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" }{TEXT 304 1 "V" }{TEXT -1 6 " is a " }{TEXT 259 29 "function of the two vari ables" }{TEXT -1 1 " " }{TEXT 266 1 "r" }{TEXT -1 5 " and " }{TEXT 267 1 "h" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 269 9 "Example 2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "The formula " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(x,y)=x+y" "6#/-%\"fG6$%\"xG%\"yG,&F'\"\"\"F(F*" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 42 "defines a function f of \+ the two variables " }{TEXT 270 1 "x" }{TEXT -1 5 " and " }{TEXT 271 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 76 "We have a real numb er value for this function for each pair of real numbers " }{TEXT 272 1 "x" }{TEXT -1 5 " and " }{TEXT 273 1 "y" }{TEXT -1 52 ", and this va lue is obtained by add the two numbers " }{TEXT 274 1 "x" }{TEXT -1 5 " and " }{TEXT 275 1 "y" }{TEXT -1 56 ". In fact we have a value of th is function at each point" }{XPPEDIT 18 0 "``(x,y)" "6#-%!G6$%\"xG%\"y G" }{TEXT -1 26 " in the coordinate plane. " }}{PARA 0 "" 0 "" {TEXT -1 78 "One way two attempt to visualize a function of two variables is to consider a " }{TEXT 259 12 "contour plot" }{TEXT -1 92 ", drawn in the coordinate plane, in which points which produce the same value ar e joined by " }{TEXT 259 8 "contours" }{TEXT -1 36 " or contour lines. For the function " }{XPPEDIT 18 0 "f(x,y)=x+y" "6#/-%\"fG6$%\"xG%\"yG ,&F'\"\"\"F(F*" }{TEXT -1 40 ", the contour associated with the value \+ " }{TEXT 276 1 "c" }{TEXT -1 36 " is the straight line with equation \+ " }{XPPEDIT 18 0 "x+y=c" "6#/,&%\"xG\"\"\"%\"yGF&%\"cG" }{TEXT -1 2 ". 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The " } {TEXT 259 5 "graph" }{TEXT -1 17 " of the function " }{XPPEDIT 18 0 "f (x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 79 " is obtained by plotting a point at a distance above or below the level of the " }{TEXT 279 1 "x " }{TEXT -1 1 "," }{TEXT 280 1 "y" }{TEXT -1 118 " plane to indicate t he value of the function at each point. We could imagine that each con tour line associated with a " }{TEXT 260 14 "positive value" }{TEXT -1 34 " (coloured red in the picture) is " }{TEXT 261 6 "raised" } {TEXT -1 24 " above the level of the " }{TEXT 281 1 "x" }{TEXT -1 1 ", " }{TEXT 282 1 "y" }{TEXT -1 83 " plane a distance equal to the value \+ associated with the contour. Thus the contour " }{XPPEDIT 18 0 "x+y=2 " "6#/,&%\"xG\"\"\"%\"yGF&\"\"#" }{TEXT -1 29 " is raised 2 units abov e the " }{TEXT 283 1 "x" }{TEXT -1 1 "," }{TEXT 284 1 "y" }{TEXT -1 7 " plane." }}{PARA 0 "" 0 "" {TEXT -1 36 "Similarly, contours associate d with " }{TEXT 256 15 "negative values" }{TEXT -1 36 " (coloured blue in the picture) are " }{TEXT 261 7 "lowered" }{TEXT -1 29 " down belo w the level of the " }{TEXT 285 1 "x" }{TEXT -1 1 "," }{TEXT 286 1 "y " }{TEXT -1 25 " plane. Thus the contour " }{XPPEDIT 18 0 "x+y=-2" "6# /,&%\"xG\"\"\"%\"yGF&,$\"\"#!\"\"" }{TEXT -1 30 " is lowered 2 units b elow the " }{TEXT 287 1 "x" }{TEXT -1 1 "," }{TEXT 288 1 "y" }{TEXT -1 33 " plane. The resulting graph is a " }{TEXT 259 7 "surface" } {TEXT -1 25 " in 3-dimensional space. " }}{PARA 0 "" 0 "" {TEXT -1 28 "We can introduce a vertical " }{TEXT 291 1 "z" }{TEXT -1 47 " axis to measure distances above and below the " }{TEXT 289 1 "x" }{TEXT -1 1 "," }{TEXT 290 1 "y" }{TEXT -1 8 " plane. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "The following 3d plot can be m anipulated with the mouse so that you can get a feel for its 3-dimensi onal \"shape\". 