{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "MS Sans Serif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "MS Sans Serif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "MS Sans Serif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "MS Sans Se rif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "MS Sans Serif " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "MS Sans Serif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "MS Sans Serif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "MS Sans Serif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "MS Sans Serif" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE " " -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Heading 3" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 18 "" 0 "" {TEXT -1 34 "Surface Integrals of Scalar Fiel ds" }}{PARA 18 "" 0 "" {TEXT -1 20 "Using Maple and the " }{TEXT 256 8 "vec_calc" }{TEXT -1 8 " Package" }}{PARA 0 "" 0 "" {TEXT -1 77 "Thi s worksheet shows how to compute surface integrals of scalar fields us ing " }{TEXT 257 5 "Maple" }{TEXT -1 9 " and the " }{TEXT 258 8 "vec_c alc" }{TEXT -1 33 " package. As examples we compute" }}{PARA 0 "" 0 " " {TEXT 260 28 "* The Net Charge on a Sphere" }}{PARA 0 "" 0 "" {TEXT 261 47 "* The Mass and Center of Mass of a Rubber Sheet" }{TEXT 262 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "To s tart the " }{TEXT 259 8 "vec_calc" }{TEXT -1 41 " package, execute the following commands:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "libname:=libname,\"C :/mylib/vector\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(libnameG6$Q=C: \\Program~Files\\Maple~6/lib6\"Q0C:/mylib/vectorF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(vec_calc): vc_aliases:" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have bee n redefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}{PARA 6 "" 1 "" {TEXT -1 33 "Package: vec_calc Version 4.3" }}{PARA 6 "" 1 "" {TEXT -1 32 "For all HELP, execute: ?vec_calc" }}{PARA 6 "" 1 "" {TEXT -1 38 "T o use aliases, execute: vc_aliases;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 264 26 "The Net Cha rge on a Sphere" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 " The charge distribution on the surface of a semiconductor sphere of ra dius " }{XPPEDIT 18 0 "15*cm;" "6#*&\"#:\"\"\"%#cmGF%" }{TEXT -1 13 " \+ is given by " }{XPPEDIT 18 0 "density := [2*sin(theta)+3*cos(phi)*cos( theta)-2*sin(phi)];" "6#>%(densityG7#,(*&\"\"#\"\"\"-%$sinG6#%&thetaGF )F)*(\"\"$F)-%$cosG6#%$phiGF)-F16#F-F)F)*&F(F)-F+6#F3F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "coul/(cm^2)" "6#*&%%coulG\"\"\"*$%#cmG\"\"#! \"\"" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 43 " is the polar angle measured down from the " }{XPPEDIT 18 0 "z; " "6#%\"zG" }{TEXT -1 10 "-axis and " }{XPPEDIT 18 0 "theta;" "6#%&the taG" }{TEXT -1 68 " is the azimuthal angle measured counterclockwise f rom the positive " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 53 "-axis. \+ We want to find the net charge on the sphere." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We first make some assump tions on the variables which will facilitate the integrations:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(showassumed=0);" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "assume(0<=phi,phi<=Pi,0<=theta,the ta<=2*Pi);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "about(phi,theta);" }} {PARA 6 "" 1 "" {TEXT -1 29 "Originally phi, renamed phi~:" }}{PARA 6 "" 1 "" {TEXT -1 35 " is assumed to be: RealRange(0,Pi)" }}{PARA 6 " " 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 33 "Originally theta, \+ renamed theta~:" }}{PARA 6 "" 1 "" {TEXT -1 37 " is assumed to be: Re alRange(0,2*Pi)" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Then we define the density:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "density:=2*sin(theta)+3*cos(phi)*cos(theta)-2*si n(phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(densityG,(-%$sinG6#%'the ta|irG\"\"#*(\"\"$\"\"\"-%$cosG6#%%phi|irGF--F/F(F-F-*&F*F--F'F0F-!\" \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 358 "Some places the charge is \+ positive and some places the charge is negative. To visualize this we make a spherical plot of the density. We take the radius to be 20 pl us the density so that the radius will be positive. (The number 20 is arbitrary.) Then the charge is positive when the radius is bigger th an 20, and negative when the radius is less than 20." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "dens_plot:=sphereplot(20+density,theta=0 ..2*Pi,phi=0..Pi, color=density, scaling=constrained, orientation=[15, 75]): dens_plot;" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6&-%%MESHG6$X.