }, Closed]], Cell["and a smaller volume at the center", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f[\[Pi]] // N\)], "Input"], Cell[BoxData[ \(0.453449841058554525`\)], "Output"] }, Closed]], Cell["which is in fact a local minimum!", "Text"], Cell[CellGroupData[{ Cell["Controle by plotting f'", "Subsubsection"], Cell["The default plotdoes not give much information", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[\(f'\)[angle], {angle, 0, 2\ Pi}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.151576 0.298927 0.494001 [ [.17539 .28643 -3 -9 ] [.17539 .28643 3 0 ] [.32696 .28643 -3 -9 ] [.32696 .28643 3 0 ] [.47854 .28643 -3 -9 ] [.47854 .28643 3 0 ] [.63011 .28643 -3 -9 ] [.63011 .28643 3 0 ] [.78169 .28643 -3 -9 ] [.78169 .28643 3 0 ] [.93327 .28643 -3 -9 ] [.93327 .28643 3 0 ] [.01131 .00253 -24 -4.5 ] [.01131 .00253 0 4.5 ] [.01131 .10133 -24 -4.5 ] [.01131 .10133 0 4.5 ] [.01131 .20013 -24 -4.5 ] [.01131 .20013 0 4.5 ] [.01131 .39773 -18 -4.5 ] [.01131 .39773 0 4.5 ] [.01131 .49653 -18 -4.5 ] [.01131 .49653 0 4.5 ] [.01131 .59533 -18 -4.5 ] [.01131 .59533 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .17539 .29893 m .17539 .30518 L s [(1)] .17539 .28643 0 1 Mshowa .32696 .29893 m .32696 .30518 L s [(2)] .32696 .28643 0 1 Mshowa .47854 .29893 m .47854 .30518 L s [(3)] .47854 .28643 0 1 Mshowa .63011 .29893 m .63011 .30518 L s [(4)] .63011 .28643 0 1 Mshowa .78169 .29893 m .78169 .30518 L s [(5)] .78169 .28643 0 1 Mshowa .93327 .29893 m .93327 .30518 L s [(6)] .93327 .28643 0 1 Mshowa .125 Mabswid .05412 .29893 m .05412 .30268 L s .08444 .29893 m .08444 .30268 L s .11476 .29893 m .11476 .30268 L s .14507 .29893 m .14507 .30268 L s .2057 .29893 m .2057 .30268 L s .23602 .29893 m .23602 .30268 L s .26633 .29893 m .26633 .30268 L s .29665 .29893 m .29665 .30268 L s .35728 .29893 m .35728 .30268 L s .38759 .29893 m .38759 .30268 L s .41791 .29893 m .41791 .30268 L s .44822 .29893 m .44822 .30268 L s .50885 .29893 m .50885 .30268 L s .53917 .29893 m .53917 .30268 L s .56948 .29893 m .56948 .30268 L s .5998 .29893 m .5998 .30268 L s .66043 .29893 m .66043 .30268 L s .69074 .29893 m .69074 .30268 L s .72106 .29893 m .72106 .30268 L s .75138 .29893 m .75138 .30268 L s .81201 .29893 m .81201 .30268 L s .84232 .29893 m .84232 .30268 L s .87264 .29893 m .87264 .30268 L s .90295 .29893 m .90295 .30268 L s .96358 .29893 m .96358 .30268 L s .9939 .29893 m .9939 .30268 L s .25 Mabswid 0 .29893 m 1 .29893 L s .02381 .00253 m .03006 .00253 L s [(-0.6)] .01131 .00253 1 0 Mshowa .02381 .10133 m .03006 .10133 L s [(-0.4)] .01131 .10133 1 0 Mshowa .02381 .20013 m .03006 .20013 L s [(-0.2)] .01131 .20013 1 0 Mshowa .02381 .39773 m .03006 .39773 L s [(0.2)] .01131 .39773 1 0 Mshowa .02381 .49653 m .03006 .49653 L s [(0.4)] .01131 .49653 1 0 Mshowa .02381 .59533 m .03006 .59533 L s [(0.6)] .01131 .59533 1 0 Mshowa .125 Mabswid .02381 .02723 m .02756 .02723 L s .02381 .05193 m .02756 .05193 L s .02381 .07663 m .02756 .07663 L s .02381 .12603 m .02756 .12603 L s .02381 .15073 m .02756 .15073 L s .02381 .17543 m .02756 .17543 L s .02381 .22483 m .02756 .22483 L s .02381 .24953 m .02756 .24953 L s .02381 .27423 m .02756 .27423 L s .02381 .32363 m .02756 .32363 L s .02381 .34833 m .02756 .34833 L s .02381 .37303 m .02756 .37303 L s .02381 .42243 m .02756 .42243 L s .02381 .44713 m .02756 .44713 L s .02381 .47183 m .02756 .47183 L s .02381 .52123 m .02756 .52123 L s .02381 .54593 m .02756 .54593 L s .02381 .57063 m .02756 .57063 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .04696 .61803 m .04793 .60813 L .05297 .57018 L .06244 .51896 L .0675 .49827 L .07289 .47958 L .08259 .45224 L .09412 .42696 L .10458 .40865 L .12383 .38259 L .14465 .36181 L .16458 .34669 L .18565 .33431 L .22362 .31847 L .24313 .31269 L .26405 .30784 L .28529 .30408 L .30541 .3014 L .32532 .29947 L .34374 .29821 L .35336 .29772 L .3624 .29737 L .37231 .29707 L .37789 .29695 L .383 .29687 L .38774 .29681 L .3904 .29678 L .39281 .29676 L .39506 .29675 L .39633 .29674 L .3975 .29674 L .39852 .29674 L .39965 .29673 L .40083 .29673 L .40193 .29673 L .40321 .29673 L .40393 .29673 L .4046 .29674 L .40583 .29674 L .40714 .29674 L .40964 .29675 L .41191 .29677 L .41743 .29681 L .42263 .29687 L .43242 .29702 L .44274 .29723 L .46123 .2977 L .49925 .2989 L .52044 .2996 L .53973 .30018 L Mistroke .54933 .30043 L .55975 .30068 L .56953 .30086 L .57869 .301 L .58323 .30105 L .58812 .30109 L .59076 .3011 L .59209 .30111 L .59354 .30111 L .59478 .30112 L .59548 .30112 L .59611 .30112 L .59734 .30112 L .5985 .30112 L .59972 .30112 L .60041 .30112 L .60106 .30112 L .60236 .30112 L .60376 .30111 L .60506 .3011 L .60629 .3011 L .6086 .30108 L .61131 .30106 L .61383 .30103 L .61953 .30095 L .62415 .30086 L .62914 .30074 L .63817 .30047 L .64891 .30003 L .65876 .2995 L .67766 .29814 L .68837 .29713 L .69835 .29601 L .71878 .29314 L .73736 .28972 L .75653 .28525 L .77731 .27907 L .79717 .27159 L .8182 .26151 L .83775 .24959 L .85606 .23544 L .87632 .21509 L .89485 .1901 L .91385 .15412 L .9246 .12602 L .93458 .09151 L .93966 .06914 L .94515 .03951 L .95034 .00391 L Mfstroke .95034 .00391 m .95078 0 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgOol001]oo`03001oogoo0?5oo`03001oogoo00ioo`006goo00<007ooOol0lGoo00<0 07ooOol03Woo000KOol00`00Oomoo`3`Ool00`00Oomoo`0?Ool001]oo`03001oogoo0?1oo`03001o ogoo00moo`006goo00<007ooOol0l7oo00<007ooOol03goo000KOol2003aOol00`00Oomoo`0?Ool0 