{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis " -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 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 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 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 "Maple O utput" -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 "Maple 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 "Normal" -1 256 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 "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 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 "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 39 "Limits of functions of two variab les " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., \+ Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "Limits and continuity for functions of two vari ables " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 28 "If the values of a function " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 72 " can be made as close as we like \+ to a fixed number L by taking the point" }{XPPEDIT 18 0 "``(x,y)" "6#- %!G6$%\"xG%\"yG" }{TEXT -1 30 " sufficiently close to a point" } {XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 17 " but not eq ual to" }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 28 ", \+ then we say that L is the " }{TEXT 259 5 "limit" }{TEXT -1 4 " of " } {XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 3 " as" } {XPPEDIT 18 0 "``(x,y)" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 11 " approaches " }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 10 ", that i s," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(g(x,y),`` (x,y) = (a, b)) = ``;" "6#/-%&LimitG6$-%\"gG6$%\"xG%\"yG/-%!G6$F*F+6$% \"aG%\"bGF." }{TEXT -1 3 "L. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 101 "In many situations of interest such a li mit is simply equal to the value of the function at the point" } {XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 3 ": " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(g(x,y),``(x,y) \+ = (a, b)) = g(a,b)" "6#/-%&LimitG6$-%\"gG6$%\"xG%\"yG/-%!G6$F*F+6$%\"a G%\"bG-F(6$F1F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 30 "When \+ this happens we say that " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\" yG" }{TEXT -1 4 " is " }{TEXT 259 10 "continuous" }{TEXT -1 3 " at" } {XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Roughly speakin g, a function of one variable is discontinous at a point " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 57 " when there is a \"break\" in the graph of the function at " }{TEXT 263 1 "a" }{TEXT -1 44 ", and s imilary, a function of two variables " }{XPPEDIT 18 0 "g(x,y)" "6#-%\" gG6$%\"xG%\"yG" }{TEXT -1 28 " is discontinuous at a point" }{XPPEDIT 18 0 "``(x,y) = (a, b);" "6#/-%!G6$%\"xG%\"yG6$%\"aG%\"bG" }{TEXT -1 113 " when there is a \"tear\" or \"contortion\" at the corresponding \+ point in the surface which constitutes the graph of " }{XPPEDIT 18 0 " g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "g(x,y)=P IECEWISE([x*y/(x^2+y^2) ,``(x,y)<>(0,0) ],[ 0,``(x,y)=(0,0)])" "6#/-% \"gG6$%\"xG%\"yG-%*PIECEWISEG6$7$*(F'\"\"\"F(F.,&*$F'\"\"#F.*$F(F1F.! \"\"0-%!G6$F'F(6$\"\"!F97$F9/-F66$F'F(6$F9F9" }{TEXT -1 50 " is conti nuous at every point except the origin. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "x=r*cos(theta )" "6#/%\"xG*&%\"rG\"\"\"-%$cosG6#%&thetaGF'" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y=r*sin(theta)" "6#/%\"yG*&%\"rG\"\"\"-%$sinG6#%&thetaG F'" }{TEXT -1 9 " so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "g(x,y)=g(r*cos(theta),r*sin(theta))" "6#/-%\"gG6$%\"xG% \"yG-F%6$*&%\"rG\"\"\"-%$cosG6#%&thetaGF-*&F,F--%$sinG6#F1F-" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "r^2*cos(theta)*sin(theta)/(r^2*cos(theta)^2 +r^2*sin(theta)^2) = cos(theta)*sin(theta)" "6#/**%\"rG\"\"#-%$cosG6#% &thetaG\"\"\"-%$sinG6#F*F+,&*&F%F&-F(6#F*F&F+*&F%F&-F-6#F*F&F+!\"\"*&- F(6#F*F+-F-6#F*F+" }{XPPEDIT 18 0 "`` = sin(2*theta)/2" "6#/%!G*&-%$si nG6#*&\"\"#\"\"\"%&thetaGF+F+F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The value of " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"x G%\"yG" }{TEXT -1 26 " is constant and equal to " }{XPPEDIT 18 0 "sin( 2*theta)/2" "6#*&-%$sinG6#*&\"\"#\"\"\"%&thetaGF)F)F(!\"\"" }{TEXT -1 37 " along the line which makes an angle " }{XPPEDIT 18 0 "theta" "6#% &thetaG" }{TEXT -1 36 " with the positive direction of the " }{TEXT 264 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 "" {TEXT -1 18 "In part icular, if " }{XPPEDIT 18 0 "proc (x, y) options operator, arrow; 0, 0 end proc" "6#f*6$%\"xG%\"yG7\"6$%)operatorG%&arrowG6\"6$\"\"!F-F+F+F+ " }{TEXT -1 11 " along the " }{TEXT 265 1 "x" }{TEXT -1 40 " axis in e ither direction, or along the " }{TEXT 266 1 "y" }{TEXT -1 27 " axis i n either direction, " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" } {TEXT -1 14 " approaches 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "On the other hand if " }{XPPEDIT 18 0 "(x,y)->( 0,0)" "6#f*6$%\"xG%\"yG7\"6$%)operatorG%&arrowG6\"6$\"\"!F-F+F+F+" } {TEXT -1 16 " along the line " }{XPPEDIT 18 0 "y=x" "6#/%\"yG%\"xG" } {TEXT -1 31 " in either direction ( so that " }{XPPEDIT 18 0 "theta=Pi /4" "6#/%&thetaG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "theta=5*Pi/4" "6#/%&thetaG*(\"\"&\"\"\"%#PiGF'\"\"%!\"\"" } {TEXT -1 9 " ), then " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 12 " approaches " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\" \"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "For this reason " } {XPPEDIT 18 0 "Limit(g(x,y),``(x,y)=(0,0))" "6#-%&LimitG6$-%\"gG6$%\"x G%\"yG/-%!G6$F)F*6$\"\"!F0" }{TEXT -1 1 " " }{TEXT 259 14 "does not ex ist" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "g(x,y" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 35 ") is not continuous \+ at the origin. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The contours of " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"x G%\"yG" }{TEXT -1 46 " are just lines which pass through the origin." }}{PARA 0 "" 0 "" {TEXT -1 26 "Here is a contour plot of " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 70 " in which each cont our is coloured according to its associated value. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " The contours of " }{XPPEDIT 18 0 " g(x,y)" "6#-%\"gG6$%\"xG%\"yG" } {TEXT -1 27 " are parabolas of the form " }{XPPEDIT 18 0 "y = s*x^2;" "6#/%\"yG*&%\"sG\"\"\"*$%\"xG\"\"#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 13 "Subsituting " }{XPPEDIT 18 0 "y=s*x^2" "6#/%\"yG*&%\"s G\"\"\"*$%\"xG\"\"#F'" }{TEXT -1 6 " in " }{XPPEDIT 18 0 "z=x^2*y/(x ^4+y^2)" "6#/%\"zG*(%\"xG\"\"#%\"yG\"\"\",&*$F&\"\"%F)*$F(F'F)!\"\"" } {TEXT -1 9 " gives " }{XPPEDIT 18 0 "z = s*x^4/(x^4+s^2*x^4);" "6#/% \"zG*(%\"sG\"\"\"*$%\"xG\"\"%F',&*$F)F*F'*&F&\"\"#F)F*F'!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "x<>0" "6#0 %\"xG\"\"!" }{TEXT -1 31 " this gives the constant value " }{XPPEDIT 18 0 "z=s/(1+s^2)" "6#/%\"zG*&%\"sG\"\"\",&F'F'*$F&\"\"#F'!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The derivative of " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$% \"xG%\"yG" }{TEXT -1 17 " with respect to " }{TEXT 271 1 "y" }{TEXT -1 6 " with " }{TEXT 270 1 "x" }{TEXT -1 46 " constant, that is, the p artial derivative of " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 17 " with respect to " }{TEXT 272 1 "y" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(``(x^2)*(x^4+y^2)- ``(x^2*y)*``(2*y))/((x^4+y^2)^2);" "6#*&,&*&-%!G6#*$%\"xG\"\"#\"\"\",& *$F*\"\"%F,*$%\"yGF+F,F,F,*&-F'6#*&F*F+F1F,F,-F'6#*&F+F,F1F,F,!\"\"F,* $,&*$F*F/F,*$F1F+F,F+F9" }{XPPEDIT 18 0 "``=x^2*(x^4-y^2)/(x^4+y^2)^2 " "6#/%!G*(%\"xG\"\"#,&*$F&\"\"%\"\"\"*$%\"yGF'!\"\"F+*$,&*$F&F*F+*$F- F'F+F'F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "g := (x,y) -> x^2*y/(x^4+y^2):\n'g( x,y)'=g(x,y);\nDiff('g(x,y)',y)=normal(diff(g(x,y),y));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"gG6$%\"xG%\"yG*(F'\"\"#F(\"\"\",&*$)F'\"\"% F+F+*$)F(F*F+F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-% \"gG6$%\"xG%\"yGF+*(F*\"\"#,&*$)F*\"\"%\"\"\"F2*$)F+F-F2!\"\"F2,&F/F2F 3F2!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 267 1 " x" }{TEXT -1 17 " is constant and " }{XPPEDIT 18 0 "x<>0" "6#0%\"xG\" \"!" }{TEXT -1 37 ", the maximum and minimum values of " }{XPPEDIT 18 0 "z=g(x,y)" "6#/%\"zG-%\"gG6$%\"xG%\"yG" }{TEXT -1 13 " occur wher e " }{XPPEDIT 18 0 "y^2=x^4" "6#/*$%\"yG\"\"#*$%\"xG\"\"%" }{TEXT -1 18 ", that is, where " }{TEXT 268 1 "y" }{TEXT -1 3 " = " }{TEXT 269 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2" "6#*$%\"xG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "Putting " }{XPPEDIT 18 0 "s=1" "6# /%\"sG\"\"\"" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "z=s/(1+s^2)" "6#/%\"z G*&%\"sG\"\"\",&F'F'*$F&\"\"#F'!\"\"" }{TEXT -1 25 " gives the maximum value " }{XPPEDIT 18 0 "z=1/2" "6#/%\"zG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 43 " which occurs everywhere along the contour " }{XPPEDIT 18 0 "y= x^2" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 24 " (except at the origin)." }} {PARA 0 "" 0 "" {TEXT -1 9 " Putting " }{XPPEDIT 18 0 "s=-1" "6#/%\"sG ,$\"\"\"!\"\"" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "z=s/(1+s^2)" "6#/%\" zG*&%\"sG\"\"\",&F'F'*$F&\"\"#F'!\"\"" }{TEXT -1 25 " gives the minimu m value " }{XPPEDIT 18 0 "-1/2" "6#,$*&\"\"\"F%\"\"#!\"\"F'" }{TEXT -1 43 " which occurs everywhere along the contour " }{XPPEDIT 18 0 "y= -x^2" "6#/%\"yG,$*$%\"xG\"\"#!\"\"" }{TEXT -1 25 " (except at the orig in). 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7%FEFE$!+r(QpM#FH7%FEFE$!+Qr&G9#FH7%FEFE$!+0bxQ>FH7%FEFE$!+sQpM1`E8FH7%FEFE$\"+_Ah I:FH7%FEFE$\"+&)QpMFH7%FEFE$\"+^r&G9#FH7%FEFE$\"+%yQ pM#FH7%FEFE$\"+(0,0)" "6#f*6$%\"xG%\"yG7\"6$%)operatorG%&arrowG6\"6$\"\"!F-F+F+F+" }{TEXT -1 16 " along the line " }{XPPEDIT 18 0 "y = m*x;" "6#/%\"yG*&%\"mG\" \"\"%\"xGF'" }{TEXT -1 12 " ( on which " }{XPPEDIT 18 0 "g(x,y)" "6#-% \"gG6$%\"xG%\"yG" }{TEXT -1 15 " has the value " }{XPPEDIT 18 0 "m*x/( x^2+m^2);" "6#*(%\"mG\"\"\"%\"xGF%,&*$F&\"\"#F%*$F$F)F%!\"\"" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 17 " approaches zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "'eval(x^2*y/(x^4+y^2),y=m*x)';\n``=value( %);\nLimit(rhs(%),x=0);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%evalG6$*(%\"xG\"\"#%\"yG\"\"\",&*$)F'\"\"%F*F**$)F)F(F*F*!\"\"/F )*&%\"mGF*F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*(%\"xG\"\"$%\"m G\"\"\",&*$)F&\"\"%F)F)*&)F(\"\"#F))F&F0F)F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*(%\"xG\"\"$%\"mG\"\"\",&*$)F'\"\"%F*F**&)F) \"\"#F*)F'F1F*F*!\"\"/F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "The plane " }{XPPEDIT 18 0 "y=m*x" "6#/%\"yG*&%\"mG\"\"\"%\"xGF'" }{TEXT -1 24 " intersects \+ the surface " }{XPPEDIT 18 0 "z=g(x,y)" "6#/%\"zG-%\"gG6$%\"xG%\"yG" } {TEXT -1 21 " in the space-curve " }{XPPEDIT 18 0 "PIECEWISE([z = m*x /(x^2+m^2), ``],[y = m*x, ``]);" "6#-%*PIECEWISEG6$7$/%\"zG*(%\"mG\"\" \"%\"xGF+,&*$F,\"\"#F+*$F*F/F+!