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0 "" 0 "" {TEXT -1 32 "The equation of this surface is " } {XPPEDIT 18 0 "z=sqrt(x^2+y^2)" "6#/%\"zG-%%sqrtG6#,&*$%\"xG\"\"#\"\" \"*$%\"yGF+F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "Note th at the equation " }{XPPEDIT 18 0 "z^2=x^2+y^2" "6#/*$%\"zG\"\"#,&*$% \"xGF&\"\"\"*$%\"yGF&F*" }{TEXT -1 18 " is equivalent to " }{TEXT 297 1 "z" }{TEXT -1 3 " = " }{TEXT 296 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(x^2+y^2)" "6#-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF)F*" } {TEXT -1 67 ", and its graph is a double cone. Since there are, in gen eral, two " }{TEXT 298 1 "z" }{TEXT -1 32 " values for each pair of va lues " }{XPPEDIT 18 0 "``(x,y)" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 14 " th e equation " }{XPPEDIT 18 0 "z^2=x^2+y^2" "6#/*$%\"zG\"\"#,&*$%\"xGF& \"\"\"*$%\"yGF&F*" }{TEXT -1 32 " does not represent a function. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x,y) = x^2+y^2;" "6#/-% \"fG6$%\"xG%\"yG,&*$F'\"\"#\"\"\"*$F(F+F," }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "The contours of \+ this function are again concentric circles with their common centre at the origin in the " }{TEXT 292 1 "x" }{TEXT -1 1 "," }{TEXT 293 1 "y " }{TEXT -1 8 " plane. " }}{PARA 0 "" 0 "" {TEXT -1 14 "This time the \+ " }{TEXT 262 20 "square of the radius" }{TEXT -1 58 " of the circle is the associated value of the function f. 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0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Functions of 3 variables " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 22 "C onsider the function " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x,y,z)=sqrt(x^2+y^2+z^2)" "6#/-%\"fG6%%\"xG%\"yG%\"zG-%%sqrtG6 #,(*$F'\"\"#\"\"\"*$F(F/F0*$F)F/F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "of the 3 variables " }{TEXT 299 1 "x" }{TEXT -1 2 ", " } {TEXT 300 1 "y" }{TEXT -1 5 " and " }{TEXT 301 1 "z" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 124 "This function can be visualised by mea ns of \"level surfaces\", that is, surfaces on which the function has \+ a constant value. " }}{PARA 0 "" 0 "" {TEXT -1 54 "The level surface o ver which f has the constant value " }{TEXT 303 1 "a" }{TEXT -1 36 " i s the surface consisting of points" }{XPPEDIT 18 0 "``(x,y,z)" "6#-%!G 6%%\"xG%\"yG%\"zG" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "sqrt(x^2 +y^2+z^2)=a" "6#/-%%sqrtG6#,(*$%\"xG\"\"#\"\"\"*$%\"yGF*F+*$%\"zGF*F+% \"aG" }{TEXT -1 5 " or " }{XPPEDIT 18 0 "x^2+y^2+z^2=a^2" "6#/,(*$%\" xG\"\"#\"\"\"*$%\"yGF'F(*$%\"zGF'F(*$%\"aGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The value of " }{XPPEDIT 18 0 "sqrt(x^2+y^2+z^2 )" "6#-%%sqrtG6#,(*$%\"xG\"\"#\"\"\"*$%\"yGF)F**$%\"zGF)F*" }{TEXT -1 32 " gives the distance of the point" }{XPPEDIT 18 0 "``(x,y,z)" "6#-% !G6%%\"xG%\"yG%\"zG" }{TEXT -1 16 " from the origin" }{XPPEDIT 18 0 "` `(0,0,0)" "6#-%!G6%\"\"!F&F&" }{TEXT -1 49 " in 3-dimensional space. H ence the level surface " }{XPPEDIT 18 0 "sqrt(x^2+y^2+z^2)=a" "6#/-%%s qrtG6#,(*$%\"xG\"\"#\"\"\"*$%\"yGF*F+*$%\"zGF*F+%\"aG" }{TEXT -1 55 " \+ is a sphere with its centre at the origin and radius " }{TEXT 302 1 " a" }{TEXT -1 3 ". 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jp$\"#&*FcjpF)F)Ffjp-%+AXESLABELSG6%%\"xG%\"yG%\"zG-%*AXESSTYLEG6#%$BO XG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 1 1 2 1 1.000000 40.000000 40.