%)anythingG6\"6\"^[[[[]\\q^`r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Try rota ting the graph by clicking in the plot and dragging the mouse." }} {PARA 0 "" 0 "" {TEXT -1 89 "We are now ready to compute the net charg e which is the surface integral of the density: " }{XPPEDIT 18 0 "Int( density,S) = Int(Int(density*Jacobian,theta = 0 .. 2*Pi),phi = 0 .. Pi );" "6#/-%$IntG6$%(densityG%\"SG-F%6$-F%6$*&F'\"\"\"%)JacobianGF./%&th etaG;\"\"!*&\"\"#F.%#PiGF./%$phiG;F3F6" }{TEXT -1 161 ". The Jacobian is the length of the coordinate normal which we now proceed to comput e. Since the radius is 15, the spherical coordinate system on the sph ere is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Rsph:=[15*sin(phi )*cos(theta),15*sin(phi)*sin(theta),15*cos(phi)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%RsphG7%,$*&-%$sinG6#%%phi|irG\"\"\"-%$cosG6#%'theta| irGF,\"#:,$*&F(F,-F)F/F,F1,$-F.F*F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The tangent vectors are:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "R[theta]:=diff(Rsph,theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%\"RG6#%'theta|irG7%,$*&-%$sinG6#%%phi|irG\"\"\"-F,F&F/!#:,$*&F+F/- %$cosGF&F/\"#:\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "R[ph i]:=diff(Rsph,phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"RG6#%%phi| irG7%,$*&-%$cosGF&\"\"\"-F,6#%'theta|irGF-\"#:,$*&F+F--%$sinGF/F-F1,$- F5F&!#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "So the normal vector i s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "N:= R[theta] &x R[phi] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG7%,$*&)-%$sinG6#%%phi|irG\" \"#\"\"\"-%$cosG6#%'theta|irGF.!$D#,$*&F(F.-F*F1F.F3,&*(F)F.)F6F-F.-F0 F+F.F3**\"$D#F.F)F.)F/F-F.F:F.!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "and the length of the normal is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "lN:=len(N); simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#lNG*$-%%sqrtG6#,(*&)-%$sinG6#%%phi|irG\"\"%\"\"\")-% $cosG6#%'theta|irG\"\"#F1\"&D1&*(F8F1F+F1)-F-F5F7F1F1*$),&*(F,F1F:F1-F 4F.F1!$D#**\"$D#F1F,F1F2F1F@F1!\"\"F7F1F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$-%%sqrtG6#,&\"\"\"F)*$)-%$cosG6#%%phi|irG\"\"#F)!\" \"F)\"$D#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Maple does not simpl ify this to the simplest form. So we enter it by hand:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "lN:=15^2*sin(phi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#lNG,$-%$sinG6#%%phi|irG\"$D#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "So the net charge is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Muint(density*lN,theta = 0 .. 2*Pi,phi = 0 .. Pi); Charge:=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,$ *&,(-%$sinG6#%'theta|irG\"\"#*(\"\"$\"\"\"-%$cosG6#%%phi|irGF2-F4F-F2F 2*&F/F2-F,F5F2!\"\"F2F9F2\"$D#/F.;\"\"!,$%#PiGF//F6;F>F@" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'ChargeG,$*$)%#PiG\"\"#\"\"\"!$]%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Another way to compute this surface integ ral is to use the " }{TEXT 266 18 "Surface_int_scalar" }{TEXT -1 23 " \+ command (or its alias " }{TEXT 267 3 "Sis" }{TEXT -1 11 ") from the " }{TEXT 268 8 "vec_calc" }{TEXT -1 135 " package which works directly w ith the parametrized surface and the scalar function. The surface mus t be a function of its parameters:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Rsphf:=MF([theta,phi],Rsph);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&RsphfG7%R6$%'theta|irG%%phi|irG6\"6$%)operatorG%&arr owGF*,$*&-%$sinG6#9%\"\"\"-%$cosG6#9$F4\"#:F*F*F*RF'F*F+F*,$*&F0F4-F1F 7F4F9F*F*F*RF'F*F+F*,$-F6F2F9F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "and the scalar must be a function of position. Looking at the formula for the density, we convert to spherical coordinates:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "densityf:=MF([x,y,z], 2*y/s qrt(x^2+y^2) + 3*z/sqrt(x^2+y^2+z^2)*x/sqrt(x^2+y^2) - 2*sqrt(x^2+y^2) /sqrt(x^2+y^2+z^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)densityfGR6 %%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,(*&9%\"\"\"*$-%%sqrtG6#,&*$ )9$\"\"#F1F1*$)F0F:F1F1F1!\"\"F:*&*(\"\"$F19&F1F9F1F1*&-F46#,(F7F1F;F1 *$)FAF:F1F1F1-F46#F6F1F=F1*&*&F:F1-F46#F6F1F1*$-F46#FEF1F=F=F*F*F*" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "We check the restriction to the s phere agrees with the given function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "dens2:=simplify(densityf(op(Rsph))); density;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dens2G*&,**&-%$sinG6#%%phi|irG\"\" \"-F)6#%'theta|irGF,\"\"#**\"\"$F,-%$cosGF*F,F(F,-F4F.F,F,F0!\"\"*&F0F ,)F3F0F,F,F,*$-%%sqrtG6#,&F,F,*$F8F,F6F,F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$sinG6#%'theta|irG\"\"#*(\"\"$\"\"\"-%$cosG6#%%phi| irGF+-F-F&F+F+*&F(F+-F%F.F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "I had some difficulty getting Mpale to show that these two are equ al, but this works:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "simp lify(dens2*sqrt(1-cos(phi)^2)-density*sin(phi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "We are no w ready to compute the integral for the charge:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 70 "Sis(densityf,Rsphf,theta = 0 .. 2*Pi,phi = 0 . . Pi); Charge:=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$- F$6$,**&-%$sinG6#%%phi|irG\"\"\"-F+6#%'theta|irGF.\"$]%**\"$v'F.-%$cos GF,F.F*F.-F6F0F.F.F2!\"\"*&F2F.)F5\"\"#F.F./F1;\"\"!,$%#PiGF;/F-;F>F@ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ChargeG,$*$)%#PiG\"\"#\"\"\"!$] %" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 " " 0 "" {TEXT 263 0 "" }{TEXT 265 33 "The Mass and Center of Mass of a \+ " }{TEXT -1 12 "Rubber Sheet" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "A \+ rubber sheet is stretched across a circular ringof radius " }{XPPEDIT 18 0 "R = 3;" "6#/%\"RG\"\"$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cm;" "6# %#cmG" }{TEXT -1 117 " and a weight is placed in the middle which pull s it downward. Consequently the sheet has the shape of the graph of \+ " }{XPPEDIT 18 0 "z = 2-2^(2-(x^2+y^2)/9);" "6#/%\"zG,&\"\"#\"\"\")F&, &F&F'*&,&*$%\"xGF&F'*$%\"yGF&F'F'\"\"*!\"\"F1F1" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "x^2+y^2 <= 9;" "6#1,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(\"\"* " }{TEXT -1 52 ". Since it has been stretched down, its density is " }{XPPEDIT 18 0 "rho = 3+(x^2+y^2)/9;" "6#/%$rhoG,&\"\"$\"\"\"*&,&*$%\" xG\"\"#F'*$%\"yGF,F'F'\"\"*!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "gm /(cm^2);" "6#*&%#gmG\"\"\"*$%#cmG\"\"#!\"\"" }{TEXT -1 116 " which is \+ less in the middle and more at the edge. We want to find the mass and center of mass of the rubber sheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "We start by parametrizing the sheet in \+ cylindrical (or polar) coordinates:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "R:=MF([r,theta],[r*cos(theta),r*sin(theta),2-2^(2-r^2 /9)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG7%R6$%\"rG%'theta|irG6 \"6$%)operatorG%&arrowGF**&9$\"\"\"-%$cosG6#9%F0F*F*F*RF'F*F+F**&F/F0- %$sinGF3F0F*F*F*RF'F*F+F*,&\"\"#F0)F;,&F;F0*&#F0\"\"*F0*$)F/F;F0F0!\" \"FCF*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "and plot it" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "plot3d(R(r,theta), r=0..3, \+ theta=0..2*Pi, scaling=constrained, style=PATCHCONTOUR, shading=ZHUE, \+ orientation=[0,70]);" }}{PARA 13 "" 1 "" {GLPLOT3D 401 265 265 {PLOTDATA 3 "6'-%%MESHG6#X.%)anythingG6\"6\"^[[[[]\\q^`r\":\":\"$00000 000000000000000000000000000C000000000000000000000000000000000000000000 00000C00000000000000000000000000000000000000000000000C0000000000000000 0000000000000000000000000000000C00000000000000000000000000000000000000 000000000C00000000000000000000000000000000000000000000000C000000000000 00000000000000000000000000000000000C0000000000000008000000000000000000 0000000000000C00000000000000080000000000000000000000000000000C00000000 000000080000000000000000000000000000000C000000000000000800000000000000 00000000000000000C00000000000000080000000000000000000000000000000C0000 0000000000080000000000000000000000000000000C00000000000000080000000000 000008000000000000000C00000000000000080000000000000008000000000000000C 00000000000000080000000000000008000000000000000C0000000000000008000000 0000000008000000000000000C00000000000000080000000000000008000000000000 000C00000000000000080000000000000008000000000000000C000000000000000000 00000000000008000000000000000C0000000000000000000000000000000800000000 0000000C00000000000000000000000000000008000000000000000C00000000000000 000000000000000008000000000000000C000000000000000000000000000000080000 00000000000C00000000000000000000000000000008000000000000000C0000000000 000003FC00000000000000000000000000000BFFFEC4BB125D6743FBEE8DD4748BF153 FA0907DC193068CBFFFEC4BB125D6743FBBB67AE8584CAD3FAFFFFFFFFFFFF8BFFFEC4 BB125D6743FB6A09E667F3BD13FB6A09E667F3BC8BFFFEC4BB125D6743FB0000000000 0083FBBB67AE8584CA6BFFFEC4BB125D6743FA0907DC19306A73FBEE8DD4748BF12BFF FEC4BB125D6743CAD1A62633145C03FC0000000000000BFFFEC4BB125D674BFA0907DC 193066F3FBEE8DD4748BF19BFFFEC4BB125D674BFAFFFFFFFFFFFE03FBBB67AE8584CB 4BFFFEC4BB125D674BFB6A09E667F3BBE3FB6A09E667F3BDCBFFFEC4BB125D674BFBBB 67AE8584C9F3FB0000000000014BFFFEC4BB125