\"\"%!G7$/%\"yG*&F*F+F,F+F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 56 "The following picture shows some of these space-curves. 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NULL:\nfor i from 0 to 40 do \n p1 := plot([r*cos(i*d),r*sin(i* d),r=0..1],\n adaptive=false,numpoints=3,\n color=C OLOR(HUE,evalf((1.22+sin(2*i*d))*0.45)),thickness=2);\n t1 := plots[ textplot]([1.1*cos(i*d),1.1*sin(i*d),\n evalf(evalf(sin(2*i *d)/2),3)],color=brown);\n cntrs := cntrs,plots[display]([p1,t1]);\n end do:\np1 := plottools[arrow]([-1.2,0],[1.3,0],.1,.03,.02,arrow,\n \+ color=COLOR(RGB,.5,.5,.5)):\np2 := plott ools[arrow]([0,-1.2],[0,1.3],.1,.03,.02,arrow,\n \+ color=COLOR(RGB,.5,.5,.5)):\nt1 := plots[textplot]([[1.25,-. 05,`x`],[-.05,1.25,`y`]],color=black):\nplots[display]([p1,p2,t1,cntrs ],scaling=constrained,tickmarks=[0,0],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 25 "code for limit animation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 534 "p1: =plot3d([r*cos(t),r*sin(t),sin(2*t)/2],t=0..2*Pi,r=0..1,\n grid=[6 0,20],style=patchcontour,color=COLOR(RGB,.8,.7,1)):\np2:=plots[spacecu rve]([0,0,t],t=-0.5..0.5,color=white):\nd := evalf(Pi/30):\nfrms := NU LL:\nfor i from 0 to 30 do \n p3:=plots[spacecurve]([r*cos(i*d),r*si n(i*d),sin(2*i*d)/2],r=-1.1..1.1,\n color=COLOR(HUE,evalf((1.22+ sin(2*i*d))*0.45)),thickness=2);\n frms := frms,plots[display]([p1,p 2,p3]);\nend do: \nplots[display]([frms],insequence=true,axes=boxed,\n labels=[`x`,`y`,`z`],lightmodel=light4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 257 "" 0 "" {TEXT -1 22 "code for contour plot " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1039 "cntrs := NULL:\nfor s in [-1,-3/4,-1/2,-1/4,0,1/4,1/2,3/4,1] do \+ \n p1 := plot([x,s*x^2,x=-1..1],\n color=COLOR(HUE,eval f((1.22+2*s/(1+s^2))*0.45)),thickness=2);\n t1 := plots[textplot]([1 .1,s,evalf(evalf(s/(1+s^2)),3)],\n color=brown);\n \+ cntrs := cntrs,plots[display]([p1,t1]);\nend do:\nfor s in [-16,-16/ 9,16/9,4,16] do \n p1 := plot([x,s*x^2,x=-1/sqrt(abs(s))..1/sqrt(abs (s))],\n color=COLOR(HUE,evalf((1.22+2*s/(1+s^2))*0.45)),th ickness=2);\n t1 := plots[textplot]([1/sqrt(abs(s)),1.1*signum(s),ev alf(evalf(s/(1+s^2)),3)],\n color=brown);\n cntrs := cntrs,plots[display]([p1,t1]);\nend do:\np1 := plottools[arrow]([- 1.2,0],[1.3,0],.1,.03,.02,arrow,\n colo r=COLOR(RGB,.5,.5,.5)):\np2 := plottools[arrow]([0,-1.2],[0,1.3],.1,.0 3,.02,arrow,\n color=COLOR(RGB,.5,.5,.5 )):\nt1 := plots[textplot]([[1.25,-.05,`x`],[-.05,1.25,`y`]],color=bla ck):\nplots[display]([p1,p2,t1,cntrs],scaling=constrained,tickmarks=[0 ,0],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 28 "c ode for 3d plot of surface " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 431 "s := 's':\np1 := plot3d([x,s*x^2,s /(1+s^2)],x=-0.3..0.3,s=-20..20,grid=[45,60],\n color=COLOR(RG B,.8,.7,1),style=patchcontour):\np2 := plot3d(x^2*y/(x^4+y^2),x=-1..1, y=-1..1,grid=[50,75],\n color=COLOR(RGB,.8,.7,1),style=patchco ntour):\np3 := plots[spacecurve]([0,0,t],t=-0.5..0.5,color=white):\npl ots[display]([p1,p2,p3],axes=boxed,lightmodel=light4,\n view=[- 1..1,-1..1,-0.5..0.5],\n labels=[`x`,`y`,`z`]);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 257 "" 0 "" {TEXT -1 22 "code for space-curves " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "crvs := NULL:\nfor m in [-2,-1,-3/4,-1/2,-1/4,0,1/4,1/2,3/4,1,2] \+ do\n p1:=plots[spacecurve]([x,m*x,m*x/(x^2+m^2)],x=-10..10,\n c olor=COLOR(HUE,evalf((1.22+m/2)*0.45)),thickness=2): \n crvs := crvs ,plots[display]([p1]);\nend do:\nplots[display]([crvs],axes=boxed,\n \+ view=[-10..10,-10..10,-1/2..1/2],labels=[`x`,`y`,`z`]);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }