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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+)-%$sinGF/F*F+F+*&F1F+)-%%coshGF(F*F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "map(simplify,%);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 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FFFFFFF4024000000000000404C1FFFFFFFFFFE4054D55555555554402400000000000 040515C71C71C71C64056EAAAAAAAAAAA4024000000000000405501C71C71C71C40590 0000000000040240000000000004059000000000000-%'COLOURG6#%*XYSHADINGG-%& STYLEG6#%&PATCHG-%*AXESSTYLEG6#%$BOXG-%+PROJECTIONG6%$\"$+\"\"\"!$\"#! )F;\"\"\"-%+AXESLABELSG6%%\"xG%\"yG%\"zG" 1 2 0 1 10 0 2 1 1 2 2 1.000000 80.000000 100.000000 1 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 " " {TEXT 271 28 "Singular just at the origin." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "jacobian([u*v,u,v^2],[u,v]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7$%\"vG%\"uG7$\"\"\"\"\"!7$F,,$F(\"\"#" } }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 287 37 "An ellipsoid x^2 + 4*y^2 + 4* z^2 = 4." }}{EXCHG {PARA 0 "" 0 "" {TEXT 272 37 "An ellipsoid x^2 + 4* y^2 + 4*z^2 = 4." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "gr1 := plot3d( [2*sin(v)*cos(u),sin(v)*sin(u),cos(v)],u=0..2*Pi,v=0..Pi,grid=[16,16], \nstyle=hidden,axes=framed,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "deriv := jacobian([2*sin(v)*cos(u),sin(v)*sin(u),cos( v)],[u,v]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&derivG-%'matrixG6#7% 7$,$*&-%$sinG6#%\"vG\"\"\"-F-6#%\"uGF0!\"#,$*&-%$cosGF.F0-F8F2F0\"\"#7 $*&F,F0F9F0*&F7F0F1F07$\"\"!,$F,!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 273 35 "Tangent to the v = constant curves." }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "T_u := submatrix(deriv,1..3,1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$T_uG-%'matrixG6#7%7#,$*&-%$sinG6#%\"vG\"\"\"-F- 6#%\"uGF0!\"#7#*&F,F0-%$cosGF2F07#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 274 35 "Tangent ot the u = constant curves." }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "T_v:= submatrix(deriv,1..3,2..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$T_vG-%'matrixG6#7%7#,$*&-%$cosG6#%\"vG\"\"\"-F-6# %\"uGF0\"\"#7#*&F,F0-%$sinGF2F07#,$-F8F.!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 275 42 "A point on the ellipsoid. (u=0,v=3*Pi/10.)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "point_1 := evalf(subs(u=0,v=3*Pi/10,[2*si n(v)*cos(u),sin(v)*sin(u),cos(v)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%(point_1G7%$\"+*)R.=;!\"*$\"\"!F*$\"+AD&y(e!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 276 110 "This doesn't work because ,matrices are not eva luated in Maple without an explicit\nrequest e.g. using \"eval\"." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "T_u_1 := evalf(subs(u=0,v=3*Pi/10,T _u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&T_u_1G%$T_uG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 277 43 "Evaluate the tangent vector T_u at point _1." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "T_u_1 := evalf(subs(u=0,v=3* Pi/10,eval(T_u)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&T_u_1G-%'matr ixG6#7%7#$!\"!\"\"!7#$\"+W*p,4)!#57#$F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 278 43 "Evaluate the tangent vector T_v at point_1." }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 46 "T_v_1 := evalf(subs(u=0,v=3*Pi/10,eval(T_v)) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&T_v_1G-%'matrixG6#7%7#$\"+/0d v6!\"*7#$\"\"!F/7#$!+W*p,4)!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 279 61 "A line segment from point_1 pointing in the direction of T_u." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "gr1_u := spacecurve(evalm(point_1 + t*T_u_1),t=0.. .4,thickness=3,color=red):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 280 61 "A line segment from point_1 pointing in the direction of T_v." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "gr1_v := spacecurve(evalm( point_1 + t*T_v_1),t=0.. .25,thickness=3,color=red):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 281 68 " A coarse grid on the ellipsoid together with \+ T_u and T_v at point1." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "display3d ([gr1,gr1_u,gr1_v],thickness=3);\n" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6'-%%MESHG6%X.%)anythingG6\"6\"^[[[[]\\q[[n\"1\"1 \"$000000000000000000000000000000003FF00000000000003FDA9CD9AC4258EE000 00000000000003FEF4CFC327A00803FEA07F921061ACA00000000000000003FED3BC3A 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