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pictures" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 25 "pictures for f(x,y)=x+y " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 440 "p1 := plot([seq([i-x ,x,x=max(i-4,-4)..min(i+4,4)],i=-7..7)],color=[blue$7,black,red$7],\n \+ labels=[``,``]):\nt1 := plots[textplot]([[4.9,-.3,`x`],[-.3,4.4,`y`] ]):\nt2 := plots[textplot]([seq([4.8,i-3.8,`x + y = `||i],i=1..7)],col or=red):\nt3 := plots[textplot]([seq([-4.8,4.2+i,`x + y = `||i],i=-7.. -1)],color=blue):\nt4 := plots[textplot]([4.8,-3.8,`x + y = 0`],color= black):\nplots[display]([p1,t1,t2,t3,t4],view=[-5..5,-4..4.5],thicknes s=2);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "pp := \{seq(plots[spacecurve]([i-x,x,i,x=max(i-4,-4) ..min(i+4,4),color=`if`(i<0,blue,`if`(i=0,black,red))]),i=-7..7)\}:\np lots[display](pp,axes=boxed,labels=[x,y,z],thickness=2);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "p1 := plot3d([x,z-x,z],x=-4..4,z=0..8,colo r=COLOR(RGB,1,.5,.5)):\np2 := plot3d([x,z-x,z],x=-4..4,z=-8..0,color=C OLOR(RGB,.5,.5,1)):\nplots[display]([p1,p2],view=[-4..4,-4..4,-8..8],a xes=boxed,\n labels=[`x`,`y`,`z`]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 " " 0 "" {TEXT -1 34 "pictures for f(x,y)=sqrt(x^2+y^2) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 452 "p1 := \+ plot([seq([i*cos(t),i*sin(t),t=0..2*Pi],i=1..4)],color=red,\n labels =[``,``]):\np2 := plot([[[0,0]]$3],style=point,symbol=[circle,diamond, cross],color=red):\nt1 := plots[textplot]([[4.5,-.3,`x`],[-.3,4.5,`y`] ]):\nt2 := plots[textplot]([[0.1,.3,`f(x,y)=0`],[1.1,.8,`f(x,y)=1`],[1 .85,1.5,`f(x,y)=2`],\n [2.6,2.2,`f(x,y)=3`],[3.4,2.9,`f(x,y)=4`]],co lor=brown):\nplots[display]([p1,p2,t1,t2],thickness=2,view=[-4.5..4.5, -4.5..4.5],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "pp := seq(plots[spacecurve] ([i*cos(t),i*sin(t),i,t=0..2*Pi],color=red),i=1..4):\np1 := plots[poin tplot3d]([[0,0,0]],color=red,symbol=circle):\nplots[display](\{pp,p1\} ,axes=boxed,labels=[x,y,z],thickness=2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 28 "pictures for f(x,y)=x^2+y^2 " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 454 "p1 := plo t([seq([i*cos(t),i*sin(t),t=0..2*Pi],i=1..4)],color=red,\n labels=[` `,``]):\np2 := plot([[[0,0]]$3],style=point,symbol=[circle,diamond,cro ss],color=red):\nt1 := plots[textplot]([[4.5,-.3,`x`],[-.3,4.5,`y`]]): \nt2 := plots[textplot]([[0.1,.3,`f(x,y)=0`],[1.1,.8,`f(x,y)=1`],\n \+ [1.85,1.5,`f(x,y)=4`],[2.6,2.2,`f(x,y)=9`],[3.4,3,`f(x,y)=16`]],col or=brown):\nplots[display]([p1,p2,t1,t2],thickness=2,view=[-4.5..4.5,- 4.5..4.5],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "pp := seq(plots[spacecurve] ([i*cos(t),i*sin(t),i^2,t=0..2*Pi],color=red),i=1..4):\np1 := plots[po intplot3d]([[0,0,0]],color=red,symbol=circle):\nplots[display](\{pp,p1 \},axes=boxed,labels=[x,y,z],thickness=2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 19 "concentric spheres " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 404 "p1 := plots [display](plottools[sphere]([0,0,0],1),\n style=wireframe, color=COLOR(RGB,.65,.2,.2)):\np2 := plots[display](plottools[sphere]([ 0,0,0],2),\n style=wireframe,color=COLOR(RGB,.8,.1,.1)):\n p3 := plots[display](plottools[sphere]([0,0,0],3),\n style =wireframe,color=COLOR(RGB,.95,0,0)):\nplots[display](\{p1,p2,p3\},axe s=boxed,scaling=constrained,labels=[`x`,`y`,`z`]);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }