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AAAAAAAAAAB401571C71C71C71F40000000000000013FE555555555555640171C71C71 C71CA40000000000000013FE8000000000001401900000000000340000000000000013 FEAAAAAAAAAAAAC401B1C71C71C71CA40000000000000013FED555555555557401D71C 71C71C72040000000000000013FF00000000000014020000000000002-%'COLOURG6#% %ZHUEG-%*AXESSTYLEG6#%&FRAMEG-%%VIEWG6%;$!\"#\"\"!$\"\"#F8;$!\"\"F8$\" \"\"F8;$F8F8$\"\"%F8" 1 2 0 1 10 0 2 4 1 3 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 205 "The \+ top 2 X 2 portion of the derivative of this parameterization\nhas rank 2, so this parameterization (like all parameterizations of\nsurfaces \+ z=f(x,y) as graphs (x,y) -> (x,y,f(x,y)) is always\nnonsingular." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "jacobian([r,s,r^2+4*s^2],[r, s]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7$\"\"\"\"\"!7$F )F(7$,$%\"rG\"\"#,$%\"sG\"\")" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 "Trigonometric Parameterization" }}{EXCHG {PARA 0 "" 0 "" {TEXT 260 39 "A reparameterization of the paraboloid." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "subs(r=2*v*cos(u),s=v*sin(u),[r,s,r^2+4*s^2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*&%\"vG\"\"\"-%$cosG6#%\"uGF'\"\"# *&F&F'-%$sinGF*F',&*&)F&F,F')F(F,F'\"\"%*(F4F'F2F')F.F,F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "map(simplify,%);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*&%\"vG\"\"\"-%$cosG6#%\"uGF'\"\"# *&F&F'-%$sinGF*F',$*$)F&F,F'\"\"%" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 261 252 "The same paraboloid parameterized via trigonometric functions .\nNotice how the grid lines in the pictures above and below are very \+ different.\nThe curves r or s constant are parabolas;\nThe curves u=co nstant are parabolas, but v=constant gives an ellipse." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 83 "plot3d([2*v*cos(u),v*sin(u),4*v^2],u=0.. 2 * P i, v= 0..1,axes=framed,shading=zhue);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 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44 "jacobian([2*v*cos(u),v*sin(u),4*v^2],[u,v]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7$,$*&%\"vG\"\"\"-%$sinG 6#%\"uGF+!\"#,$-%$cosGF.\"\"#7$*&F*F+F2F+F,7$\"\"!,$F*\"\")" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 28 "A Rational Parameterization." }} {EXCHG {PARA 0 "" 0 "" {TEXT 263 94 "The substitution u =2* arctan(t) \+ (or t = tan(u/2) gives another interesting \nparameterization." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "subs(u=2*arctan(t),[2*v*cos(u),v*si n(u),4*v^2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*&%\"vG\"\"\"-%$c osG6#,$-%'arctanG6#%\"tG\"\"#F'F0*&F&F'-%$sinGF*F',$*$)F&F0F'\"\"%" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "param3 := map(simplify,map( expand,%));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'param3G7%,$*&*&%\"vG\"\"\",&!\"\"F**$)%\"tG\"\"#F*F* F*F*,&F*F*F-F*F,!\"#,$*&*&F)F*F/F*F*F1F,F0,$*$)F)F0F*\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 172 "If we let t range from -infinity to + i nfinity, u would range from -1 to 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algebra:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "simplify((sin h(u))^2-(cosh(u))^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "subs(x=2*sinh(u)*cos(v),y=3* sinh(u)*sin(v),9*x^2+4*y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&)- %%sinhG6#%\"uG\"\"#\"\"\")-%$cosG6#%\"vGF*F+\"#O*(F1F+F%F+)-%$sinGF/F* F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)-%%coshG6#%\"uG\"\"#\"\"\" \"#OF,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "subs(x=2*sin h(u)*cos(v),y=3*sinh(u)*sin(v),z=6*cosh(u),hpb_fcn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&)-%%sinhG6#%\"uG\"\"#\"\"\")-%$cosG6#%\"vGF*F+\" #O*(F1F+F%F+)-%$sinGF/F*F+F+*&F1F+)-%%coshGF(F*F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "map(simplify,%);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 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