01]oo`03001oogoo0?1oo`03001oogoo00moo`006goo00<007ooOol0kgoo00<007ooOol047oo000K Ool00`00Oomoo`3_Ool00`00Oomoo`0@Ool001]oo`03001oogoo0>moo`03001oogoo011oo`0027oo 0P0017oo0P0017oo0`0017oo00<007ooOol0kgoo00<007ooOol047oo0007Ool01000Oomoo`000goo 0P001Goo00<007ooOol00goo00<007ooOol0kWoo00<007ooOol04Goo00000goo0000000300001Goo 001oogoo000027oo100017oo0P00kgoo00<007ooOol04Goo0007Ool01000Oomoo`0027oo00<007oo 00001Goo00<007ooOol0kWoo00<007ooOol04Goo0007Ool01000Oomoo`0027oo00<007oo00001Goo 00<007ooOol0kGoo00<007ooOol04Woo0008Ool2000:Ool20005Ool00`00Oomoo`3]Ool00`00Oomo o`0BOol001]oo`03001oogoo0>eoo`03001oogoo019oo`006goo00<007ooOol0k7oo00<007ooOol0 4goo000KOol2003]Ool00`00Oomoo`0COol001]oo`03001oogoo0>aoo`03001oogoo01=oo`006goo 00<007ooOol0k7oo00<007ooOol04goo000KOol00`00Oomoo`3[Ool00`00Oomoo`0DOol001]oo`03 001oogoo0>]oo`03001oogoo01Aoo`006goo00<007ooOol0jgoo00<007ooOol057oo000KOol00`00 Oomoo`3[Ool00`00Oomoo`0DOol001]oo`800>]oo`03001oogoo01Eoo`006goo00<007ooOol0jWoo 00<007ooOol05Goo000KOol00`00Oomoo`3ZOol00`00Oomoo`0EOol001]oo`03001oogoo0>Uoo`03 001oogoo01Ioo`006goo00<007ooOol0jGoo00<007ooOol05Woo000KOol00`00Oomoo`3XOol00`00 Oomoo`0GOol001]oo`800>Qoo`03001oogoo01Qoo`006goo00<007ooOol0igoo00<007ooOol067oo 000KOol00`00Oomoo`3VOol00`00Oomoo`0IOol001]oo`03001oogoo0>Ioo`03001oogoo01Uoo`00 6goo00<007ooOol0iGoo00<007ooOol06Woo0008Ool20004Ool20003Ool40004Ool00`00Oomoo`3U Ool00`00Oomoo`0JOol000Moo`04001oogoo0003Ool20004Ool00`00Oomoo`04Ool00`00Oomoo`3T Ool00`00Oomoo`0KOol00003Ool0000000<00005Ool007ooOol0000:Ool00`00Oomoo`03Ool2003T Ool00`00Oomoo`0LOol000Moo`04001oogoo000;Ool00`00Oomoo`02Ool00`00Oomoo`3ROol00`00 Oomoo`0MOol000Moo`04001oogoo0008Ool01000Oomoo`0017oo00<007ooOol0hWoo00<007ooOol0 7Goo0008Ool2000:Ool20005Ool00`00Oomoo`3QOol00`00Oomoo`0NOol001]oo`03001oogoo0=mo o`80025oo`006goo00<007ooOol0gWoo00<007ooOol08Goo000KOol00`00Oomoo`3MOol00`00Oomo o`0ROol001]oo`800=eoo`03001oogoo02=oo`006goo00<007ooOol0fgoo00<007ooOol097oo000K Ool00`00Oomoo`3IOol2000WOol001]oo`03001oogoo0=Qoo`03001oogoo02Moo`006goo00<007oo Ool0egoo00<007ooOol0:7oo000KOol00`00Oomoo`3FOol00`00Oomoo`0YOol001]oo`800=Eoo`80 02aoo`006goo00<007ooOol0dWoo0P00;Woo000KOol00`00Oomoo`3AOol00`00Oomoo`0^Ool001]o o`03001oogoo02=oo`<002Eoo`@002Eoo`<002Ioo`<002Aoo`<000Qoo`8001eoo`80019oo`006goo 00<007ooOol097oo00<007ooOol09Goo00<007ooOol0:7oo00<007ooOol097oo00<007ooOol09Woo 00<007ooOol00goo0P007Woo00@007ooOol0015oo`006goo00<007ooOol097oo00<007ooOol09Woo 00<007ooOol09goo00<007ooOol08Woo10009goo00<007ooOol00Woo00<007ooOol07Woo00@007oo Ool0015oo`006goo00<007ooOol097oo00<007ooOol09goo00<007ooOol097oo0P009Goo00<007oo 00009Goo0`000goo0P008Goo0`004Woo000KOol2000UOol00`00Oomoo`0TOol01000Oomoo`00:7oo 00<007ooOol08Woo00<007oo00009Goo00<007ooOol00`008goo00<007ooOol04Woo000KOol00`00 Oomoo`0SOol2000WOol2000VOol3000VOol2000UOol4000VOol3000AOol001]oo`03001oogoo0<5o o`<003ioo`006goo00<007ooOol0_Woo0`00@Goo000KOol00`00Oomoo`2iOol50014Ool001]oo`03 001oogoo0:moo`X004Uoo`0057ooo`003@00000KOol00`00Oomoo`05Ool00`00Oomoo`05Ool00`00 Oomoo`05Ool00`00Oomoo`05Ool00`00Oomoo`05Ool00`00Oomoo`05Ool00`00Oomoo`05Ool00`00 Oomoo`05Ool00`00Oomoo`05Ool60002Ool00`00Oomoo`06Ool00`00Oomoo`05Ool00`00Oomoo`05 Ool00`00Oomoo`05Ool00`00Oomoo`05Ool00`00Oomoo`05Ool00`00Oomoo`04OolP00000goo001o o`06Ool00`00Oomoo`05Ool00`00Oomoo`05Ool00`00Oomoo`05Ool00`00Oomoo`05Ool00`00Oomo o`06Ool00`00Oomoo`05Ool00`00Oomoo`05Ool00`00Oomoo`05Ool00`00Oomoo`05Ool00`00Oomo o`05Ool10001Ool1Ool001]oo`03001oogoo02Eoo`03001oogoo01Qoo`D000Qoo`03001oogoo02Io o`03001oogoo02Eoo`03001oogoo02Eoo`03001oogoo02Ioo`03001oogoo011oo`006goo00<007oo Ool0>Woo1P00`Woo000KOol00`00Oomoo`0eOol50038Ool001]oo`03001oogoo035oo`@009oo`006goo00<007ooOol0 77oo00<007ooOol0hgoo000KOol00`00Oomoo`0KOol00`00Oomoo`3TOol001]oo`8001]oo`03001o ogoo0>Eoo`006goo00<007ooOol06Goo00<007ooOol0iWoo000KOol00`00Oomoo`0HOol00`00Oomo o`3WOol001]oo`03001oogoo01Moo`03001oogoo0>Qoo`006goo00<007ooOol05goo00<007ooOol0 j7oo0008Ool20004Ool20003Ool40004Ool00`00Oomoo`0FOol00`00Oomoo`3YOol000Moo`04001o ogoo0003Ool20004Ool00`00Oomoo`04Ool00`00Oomoo`0EOol00`00Oomoo`3ZOol000Moo`04001o ogoo000:Ool00`00Oomoo`03Ool2000EOol00`00Oomoo`3[Ool000Moo`04001oogoo000;Ool00`00 Oomoo`02Ool00`00Oomoo`0COol00`00Oomoo`3/Ool000Moo`04001oogoo0008Ool01000Oomoo`00 17oo00<007ooOol04Woo00<007ooOol0kGoo0008Ool2000:Ool20005Ool00`00Oomoo`0BOol00`00 Oomoo`3]Ool001]oo`03001oogoo015oo`03001oogoo0>ioo`006goo00<007ooOol04Goo00<007oo Ool0kWoo000KOol2000BOol00`00Oomoo`3^Ool001]oo`03001oogoo011oo`03001oogoo0>moo`00 6goo00<007ooOol047oo00<007ooOol0kgoo000KOol00`00Oomoo`0?Ool00`00Oomoo`3`Ool001]o o`03001oogoo00moo`03001oogoo0?1oo`006goo00<007ooOol03Woo00<007ooOol0lGoo000KOol0 0`00Oomoo`0=Ool00`00Oomoo`3bOol001]oo`8000ioo`03001oogoo0?9oo`006goo00<007ooOol0 37oo00<007ooOol0lgoo000KOol00`00Oomoo`0"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.677733, -0.667382, 0.0248015, 0.00760993}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]], Cell["Vertical stretching gives the full information:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Show[%, PlotRange -> {\(-0.02\), 0.02}, Ticks -> {{0, Pi/2, Pi, 3 Pi/2, 2\ Pi}, None}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.151576 0.309017 15.4508 [ [.2619 .29652 -8.5 -24 ] [.2619 .29652 8.5 0 ] [.5 .29652 -5 -14 ] [.5 .29652 5 0 ] [.7381 .29652 -13 -26 ] [.7381 .29652 13 0 ] [.97619 .29652 -9.5 -14 ] [.97619 .29652 9.5 0 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .2619 .30902 m .2619 .31527 L s gsave .2619 .29652 -69.5 -28 Mabsadd m 1 1 Mabs scale currentpoint translate 0 32 translate 1 -1 scale 66.062500 10.000000 moveto %%IncludeResource: font Math1Mono %%IncludeFont: Math1Mono /Math1Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (p) show 1.000000 setlinewidth 0.000000 0.000000 0.000000 setrgbcolor 64.500000 15.500000 moveto 7.687500 0.000000 rlineto stroke 66.062500 26.500000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (2) show 1.000000 setlinewidth %%DocumentNeededResources: font Math1Mono %%+ font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Math1Mono %%+ Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Math1Mono %%+ font Courier grestore .5 .30902 m .5 .31527 L s gsave .5 .29652 -66 -18 Mabsadd m 1 1 Mabs scale currentpoint translate 0 22 translate 1 -1 scale 63.000000 15.000000 moveto %%IncludeResource: font Math1Mono %%IncludeFont: Math1Mono /Math1Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (p) show 1.000000 setlinewidth %%DocumentNeededResources: font Math1Mono %%DocumentSuppliedResources: %%DocumentNeededFonts: Math1Mono %%DocumentSuppliedFonts: %%DocumentFonts: font Math1Mono grestore .7381 .30902 m .7381 .31527 L s gsave .7381 .29652 -74 -30 Mabsadd m 1 1 Mabs scale currentpoint translate 0 34 translate 1 -1 scale 66.062500 12.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (3) show 75.062500 12.000000 moveto %%IncludeResource: font Math1Mono %%IncludeFont: Math1Mono /Math1Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (p) show 1.000000 setlinewidth 0.000000 0.000000 0.000000 setrgbcolor 64.500000 17.500000 moveto 16.687500 0.000000 rlineto stroke 70.562500 28.500000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (2) show 1.000000 setlinewidth %%DocumentNeededResources: font Math1Mono %%+ font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Math1Mono %%+ Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Math1Mono %%+ font Courier grestore .97619 .30902 m .97619 .31527 L s gsave .97619 .29652 -70.5 -18 Mabsadd m 1 1 Mabs scale currentpoint translate 0 22 translate 1 -1 scale 63.000000 15.000000 moveto %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (2) show 72.000000 15.000000 moveto %%IncludeResource: font Math1Mono %%IncludeFont: Math1Mono /Math1Mono findfont 10.000000 scalefont [1 0 0 -1 0 0 ] makefont setfont 0.000000 0.000000 0.000000 setrgbcolor (p) show 1.000000 setlinewidth %%DocumentNeededResources: font Math1Mono %%+ font Courier %%DocumentSuppliedResources: %%DocumentNeededFonts: Math1Mono %%+ Courier %%DocumentSuppliedFonts: %%DocumentFonts: font Math1Mono %%+ font Courier grestore 0 .30902 m 1 .30902 L s .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .25989 .61803 m .26405 .58789 L .28529 .47013 L .30541 .38646 L .32532 .32591 L .34374 .28646 L .35336 .27133 L .3624 .26018 L .37231 .25106 L .37789 .24726 L .383 .24455 L .38774 .24267 L .3904 .24187 L .39281 .24129 L .39506 .24089 L .39633 .24071 L .3975 .24058 L .39852 .2405 L .39965 .24043 L .40083 .24039 L .40193 .24038 L .40321 .2404 L .40393 .24043 L .4046 .24046 L .40583 .24055 L .40714 .24069 L .40964 .24103 L .41191 .24146 L .41743 .24291 L .42263 .24477 L .43242 .24949 L .44274 .25599 L .46123 .2707 L .49925 .30824 L .52044 .32999 L .53973 .34818 L .54933 .35617 L .55975 .36374 L .56953 .36959 L .57869 .37378 L .58323 .37533 L .58812 .37658 L .59076 .37706 L .59209 .37725 L .59354 .37742 L .59478 .37753 L .59548 .37758 L .59611 .37761 L .59734 .37765 L .5985 .37766 L Mistroke .59972 .37763 L .60041 .3776 L .60106 .37757 L .60236 .37747 L .60376 .37731 L .60506 .37713 L .60629 .37692 L .6086 .37642 L .61131 .37567 L .61383 .37481 L .61953 .37224 L .62415 .36949 L .62914 .36583 L .63817 .35724 L .64891 .34345 L .65876 .32706 L .67766 .28433 L .68837 .25268 L .69835 .21779 L .71878 .12786 L .73736 .02112 L Mfstroke .73736 .02112 m .74026 0 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgOol0 00Ioo`03001oogoo0Ool000Ioo`03001oogoo0Ool000Ioo`03001oogoo03moo`04001oogoo000FOol00`00Oomoo`0POol00`00 Oomoo`0hOol00`00Oomoo`08Ool20007Ool00`00Ool00019Ool000Ioo`03001oogoo041oo`03001o o`0001Ioo`03001oogoo025oo`03001oogoo03Moo`03001oogoo00Yoo`03001oogoo00=oo`D004Qo o`001Woo00<007ooOol0@7oo00<007oo00005Goo00<007ooOol08goo00<007ooOol0=Woo00<007oo Ool01goo0`00Dgoo0006Ool00`00Oomoo`0oOol5000DOol00`00Oomoo`0TOol00`00Oomoo`0dOol0 0`00Oomoo`1NOol000Ioo`03001oogoo05Moo`03001oogoo02Ioo`03001oogoo03=oo`03001oogoo 05ioo`001Woo00<007ooOol0Egoo00<007ooOol09goo00<007ooOol0Goo00<007ooOol07goo00<007ooOol0I7oo0006Ool00`00 Oomoo`1AOol00`00Oomoo`0kOol00`00Oomoo`0MOol00`00Oomoo`1UOol000Ioo`03001oogoo055o o`03001oogoo03aoo`8001eoo`03001oogoo06Eoo`001Woo00<007ooOol0DGoo00<007ooOol0?Woo 00<007ooOol06Goo00<007ooOol0IWoo0006Ool00`00Oomoo`1@Ool00`00Oomoo`10Ool00`00Oomo o`0GOol00`00Oomoo`1WOol000Ioo`03001oogoo051oo`03001oogoo045oo`03001oogoo01Aoo`80 06Yoo`001Woo00<007ooOol0D7oo00<007ooOol0@Woo00<007ooOol04Woo00<007ooOol0JWoo0006 Ool00`00Oomoo`1?Ool00`00Oomoo`14Ool2000AOol00`00Oomoo`1[Ool000Ioo`03001oogoo04mo o`03001oogoo04Ioo`<000eoo`03001oogoo06aoo`001Woo00<007ooOol0Cgoo00<007ooOol0BGoo 00<007ooOol027oo0P00Kgoo0006Ool00`00Oomoo`1>Ool00`00Oomoo`1;Ool40002Ool4001aOol0 00Ioo`03001oogoo04ioo`03001oogoo04moo`8007Eoo`001Woo00<007ooOol0CWoo00<007ooOol0 aWoo0006Ool00`00Oomoo`1=Ool00`00Oomoo`37Ool000Ioo`03001oogoo04eoo`03001oogoo0Ool000Ioo`03001oogoo04Ioo`03001oogoo0Ool000Io o`03001oogoo04Ioo`03001oogoo0"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.157112, -0.0201427, 0.0232084, 0.00022768}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Closed]] }, Closed]] }, Closed]] }, Open ]] }, Open ]] }, FrontEndVersion->"4.1 for Macintosh", ScreenRectangle->{{0, 1152}, {0, 746}}, WindowSize->{587, 662}, WindowMargins->{{20, Automatic}, {Automatic, 28}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1727, 52, 42, 0, 65, "Subtitle"], Cell[1772, 54, 50, 0, 53, "Subsubtitle"], Cell[CellGroupData[{ Cell[1847, 58, 34, 0, 56, "Section"], Cell[CellGroupData[{ Cell[1906, 62, 33, 0, 46, "Subsection"], Cell[1942, 64, 161, 4, 46, "Text"], Cell[CellGroupData[{ Cell[2128, 72, 38, 1, 27, "Input"], Cell[2169, 75, 35, 1, 27, "Output"] }, Closed]], Cell[2219, 79, 101, 3, 30, "Text"], Cell[2323, 84, 73, 2, 43, "Input"], Cell[2399, 88, 128, 3, 43, "Input"], Cell[2530, 93, 91, 1, 27, "Input"], Cell[CellGroupData[{ Cell[2646, 98, 107, 2, 27, "Input"], Cell[2756, 102, 8977, 374, 66, 4524, 315, "GraphicsData", "PostScript", \ "Graphics"], Cell[11736, 478, 130, 3, 27, "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[11915, 487, 31, 0, 30, "Subsection"], Cell[11949, 489, 71, 0, 30, "Text"], Cell[CellGroupData[{ Cell[12045, 493, 125, 2, 43, "Input"], Cell[12173, 497, 8817, 371, 66, 4493, 313, "GraphicsData", "PostScript", \ "Graphics"], Cell[20993, 870, 130, 3, 27, "Output"] }, Closed]], Cell[21138, 876, 24, 0, 30, "Text"], Cell[21165, 878, 103, 2, 27, "Input"], Cell[CellGroupData[{ Cell[21293, 884, 104, 2, 43, "Input"], Cell[21400, 888, 26480, 670, 169, 5909, 412, "GraphicsData", "PostScript", \ "Graphics"], Cell[47883, 1560, 130, 3, 27, "Output"] }, Closed]], Cell[48028, 1566, 57, 0, 30, "Text"], Cell[48088, 1568, 64, 1, 27, "Input"], Cell[48155, 1571, 63, 0, 30, "Text"], Cell[CellGroupData[{ Cell[48243, 1575, 53, 1, 27, "Input"], Cell[48299, 1578, 55, 1, 27, "Output"] }, Closed]], Cell[48369, 1582, 50, 0, 30, "Text"], Cell[CellGroupData[{ Cell[48444, 1586, 46, 1, 27, "Input"], Cell[48493, 1589, 55, 1, 27, "Output"] }, Closed]], Cell[48563, 1593, 49, 0, 30, "Text"], Cell[CellGroupData[{ Cell[48637, 1597, 48, 0, 42, "Subsubsection"], Cell[48688, 1599, 62, 0, 30, "Text"], Cell[CellGroupData[{ Cell[48775, 1603, 71, 1, 27, "Input"], Cell[48849, 1606, 13714, 466, 186, 5138, 356, "GraphicsData", "PostScript", \ "Graphics"], Cell[62566, 2074, 130, 3, 27, "Output"] }, Closed]], Cell[62711, 2080, 63, 0, 30, "Text"], Cell[CellGroupData[{ Cell[62799, 2084, 127, 2, 43, "Input"], Cell[62929, 2088, 14411, 380, 186, 5295, 263, "GraphicsData", "PostScript", \ "Graphics"], Cell[77343, 2470, 130, 3, 27, "Output"] }, Closed]] }, Closed]] }, Closed]] }, Open ]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)