{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "Vessiot_Text" -1 256 "Intrepid" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Vess_Title2" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 128 0 64 1 2 2 0 0 0 2 0 0 0 }1 0 0 0 4 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Vess_IO" -1 257 1 {CSTYLE "" -1 -1 " Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 1 0 0 }1 0 0 -1 -1 -1 3 30 0 0 0 0 -1 3 }{PSTYLE "Vess_Title1" -1 258 1 {CSTYLE "" -1 -1 "Helvetica" 1 18 128 0 64 1 0 0 0 0 0 0 3 0 0 }2 1 0 0 10 10 3 6 3 30 0 0 -1 0 } {PSTYLE "Example" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 256 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 258 "" 0 "" {TEXT -1 83 " \+ Vessiot Tutorial: Polynomial Harmonic Maps " }}{PARA 260 "" 0 "" {TEXT 257 7 "Purpose" }}{PARA 257 "" 0 "" {TEXT -1 76 "We check t hat some polynomial maps define harmonic maps between spheres." }} {PARA 257 "" 0 "" {TEXT -1 29 "See J. Eells and A. Ratto, " }{TEXT 264 52 "Harmonic Maps and Minimal Immersions with Symmetrics" }{TEXT -1 13 ", Chapter VII" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT 258 22 "Procedures Illustrated" }}{PARA 257 "" 0 "" {TEXT -1 11 "harmonicSS " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( Vessiot):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(tensors): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(harmonic_maps):" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 259 18 "Veronese Type Maps" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 51 "A. The classical Ver onese map s from S^2 to S^4. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "coord_frame([x,y,z],[u1,u2 ,u3,u4,u5]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Define the map s \+ as a map from R^3 to R^5." }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 92 "s:=[x*y/sqrt(3), x*z/sqrt(3), y*z/sqrt(3), 1/(2*sqrt(3))*(x^2 \+ -y^2), 1/6*(x^2 +y^2 -2*z^2)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" sG7',$*(%\"xG\"\"\"%\"yGF)-%%sqrtG6#\"\"$\"\"\"#F)F.,$*(F(F/%\"zGF)F+F /F0,$*(F*F/F3F/F+F/F0,$*&F+F/,&*$)F(\"\"#F/F)*$)F*F;F/!\"\"F)#F)\"\"', (F9F?FF." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Check \+ that the image of the map lies in S^4." }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 46 "factor(s[1]^2 +s[2]^2 +s[3]^2+s[4]^2 +s[5]^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$),(*$)%\"yG\"\"#\"\"\"\"\"\"*$)% \"zGF*F+F,*$)%\"xGF*F+F,F*F+#F,\"\"*" }}}{EXCHG {PARA 0 "euclid>" 0 " " {MPLTEXT 1 0 25 "H:=factor(harmonicSS(s)):" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 45 "Check that H vanishes when restrictes to S^2." }}} {EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 48 "show(helmsimp(map2(su bs,\{x^2= 3- y^2 -z^2\},H)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0 ~G\"\"\"&&%%D_u1G6#7%\"\"!F+F+6#%!GF%" }}}{EXCHG {PARA 0 "euclid>" 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 35 "B. The Veronese map from S^ 2 to S^6" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 45 "coord_frame([x,y,z], \+ [u1,u2,u3,u4,u5,u6,u7]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "s:=[1/2*y^3 -3/2*x^2*y, sqrt(6)/2*(y^2-x^2)*z, sqrt(15)/10*(x^2+y^2-4 *z^2)*y," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "sqrt(15)/10*(x^2 +y^2 - 4*z^2)*x,sqrt(6)*x*y*z,1/2*x^3 -3/2*x*y^2,sqrt(10)/10* " }}{PARA 0 "eu clid>" 0 "" {MPLTEXT 1 0 24 "(3*x^2+3*y^2 -2*z^2)*z];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"sG7),&*$)%\"yG\"\"$\"\"\"#\"\"\"\"\"#*&)%\"xGF .F+F)F-#!\"$F.,$*(-%%sqrtG6#\"\"'F+,&*$)F)F.F+F-*$F0F+!\"\"F-%\"zGF-F, ,$*(-F76#\"#:F+,(F=F-F;F-*$)F?F.F+!\"%F-F)F+#F-\"#5,$*(FBF+FEF+F1F-FI* *F6F+F1F+F)F+F?F+,&*$)F1F*F+F,*&F1F+F" 0 "" {MPLTEXT 1 0 27 "f actor(add(s[i]^2,i=1..7));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$),(* $)%\"yG\"\"#\"\"\"\"\"\"*$)%\"zGF*F+F,*$)%\"xGF*F+F,\"\"$F+#F*\"\"&" } }}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 25 "H:=factor(harmonicSS( s)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Check that H vanishes whe n restrictes to S^2." }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 66 "show(helmsimp(subs(\{y^6 = 5/2- expand((x^2+y^2+z^2)^3) +y^6\},H)) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G\"\"\"&&%%D_u1G6#7%\"\"!F +F+6#%!GF%" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 256 "" 0 "" {TEXT 260 12 "Eiconal Maps" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT 265 35 "A. The Eiconal Map from S^4 to S^ 4:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euclid > " 0 " " {MPLTEXT 1 0 42 "coord_frame([x,y,z,u,v],[u1,u2,u3,u4,u5]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "P1:=1/3*u^3 -u*v^2+u/2*(x^2+ y^2-2*z^2)" }}{PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 38 "+sqrt(3)/2*v*(x^ 2-y^2) +sqrt(3)*x*y*z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G,,*$)% \"uG\"\"$\"\"\"#\"\"\"F)*&F(F,)%\"vG\"\"#F*!\"\"*&F(F*,(*$)%\"xGF0F*F, *$)%\"yGF0F*F,*$)%\"zGF0F*!\"#F,#F,F0*(-%%sqrtG6#F)F*F/F,,&F4F,F7F1F,F >**F@F*F6F,F9F,F" 0 "" {MPLTEXT 1 0 37 "alpha:= ext_d(P1&mult scalar_form()):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 32 "s:=[seq(alpha[2][i][2],i=1..5)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG7',(*&%\"uG\"\"\"%\"xGF)F)*(-%%sqrtG6#\" \"$\"\"\"%\"vGF)F*F0F)*(F,F0%\"yGF)%\"zGF)F),(*&F(F0F3F0F)*(F,F0F1F0F3 F0!\"\"*(F,F0F*F0F4F0F),&*&F(F0F4F0!\"#*(F,F0F*F0F3F0F),,*$)F(\"\"#F0F )*$)F1FAF0F8*$)F*FAF0#F)FA*$)F3FAF0FF*$)F4FAF0F8,(*&F(F0F1F0F<*&F,F0FE F0FF*&F,F0FHF0#F8FA" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Check that the image of s is in S^4." }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 20 "add(s[i]^2, i=1..5):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$ ),,*$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"vGF)F*F+*$)%\"yGF)F*F+*$)%\"zGF)F*F+ *$)%\"uGF)F*F+F)F*" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 25 "H:=factor(harmonicSS(s)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 " Check that H vanishes when restricted to S^4." }}}{EXCHG {PARA 0 "euc lid > " 0 "" {MPLTEXT 1 0 49 "show(helmsimp(subs(\{z^2=1-x^2-y^2 -u^2- v^2\},H)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G\"\"\"&&%%D_u1G6 #7'\"\"!F+F+F+F+6#%!GF%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 34 "B. Th e Eiconal Map from S^7 to S^7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 63 "coord_frame([u,v,x1,y1,x2,y2,z1,z2],[u1,u2,u3, u4,u5,u6,u7,u8]):" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 34 "X :=x1+I*x2:Y:=y1+I*y2: Z:=z1+I*z2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "cX:=x1-I*x2:cY:=y1-I*y2:cZ:=z1-I*z2:" }}{PARA 0 "eucl id>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 106 "P2:=expand(1/3*u^3 -u*v^2 +u/2*(X*cX +Y*cY -2*Z*cZ) +sqrt(3)/ 2*v*(X*cX-Y*cY)+sqrt(3)/2*(X*Y*Z +cX*cY*cZ));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#P2G,B*$)%\"uG\"\"$\"\"\"#\"\"\"F)*&F(F,)%\"vG\"\"#F* !\"\"*&F(F*)%#x1GF0F*#F,F0*&F(F*)%#x2GF0F*F5*&F(F*)%#y1GF0F*F5*&F(F*)% #y2GF0F*F5*&F(F*)%#z1GF0F*F1*&F(F*)%#z2GF0F*F1*(-%%sqrtG6#F)F*F/F,F3F* F5*(FFF*F/F*F7F*F5*(FFF*F/F*F:F*#F1F0*(FFF*F/F*F=F*FK**FFF*F4F,F;F,FAF ,F,**FFF*F4F*F>F,FDF,F1**FFF*F8F,F;F*FDF*F1**FFF*F8F*F>F*FAF*F1" }}} {EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 37 "alpha:= ext_d(P2&mult s calar_form()):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 32 "s:=[seq(alpha[2][i] [2],i=1..8)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"sG7*,2*$)%\"uG\" \"#\"\"\"\"\"\"*$)%\"vGF*F+!\"\"*$)%#x1GF*F+#F,F**$)%#x2GF*F+F4*$)%#y1 GF*F+F4*$)%#y2GF*F+F4*$)%#z1GF*F+F0*$)%#z2GF*F+F0,,*&F)F,F/F,!\"#*&-%% sqrtG6#\"\"$F+F2F+F4*&FHF+F6F+F4*&FHF+F9F+#F0F**&FHF+F " 0 "" {MPLTEXT 1 0 20 "add(s[k]^2, k=1..8):" }}} {EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),2*$)%#z2G\"\"#\"\"\"\"\"\"*$)%#z1GF)F*F+ *$)%#y2GF)F*F+*$)%#y1GF)F*F+*$)%#x2GF)F*F+*$)%#x1GF)F*F+*$)%\"uGF)F*F+ *$)%\"vGF)F*F+F)F*" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 17 "H:=harmonicSS(s):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 14 "H1:=factor(H):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Check tha t H vanishes when restricted to S^7." }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 64 "show(helmsimp(subs(\{u^2=1 -v^2 -expand(X*cX +Y* cY +Z*cZ)\},H1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G\"\"\"&&% %D_u1G6#7*\"\"!F+F+F+F+F+F+F+6#%!GF%" }}}{EXCHG {PARA 0 "euclid>" 0 " " {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 256 "" 0 "" {TEXT 261 8 "Hopf Map" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The classical Hopf map from S^3 to S^2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "coord_frame([x,y,u,v],[u1,u2,u3]):" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 56 "s:=[x*u+y*v,x*v-y*u,1/2*x^2 +1/2*y^2 -1/2*u^2 -1/2*v^ 2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG7%,&*&%\"xG\"\"\"%\"uGF)F )*&%\"yGF)%\"vGF)F),&*&F(\"\"\"F-F0F)*&F,F0F*F0!\"\",**$)F(\"\"#F0#F)F 6*$)F,F6F0F7*$)F*F6F0#F2F6*$)F-F6F0F<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Check that the image of s is in S^2." }}}{EXCHG {PARA 0 " euclid>" 0 "" {MPLTEXT 1 0 31 "factor(s[1]^2 +s[2]^2 +s[3]^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$),**$)%\"uG\"\"#\"\"\"\"\"\"*$)%\" vGF*F+F,*$)%\"xGF*F+F,*$)%\"yGF*F+F,F*F+#F,\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "H:=factor(harmonicSS(s)):" }}{PARA 0 "" 0 "" {TEXT -1 44 "Check that s vanishes when restricted to S^3" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 49 "show(helmsimp(subs( \{v^2= 2 - x^2 -y^2 -u^2\},H)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G\"\" \"&&%%D_u1G6#7&\"\"!F+F+F+6#%!GF%" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "8 0" 0 }{VIEWOPTS 1 1 0 3 4 1802 }