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It measures the rate of change of " }{TEXT 266 1 "z" }{TEXT -1 17 " with respect to " }{TEXT 267 1 "x" }{TEXT -1 7 " while " }{TEXT 268 1 "y" }{TEXT -1 129 " is kept constant. This essentially means that, if you \+ know how to take a derivative, you know how to take a partial derivati ve. " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 21 "partial deriv ative of" }{TEXT -1 2 " " }{XPPEDIT 18 0 "z=g(x,y)" "6#/%\"zG-%\"gG6$ %\"xG%\"yG" }{TEXT 304 1 " " }{TEXT -1 1 " " }{TEXT 259 15 "with respe ct to" }{TEXT -1 1 " " }{TEXT 269 1 "x" }{TEXT -1 16 " is denoted by \+ " }{XPPEDIT 18 0 "Diff(z,x) = Diff(g(x,y),x);" "6#/-%%DiffG6$%\"zG%\"x G-F%6$-%\"gG6$F(%\"yGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = g[x](x,y );" "6#/%!G-&%\"gG6#%\"xG6$F)%\"yG" }{TEXT -1 29 ", and is defined for mally by " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "g[x](x,y)=``" "6#/-&%\"g G6#%\"xG6$F(%\"yG%!G" }{XPPEDIT 18 0 "Diff(g(x,y),x) = Limit((g(x+h,y) -g(x,y))/h,h = 0);" "6#/-%%DiffG6$-%\"gG6$%\"xG%\"yGF*-%&LimitG6$*&,&- F(6$,&F*\"\"\"%\"hGF4F+F4-F(6$F*F+!\"\"F4F5F8/F5\"\"!" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Simil arly, the " }{TEXT 259 21 "partial derivative of" }{TEXT -1 2 " " } {XPPEDIT 18 0 "z=g(x,y)" "6#/%\"zG-%\"gG6$%\"xG%\"yG" }{TEXT -1 1 " " }{TEXT 259 15 "with respect to" }{TEXT -1 1 " " }{TEXT 270 1 "y" } {TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g [y](x,y)=``" "6#/-&%\"gG6#%\"yG6$%\"xGF(%!G" }{XPPEDIT 18 0 "Diff(g(x, y),y)=Limit((g(x,y+k)-g(x,y))/k,k = 0)" "6#/-%%DiffG6$-%\"gG6$%\"xG%\" yGF+-%&LimitG6$*&,&-F(6$F*,&F+\"\"\"%\"kGF4F4-F(6$F*F+!\"\"F4F5F8/F5\" \"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "g(x,y) = x^2+y^2;" "6#/-%\"gG6$%\"xG%\"yG,&*$F'\"\"#\"\"\"*$F(F+F, " }{TEXT -1 11 ", we have " }{XPPEDIT 18 0 "Diff(g(x,y),x) = 2*x;" "6 #/-%%DiffG6$-%\"gG6$%\"xG%\"yGF**&\"\"#\"\"\"F*F." }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "f := (x, y) -> x^2+y^2:\n'f(x,y)'=f(x,y);\nDiff('f(x,y)',x)=diff(f(x,y),x);\nDi ff('f(x,y)',y)=diff(f(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"fG6$%\"xG%\"yG,&*$)F'\"\"#\"\"\"F-*$)F(F,F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yGF*,$F*\"\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yGF+,$F+\"\"#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Computing partial deriv atives " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT -1 66 "As an example we compute the partial derivatives \+ of the function " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " g(x,y) = x^4*y^2-x*y+3;" "6#/-%\"gG6$%\"xG%\"yG,(*&F'\"\"%F(\"\"#\"\" \"*&F'F-F(F-!\"\"\"\"$F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "at the point" }{XPPEDIT 18 0 "``(x,y)" "6#-%!G6$%\"xG%\"yG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "This can be done by usi ng the limit definition. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "g := (x,y) -> x^4*y^2-x*y+3:\n'g(x ,y)'=g(x,y);\nLimit(('g(x+h,y)-g(x,y)')/h,h=0)=limit((g(x+h,y)-g(x,y)) /h,h=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6$%\"xG%\"yG,(*&)F' \"\"%\"\"\")F(\"\"#F-F-*&F'F-F(F-!\"\"\"\"$F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&,&-%\"gG6$,&%\"xG\"\"\"%\"hGF.%\"yGF.-F*6 $F-F0!\"\"F.F/F3/F/\"\"!*&F0F.,&*&F0F.)F-\"\"$F.\"\"%F.F3F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "g := (x,y) -> x^4*y^2-x*y+3:\n'g(x,y)'=g(x,y);\nLimit(('g(x,y+k)-g(x,y) ')/k,k=0)=limit((g(x,y+k)-g(x,y))/k,k=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6$%\"xG%\"yG,(*&)F'\"\"%\"\"\")F(\"\"#F-F-*&F'F- F(F-!\"\"\"\"$F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&LimitG6$*&,&-% \"gG6$%\"xG,&%\"yG\"\"\"%\"kGF/F/-F*6$F,F.!\"\"F/F0F3/F0\"\"!*&F,F/,&* &F.F/)F,\"\"$F/\"\"#F/F3F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 " The partial derivative of g with respect to " }{TEXT 291 1 "x" }{TEXT -1 5 " at (" }{XPPEDIT 18 0 "x,y" "6$%\"xG%\"yG" }{TEXT -1 4 ") is" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Diff(g(x,y),x) = 4*x ^3*y^2-y;" "6#/-%%DiffG6$-%\"gG6$%\"xG%\"yGF*,&*(\"\"%\"\"\"*$F*\"\"$F /F+\"\"#F/F+!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 48 "and the partial derivative of g with respect to " }{TEXT 292 1 "y" } {TEXT -1 5 " at (" }{XPPEDIT 18 0 "x,y" "6$%\"xG%\"yG" }{TEXT -1 4 ") \+ is" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Diff(g(x,y),y) = 2*x^4*y-x;" "6#/-%%DiffG6$-%\"gG6$%\"xG%\"yGF+,&*(\"\"#\"\"\"*$F*\" \"%F/F+F/F/F*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 51 "The partial derivatives can be computed d irectly. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "D iff" {MPLTEXT 1 0 111 "g := (x,y) -> x^4*y^2-x*y+3:\n'g(x,y)'=g(x,y); \nDiff('g(x,y)',x)=diff(g(x,y),x);\nDiff('g(x,y)',y)=diff(g(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6$%\"xG%\"yG,(*&)F'\"\"%\"\" \")F(\"\"#F-F-*&F'F-F(F-!\"\"\"\"$F-" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%%DiffG6$-%\"gG6$%\"xG%\"yGF*,&*&)F*\"\"$\"\"\")F+\"\"#F0\"\"%F+! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"gG6$%\"xG%\"yGF +,&*&)F*\"\"%\"\"\"F+F0\"\"#F*!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Examples of partial derivatives " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Ex ample 1 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x,y)=x^2 -2*y^2" "6#/-%\"fG6$%\"xG%\"yG,&*$F'\"\"#\"\"\"*&F+F,*$F(F+F,!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "f := (x,y) -> x^2-2*y^2:\n'f(x,y)'=f(x,y);\nDif f('f(x,y)',x)=diff(f(x,y),x);\nDiff('f(x,y)',y)=diff(f(x,y),y);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6$%\"xG%\"yG,&*$)F'\"\"#\"\"\"F -*&F,F-)F(F,F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\" fG6$%\"xG%\"yGF*,$F*\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG 6$-%\"fG6$%\"xG%\"yGF+,$F+!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "f(x,y) = x^2*y;" "6#/-%\"fG6$%\"xG%\"yG*&F'\"\"#F(\"\" \"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 103 "f := (x,y) -> x^2*y:\n'f(x,y)'=f(x,y);\nDif f('f(x,y)',x)=diff(f(x,y),x);\nDiff('f(x,y)',y)=diff(f(x,y),y);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6$%\"xG%\"yG*&)F'\"\"#\"\"\"F(F ," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yGF*,$ *&F*\"\"\"F+F.\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\" fG6$%\"xG%\"yGF+*$)F*\"\"#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "f(x,y) = x/y;" "6#/-%\"fG6$%\"xG%\"yG*&F'\"\"\"F(!\"\" " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 101 "f := (x,y) -> x/y:\n'f(x,y)'=f(x,y);\nDiff('f (x,y)',x)=diff(f(x,y),x);\nDiff('f(x,y)',y)=diff(f(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6$%\"xG%\"yG*&F'\"\"\"F(!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yGF**&\"\" \"F-F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"x G%\"yGF+,$*&F*\"\"\"F+!\"#!\"\"" }}}{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 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "f(x,y) = x*sin(3*y);" "6#/-%\" fG6$%\"xG%\"yG*&F'\"\"\"-%$sinG6#*&\"\"$F*F(F*F*" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "f := (x,y) -> x*sin(3*y):\n'f(x,y)'=f(x,y);\nDiff('f(x,y)',x)=dif f(f(x,y),x);\nDiff('f(x,y)',y)=diff(f(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6$%\"xG%\"yG*&F'\"\"\"-%$sinG6#,$F(\"\"$F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yGF*-%$sinG 6#,$F+\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"x G%\"yGF+,$*&F*\"\"\"-%$cosG6#,$F+\"\"$F.F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "f(x,y) = cos(x*y );" "6#/-%\"fG6$%\"xG%\"yG-%$cosG6#*&F'\"\"\"F(F-" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "f := (x,y) -> cos(x*y):\n'f(x,y)'=f(x,y);\nDiff('f(x,y)',x)=diff( f(x,y),x);\nDiff('f(x,y)',y)=diff(f(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6$%\"xG%\"yG-%$cosG6#*&F'\"\"\"F(F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yGF*,$*&-%$sinG6#*&F *\"\"\"F+F2F2F+F2!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$- %\"fG6$%\"xG%\"yGF+,$*&-%$sinG6#*&F*\"\"\"F+F2F2F*F2!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 6 " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "f(x,y) = exp(x*y)+y*sin(x);" "6#/-%\" fG6$%\"xG%\"yG,&-%$expG6#*&F'\"\"\"F(F.F.*&F(F.-%$sinG6#F'F.F." } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "f := (x,y) -> exp(x*y)+y*sin(x):\n'f(x,y)'=f(x, y);\nDiff('f(x,y)',x)=diff(f(x,y),x);\nDiff('f(x,y)',y)=diff(f(x,y),y) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6$%\"xG%\"yG,&-%$expG6#*&F '\"\"\"F(F.F.*&F(F.-%$sinG6#F'F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%%DiffG6$-%\"fG6$%\"xG%\"yGF*,&*&F+\"\"\"-%$expG6#*&F*F.F+F.F.F.*&F+ F.-%$cosG6#F*F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG 6$%\"xG%\"yGF+,&*&F*\"\"\"-%$expG6#*&F*F.F+F.F.F.-%$sinG6#F*F." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 7 " }}{PARA 0 " " 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "f(x,y) = exp(x^2*y)*cos(2*x); " "6#/-%\"fG6$%\"xG%\"yG*&-%$expG6#*&F'\"\"#F(\"\"\"F/-%$cosG6#*&F.F/F 'F/F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "f := (x,y) -> exp(x^2*y)*cos(2*x):\n'f(x ,y)'=f(x,y);\nDiff('f(x,y)',x)=diff(f(x,y),x);\nDiff('f(x,y)',y)=diff( f(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6$%\"xG%\"yG*&-%$ expG6#*&)F'\"\"#\"\"\"F(F0F0-%$cosG6#,$F'F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yGF*,&**F*\"\"\"F+F.-%$expG6 #*&)F*\"\"#F.F+F.F.-%$cosG6#,$F*F4F.F4*(F4F.F/F.-%$sinGF7F.!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yGF+*()F*\" \"#\"\"\"-%$expG6#*&F-F/F+F/F/-%$cosG6#,$F*F.F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 8 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x,y) = (4*x+y^2)^3;" "6#/-%\"fG6$%\"xG%\"yG*$ ,&*&\"\"%\"\"\"F'F-F-*$F(\"\"#F-\"\"$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "f := (x, y) -> (4*x+y^2)^3:\n'f(x,y)'=f(x,y);\nDiff('f(x,y)',x)=diff(f(x,y),x); \nDiff('f(x,y)',y)=diff(f(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"fG6$%\"xG%\"yG*$),&F'\"\"%*$)F(\"\"#\"\"\"F0\"\"$F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yGF*,$*$),&F*\"\"%*$) F+\"\"#\"\"\"F4F3F4\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$ -%\"fG6$%\"xG%\"yGF+,$*&),&F*\"\"%*$)F+\"\"#\"\"\"F4F3F4F+F4\"\"'" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 9 " }}{PARA 0 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x,y) = arctan(y/x);" "6#/-%\" fG6$%\"xG%\"yG-%'arctanG6#*&F(\"\"\"F'!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "f := (x,y) -> arctan(y/x):\n'f(x,y)'=f(x,y);\nDiff('f(x,y)',x)=diff(f(x ,y),x);\n``=simplify(rhs(%));\nDiff('f(x,y)',y)=diff(f(x,y),y);\n``=si mplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6$%\"xG%\"yG -%'arctanG6#*&F(\"\"\"F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%D iffG6$-%\"fG6$%\"xG%\"yGF*,$*(F+\"\"\"F*!\"#,&F.F.*&F+\"\"#F*F/F.!\"\" F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&%\"yG\"\"\",&*$)%\"xG\" \"#F(F(*$)F'F-F(F(!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG 6$-%\"fG6$%\"xG%\"yGF+*&\"\"\"F-*&F*F-,&F-F-*&F+\"\"#F*!\"#F-F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&%\"xG\"\"\",&*$)F&\"\"#F'F'*$) %\"yGF+F'F'!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Ex ample 10 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x,y) = \+ ln(1+x^2-2*x*y+y^2);" "6#/-%\"fG6$%\"xG%\"yG-%#lnG6#,*\"\"\"F-*$F'\"\" #F-*(F/F-F'F-F(F-!\"\"*$F(F/F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "f := (x,y) \+ -> ln(1+x^2-2*x*y+y^2):\n'f(x,y)'=f(x,y);\nDiff('f(x,y)',x)=diff(f(x,y ),x);\nDiff('f(x,y)',y)=diff(f(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6$%\"xG%\"yG-%#lnG6#,*\"\"\"F-*$)F'\"\"#F-F-*(F0F-F'F-F(F -!\"\"*$)F(F0F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG 6$%\"xG%\"yGF**&,&F*\"\"#*&F.\"\"\"F+F0!\"\"F0,*F0F0*$)F*F.F0F0*(F.F0F *F0F+F0F1*$)F+F.F0F0F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$- %\"fG6$%\"xG%\"yGF+*&,&F*!\"#*&\"\"#\"\"\"F+F1F1F1,*F1F1*$)F*F0F1F1*(F 0F1F*F1F+F1!\"\"*$)F+F0F1F1F6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "The fun ction " }{XPPEDIT 18 0 "f(x,y)=x*y" "6#/-%\"fG6$%\"xG%\"yG*&F'\"\"\"F( F*" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 22 "Consider the function " }{XPPEDIT 18 0 "f (x,y)=x*y" "6#/-%\"fG6$%\"xG%\"yG*&F'\"\"\"F(F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 45 "The contours of f are rectangular hyperbo las " }{XPPEDIT 18 0 "x*y=c" "6#/*&%\"xG\"\"\"%\"yGF&%\"cG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "c<>0" "6#0%\"cG\"\"!" }{TEXT -1 7 ". When \+ " }{XPPEDIT 18 0 "c=0" "6#/%\"cG\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x*y=0" "6#/*&%\"xG\"\"\"%\"yGF&\"\"!" }{TEXT -1 7 " gives " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 9 ", so the " }{TEXT 271 1 "x" }{TEXT -1 5 " and " }{TEXT 272 1 "y" }{TEXT -1 32 " axes constitute the 0 con tour. 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is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot3d(x*y,x=-5..5,y=-5..5,view=[-5..5,-5 ..5,-5..5],axes=boxed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "f(x,y)=x*y" " 6#/-%\"fG6$%\"xG%\"yG*&F'\"\"\"F(F*" }{TEXT -1 27 " the partial deriva tive of " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 17 " with respect to " }{TEXT 273 1 "x" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "Diff(f(x,y),x)=y" "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yGF*F+" }{TEXT -1 32 ", and the partial derivative of " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"f G6$%\"xG%\"yG" }{TEXT -1 17 " with respect to " }{TEXT 274 1 "x" } {TEXT -1 4 " is " }{XPPEDIT 18 0 "Diff(f(x,y),y)=x" "6#/-%%DiffG6$-%\" fG6$%\"xG%\"yGF+F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "f := (x,y) -> x*y:\n'f(x,y) '=f(x,y);\nDiff('f(x,y)',x)=diff(f(x,y),x);\nDiff('f(x,y)',y)=diff(f(x ,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6$%\"xG%\"yG*&F'\"\" \"F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"y GF*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yG F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 " Diff(f(x,y),y)=x" "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yGF+F*" }{TEXT -1 98 " , this partial derivative has the constant value 2 along the \"curve\" of intersection of the plane " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" } {TEXT -1 18 " with the surface " }{XPPEDIT 18 0 "z=x*y" "6#/%\"zG*&%\" xG\"\"\"%\"yGF'" }{TEXT -1 71 ", that is, it is the straight line dete rmined by the pair of equations " }{XPPEDIT 18 0 "PIECEWISE([x=2,``],[ z=2*y,``])" "6#-%*PIECEWISEG6$7$/%\"xG\"\"#%!G7$/%\"zG*&F)\"\"\"%\"yGF /F*" }{TEXT -1 3 ". 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-1 45 " axis, so that, when it has moved a distance " }{TEXT 280 1 "a" }{TEXT -1 19 " (intersecting the " }{TEXT 278 1 "x" }{TEXT -1 9 " axis at " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 33 "), its gradient within the plane " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 14 " is precisely " }{TEXT 279 1 "a" }{TEXT -1 53 ". Note t hat, at this stage, after moving a distance " }{TEXT 281 1 "a" }{TEXT -1 45 ", it is determined by the pair of equations " }{XPPEDIT 18 0 " PIECEWISE([x = a, ``],[z = a*y, ``])" "6#-%*PIECEWISEG6$7$/%\"xG%\"aG% !G7$/%\"zG*&F)\"\"\"%\"yGF/F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 102 "This idea is illustrated in the following animation whic h shows the line of intersection of the plane " }{XPPEDIT 18 0 "x=a" " 6#/%\"xG%\"aG" }{TEXT -1 18 " with the surface " }{XPPEDIT 18 0 "z=x*y " "6#/%\"zG*&%\"xG\"\"\"%\"yGF'" }{TEXT -1 4 " as " }{TEXT 282 1 "a" } {TEXT -1 24 " increases from 0 to 5. " }}{PARA 0 "" 0 "" {TEXT -1 91 " (Click on the graphic and use the controls in the context bar, or use \+ the animation menu.) 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" }} {PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "y = b;" "6 #/%\"yG%\"bG" }{TEXT -1 46 " represents a vertical plane, parallel to \+ the " }{TEXT 284 1 "x" }{TEXT -1 1 "," }{TEXT 283 1 "z" }{TEXT -1 49 " plane, and intersecting the surface in a curve. " }}{PARA 0 "" 0 "" {TEXT -1 23 "The partial derivative " }{XPPEDIT 18 0 "Diff(f(x,y),x)" "6#-%%DiffG6$-%\"fG6$%\"xG%\"yGF)" }{TEXT -1 24 ", evaluated at the po int" }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 64 ", is \+ the gradient of the tangent to this curve of intersection. " }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus " }{XPPEDIT 18 0 "Diff(f(x,y),x)" "6#-%%Di ffG6$-%\"fG6$%\"xG%\"yGF)" }{TEXT -1 1 " " }{TEXT 289 1 "|" }{XPPEDIT 18 0 "``[``(a,b)] = tan(alpha);" "6#/&%!G6#-F%6$%\"aG%\"bG-%$tanG6#%&a lphaG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" } {TEXT -1 46 " is the angle shown in the following picture. 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" }}{PARA 0 "" 0 "" {TEXT -1 23 "The partial derivative " }{XPPEDIT 18 0 "Diff(f(x,y),y);" "6#-%%Di ffG6$-%\"fG6$%\"xG%\"yGF*" }{TEXT -1 24 ", evaluated at the point" } {XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 71 ", is the gr adient of the tangent to this second curve of intersection. " }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus " }{XPPEDIT 18 0 "Diff(f(x,y),y);" "6#-%%D iffG6$-%\"fG6$%\"xG%\"yGF*" }{TEXT -1 1 " " }{TEXT 290 1 "|" } {XPPEDIT 18 0 "``[``(a,b)] = tan(beta);" "6#/&%!G6#-F%6$%\"aG%\"bG-%$t anG6#%%betaG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "beta;" "6#%%betaG " }{TEXT -1 35 " is the angle shown in the picture." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "2nd order partial derivatives " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 34 "Given a function of two variables " }{XPPEDIT 18 0 "phi(x,y)" " 6#-%$phiG6$%\"xG%\"yG" }{TEXT -1 26 ", the partial derivatives " } {XPPEDIT 18 0 "Diff(phi(x,y),x)" "6#-%%DiffG6$-%$phiG6$%\"xG%\"yGF)" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "Diff(phi(x,y),y)" "6#-%%DiffG6$-%$p hiG6$%\"xG%\"yGF*" }{TEXT -1 5 " are " }{TEXT 259 31 "first order part ial derivatives" }{TEXT -1 116 ". It may be possible to find the two p artial derivatives of each of these two functions, leading to the foll owing 4 " }{TEXT 259 32 "second order partial derivatives" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Diff(phi(x, y),`$`(x,2)) = Diff(Diff(phi(x,y),x),x);" "6#/-%%DiffG6$-%$phiG6$%\"xG %\"yG-%\"$G6$F*\"\"#-F%6$-F%6$-F(6$F*F+F*F*" }{XPPEDIT 18 0 "`` = phi[ xx](x,y);" "6#/%!G-&%$phiG6#%#xxG6$%\"xG%\"yG" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Diff(phi(x,y),x,y) = Diff(Diff(phi(x,y),x),y);" "6#/-%%DiffG6%-%$phiG6$%\"xG%\"yGF*F+-F%6$ -F%6$-F(6$F*F+F*F+" }{XPPEDIT 18 0 "`` = phi[xy](x,y);" "6#/%!G-&%$phi G6#%#xyG6$%\"xG%\"yG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Diff(phi(x,y),y,x) = Diff(Diff(phi(x,y),y),x);" "6 #/-%%DiffG6%-%$phiG6$%\"xG%\"yGF+F*-F%6$-F%6$-F(6$F*F+F+F*" }{XPPEDIT 18 0 "`` = phi[yx](x,y);" "6#/%!G-&%$phiG6#%#yxG6$%\"xG%\"yG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Diff(phi( x,y),`$`(y,2)) = Diff(Diff(phi(x,y),y),y);" "6#/-%%DiffG6$-%$phiG6$%\" xG%\"yG-%\"$G6$F+\"\"#-F%6$-F%6$-F(6$F*F+F+F+" }{XPPEDIT 18 0 "``=phi[ yy](x,y)" "6#/%!G-&%$phiG6#%#yyG6$%\"xG%\"yG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 300 55 "Subscript nota tion for higher order partial derivatives" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 112 "The order of differentiation corresponds to read ing the subscripts from left to right, (as in reading English). " }} {PARA 0 "" 0 "" {TEXT -1 11 "To compute " }{XPPEDIT 18 0 "phi[xy](x,y) " "6#-&%$phiG6#%#xyG6$%\"xG%\"yG" }{TEXT -1 22 ", first differentiate \+ " }{XPPEDIT 18 0 "phi(x,y)" "6#-%$phiG6$%\"xG%\"yG" }{TEXT -1 17 " wit h respect to " }{TEXT 296 1 "x" }{TEXT -1 48 ", then differentiate the result with respect to " }{TEXT 295 1 "y" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 1 "\n" }{TEXT 299 70 "Curly-d notation for higher orde r partial derivatives curly-d notation" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 132 "The order of differentiation corresponds to reading the partials in the \"denominator\" right to left, (as in read Hebrew or Arabic). " }}{PARA 0 "" 0 "" {TEXT -1 11 "To compute " }{XPPEDIT 18 0 "Diff(phi(x,y),x,y);" "6#-%%DiffG6%-%$phiG6$%\"xG%\"yGF)F*" } {TEXT -1 30 ", which is the same object as " }{XPPEDIT 18 0 "phi[xy](x ,y)" "6#-&%$phiG6#%#xyG6$%\"xG%\"yG" }{TEXT -1 22 ", first differentia te " }{XPPEDIT 18 0 "phi(x,y)" "6#-%$phiG6$%\"xG%\"yG" }{TEXT -1 17 " \+ with respect to " }{TEXT 298 1 "x" }{TEXT -1 48 ", then differentiate \+ the result with respect to " }{TEXT 297 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 9 "Example 1" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x, y)=x^3*y^2" "6#/-%\"fG6$%\"xG%\"yG*&F'\"\"$F(\"\"#" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "f := (x,y) -> x^3*y^2;\n'f(x,y)'=f(x,y);\nDiff('f(x,y)',x)=diff(f (x,y),x);\nDiff('f(x,y)',x$2)=diff(f(x,y),x$2);\nDiff('f(x,y)',x,y)=di ff(f(x,y),x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"y G6\"6$%)operatorG%&arrowGF)*&)9$\"\"$\"\"\")9%\"\"#F1F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6$%\"xG%\"yG*&)F'\"\"$\"\"\")F(\"\"#F ," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yGF*,$ *&)F*\"\"#\"\"\")F+F/F0\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%Di ffG6$-%\"fG6$%\"xG%\"yG-%\"$G6$F*\"\"#,$*&F*\"\"\")F+F/F2\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6%-%\"fG6$%\"xG%\"yGF*F+,$*&) F*\"\"#\"\"\"F+F0\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "f := (x,y) -> x^3*y^2;\n'f(x,y)'=f (x,y);\nDiff('f(x,y)',y)=diff(f(x,y),y);\nDiff('f(x,y)',y,x)=diff(f(x, y),y,x);\nDiff('f(x,y)',y$2)=diff(f(x,y),y$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&)9$\"\" $\"\"\")9%\"\"#F1F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6$% \"xG%\"yG*&)F'\"\"$\"\"\")F(\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%%DiffG6$-%\"fG6$%\"xG%\"yGF+,$*&)F*\"\"$\"\"\"F+F0\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6%-%\"fG6$%\"xG%\"yGF+F*,$*&)F*\"\" #\"\"\"F+F0\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6 $%\"xG%\"yG-%\"$G6$F+\"\"#,$*$)F*\"\"$\"\"\"F/" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 6 ": The " }{TEXT 261 25 "mixed partial derivatives" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Diff(f(x,y),x,y)" "6#-%%DiffG6%-%\"fG6$%\"xG% \"yGF)F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Diff(f(x,y),y,x)" "6#-%% DiffG6%-%\"fG6$%\"xG%\"yGF*F)" }{TEXT -1 1 " " }{TEXT 259 9 "are equal " }{TEXT -1 100 ", that is, the order of differentiation in computing \+ the mixed partial derivatives does not matter. " }}{PARA 0 "" 0 "" {TEXT -1 55 "In fact this happens for all \"well-behaved\" functions. \+ " }}{PARA 0 "" 0 "" {TEXT -1 119 "More precisely, whenever all the par tial derivatives concerned are continuous functions of the two variabl es, we have " }{XPPEDIT 18 0 "Diff(f(x,y),x,y)=Diff(f(x,y),y,x)" "6#/ -%%DiffG6%-%\"fG6$%\"xG%\"yGF*F+-F%6%-F(6$F*F+F+F*" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 294 9 "Example \+ 2" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " f(x,y) = sin(x*y);" "6#/-%\"fG6$%\"xG%\"yG-%$sinG6#*&F'\"\"\"F(F-" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "f := (x,y) -> sin(x*y);\n'f(x,y)'=f(x,y);\nDiff ('f(x,y)',x)=factor(diff(f(x,y),x));\nDiff('f(x,y)',x$2)=factor(diff(f (x,y),x$2));\nDiff('f(x,y)',x,y)=factor(diff(f(x,y),x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowG F)-%$sinG6#*&9$\"\"\"9%F2F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"fG6$%\"xG%\"yG-%$sinG6#*&F'\"\"\"F(F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yGF**&-%$cosG6#*&F*\"\"\"F+F1F1F+F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yG-%\"$G6 $F*\"\"#,$*&-%$sinG6#*&F*\"\"\"F+F6F6)F+F/F6!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6%-%\"fG6$%\"xG%\"yGF*F+,&*(-%$sinG6#*&F*\"\" \"F+F2F2F*F2F+F2!\"\"-%$cosGF0F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "f := (x,y) -> sin(x*y);\n'f (x,y)'=f(x,y);\nDiff('f(x,y)',y)=factor(diff(f(x,y),y));\nDiff('f(x,y) ',y,x)=factor(diff(f(x,y),y,x));\nDiff('f(x,y)',y$2)=factor(diff(f(x,y ),y$2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$% )operatorG%&arrowGF)-%$sinG6#*&9$\"\"\"9%F2F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6$%\"xG%\"yG-%$sinG6#*&F'\"\"\"F(F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yGF+*&-%$cosG6#*&F* \"\"\"F+F1F1F*F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6%-%\"fG6 $%\"xG%\"yGF+F*,&*(-%$sinG6#*&F*\"\"\"F+F2F2F*F2F+F2!\"\"-%$cosGF0F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6$%\"xG%\"yG-%\"$G6 $F+\"\"#,$*&-%$sinG6#*&F*\"\"\"F+F6F6)F*F/F6!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Again, the mixed partial \+ derivatives are equal." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 303 9 "Example 3" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x,y)=x^4*y^2-x*y+3" "6#/-%\"gG6$%\"xG%\"yG, (*&F'\"\"%F(\"\"#\"\"\"*&F'F-F(F-!\"\"\"\"$F-" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "If we wis h to obtain the partial derivatives as functions, then the " }{TEXT 0 1 "D" }{TEXT -1 22 " operator can be used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "g := (x,y) -> x^4* y^2-x*y+3;\ng_x := D[1](g);\ng_y := D[2](g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),(*&)9$\" \"%\"\"\")9%\"\"#F2F2*&F0F2F4F2!\"\"\"\"$F2F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$g_xGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&)9$ \"\"$\"\"\")9%\"\"#F2\"\"%F4!\"\"F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$g_yGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&)9$\"\"%\"\" \"9%F2\"\"#F0!\"\"F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 53 "The second order partial derivatives with respect to " }{TEXT 301 1 "x" }{TEXT -1 5 " and " }{TEXT 302 1 "y" }{TEXT -1 29 " can be obtained as follows: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 2 ": " }{TEXT 262 13 "D[i ](D[j](g))" }{TEXT -1 3 " = " }{TEXT 262 9 "D[i,j](g)" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "g_xx := D[1,1](g);\ng_yy := D[2,2](g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%g_xxGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*&)9$ \"\"#\"\"\")9%F1F2\"#7F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%g_y yGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*$)9$\"\"%\"\"\"\"\"#F)F)F )" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "As \+ before, " }{TEXT 259 56 "the two mixed second order partial derivative s are equal" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "g_yx := D[1,2](g);\ng_xy := \+ D[2,1](g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%g_yxGf*6$%\"xG%\"yG6 \"6$%)operatorG%&arrowGF),&*&)9$\"\"$\"\"\"9%F2\"\")F2!\"\"F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%g_xyGf*6$%\"xG%\"yG6\"6$%)operatorG %&arrowGF),&*&)9$\"\"$\"\"\"9%F2\"\")F2!\"\"F)F)F)" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Ta sks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 5 "Find " } {XPPEDIT 18 0 "Diff(g(x,y),x)" "6#-%%DiffG6$-%\"gG6$%\"xG%\"yGF)" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "Diff(g(x,y),y)" "6#-%%DiffG6$-%\"gG 6$%\"xG%\"yGF*" }{TEXT -1 38 " for each of the following functions. " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "g(x,y)=x^2+2*x*y^ 2-y^3" "6#/-%\"gG6$%\"xG%\"yG,(*$F'\"\"#\"\"\"*(F+F,F'F,F(F+F,*$F(\"\" $!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "g(x,y)=x^3*y-3*x^2*y^2+x*y^3" "6#/-%\"gG6$%\"xG%\"yG,(*&F'\"\"$F (\"\"\"F,*(F+F,*$F'\"\"#F,F(F/!\"\"*&F'F,*$F(F+F,F," }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 5 "(c) " }{XPPEDIT 18 0 "g(x,y)=exp(x)*sin( y)" "6#/-%\"gG6$%\"xG%\"yG*&-%$expG6#F'\"\"\"-%$sinG6#F(F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(d) " }{XPPEDIT 18 0 "g(x,y)=x/(1 +x^2-y^2)" "6#/-%\"gG6$%\"xG%\"yG*&F'\"\"\",(F*F**$F'\"\"#F**$F(F-!\" \"F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(e) " }{XPPEDIT 18 0 "g(x,y)= exp(x*ln(y))" "6#/-%\"gG6$%\"xG%\"yG-%$expG6#*&F'\"\"\"- %#lnG6#F(F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(f) " } {XPPEDIT 18 0 "g(x,y)=(x^2+y^2)*exp(-x*y)" "6#/-%\"gG6$%\"xG%\"yG*&,&* $F'\"\"#\"\"\"*$F(F,F-F--%$expG6#,$*&F'F-F(F-!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "____________ ____________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "g(x,y)=8/3*x^3+4*y^3-x^4-y^4" "6#/-%\"gG6$%\" xG%\"yG,**(\"\")\"\"\"\"\"$!\"\"F'F-F,*&\"\"%F,*$F(F-F,F,*$F'F0F.*$F(F 0F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "(a) Find " } {XPPEDIT 18 0 "Diff(g(x,y),x)" "6#-%%DiffG6$-%\"gG6$%\"xG%\"yGF)" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "Diff(g(x,y),y)" "6#-%%DiffG6$-%\"gG 6$%\"xG%\"yGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "(b) Fi nd all the points " }{XPPEDIT 18 0 "``(x,y)" "6#-%!G6$%\"xG%\"yG" } {TEXT -1 11 " for which " }{XPPEDIT 18 0 "Diff(g(x,y),x)=0" "6#/-%%Dif fG6$-%\"gG6$%\"xG%\"yGF*\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Dif f(g(x,y),y)=0" "6#/-%%DiffG6$-%\"gG6$%\"xG%\"yGF+\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 32 "________________________________" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 \+ " }}{PARA 0 "" 0 "" {TEXT -1 12 "Check that " }{XPPEDIT 18 0 "Diff(g( x,y),x,y) = Diff(g(x,y),y,x);" "6#/-%%DiffG6%-%\"gG6$%\"xG%\"yGF*F+-F% 6%-F(6$F*F+F+F*" }{TEXT -1 31 " for the following functions. " }} {PARA 0 "" 0 "" {TEXT -1 14 " (a) " }{XPPEDIT 18 0 "f(x,y) = \+ y^5-10*x^2*y^3+5*x^4*y;" "6#/-%\"fG6$%\"xG%\"yG,(*$F(\"\"&\"\"\"*(\"#5 F,*$F'\"\"#F,F(\"\"$!\"\"*(F+F,*$F'\"\"%F,F(F,F," }{TEXT -1 15 " \+ (b) " }{XPPEDIT 18 0 "f(x,y)=x/y^2" "6#/-%\"fG6$%\"xG%\"yG*&F'\" \"\"*$F(\"\"#!\"\"" }{TEXT -1 15 " (c) " }{XPPEDIT 18 0 "f(x ,y) = x*sin(x-y);" "6#/-%\"fG6$%\"xG%\"yG*&F'\"\"\"-%$sinG6#,&F'F*F(! \"\"F*" }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 14 " (d ) " }{XPPEDIT 18 0 "f(x,y) = exp(y)*ln(x+y);" "6#/-%\"fG6$%\"xG%\"yG* &-%$expG6#F(\"\"\"-%#lnG6#,&F'F-F(F-F-" }{TEXT -1 30 " \+ (e) " }{XPPEDIT 18 0 "f(x,y) = x*y/(x^2+y^2);" "6#/-%\"fG6$% \"xG%\"yG*(F'\"\"\"F(F*,&*$F'\"\"#F**$F(F-F*!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 36 "A steady state tempe rature function " }{XPPEDIT 18 0 "u=u(x,y)" "6#/%\"uG-F$6$%\"xG%\"yG" }{TEXT -1 33 " for a thin flat plate satisfies " }{TEXT 262 17 "Laplac es equation" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Diff(u,x$2)+Diff(u,y$2)=0" "6#/,&-%%DiffG6$%\"uG-%\"$G6 $%\"xG\"\"#\"\"\"-F&6$F(-F*6$%\"yGF-F.\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "Determine which of the following functions sati sfy Laplace's equation: " }}{PARA 0 "" 0 "" {TEXT -1 11 " (a) " }{XPPEDIT 18 0 "u(x,y) = y^4-6*y^2*x^2+x^4" "6#/-%\"uG6$%\"xG%\"yG,(*$ F(\"\"%\"\"\"*(\"\"'F,*$F(\"\"#F,F'F0!\"\"*$F'F+F," }{TEXT -1 8 " ( b) " }{XPPEDIT 18 0 "u(x,y) = sin*2*x*cos*2*y;" "6#/-%\"uG6$%\"xG%\"yG *.%$sinG\"\"\"\"\"#F+F'F+%$cosGF+F,F+F(F+" }{TEXT -1 8 " (c) " } {XPPEDIT 18 0 "u(x,y) = exp(-x)*sin*y;" "6#/-%\"uG6$%\"xG%\"yG*(-%$exp G6#,$F'!\"\"\"\"\"%$sinGF/F(F/" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " (d) " }{XPPEDIT 18 0 "u(x,y)=ln(sqrt(x^2+y^2))" "6#/-%\"uG6$%\"xG%\"yG-%#lnG6#-%%sqrtG 6#,&*$F'\"\"#\"\"\"*$F(F1F2" }{TEXT -1 10 " (e) " }{XPPEDIT 18 0 "u(x,y)=arctan(y/x)" "6#/-%\"uG6$%\"xG%\"yG-%'arctanG6#*&F(\"\"\"F'!\" \"" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "_________________ _______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "____________________ ____________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {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 31 "1st pa rtial derivative picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 864 "p1 := plot(x^2,x=-1.9..1.9,color=C OLOR(RGB,.4,.5,.9)):\np2 := plot([[1.5*cos(t),2.25+.3*sin(t),t=0..2*Pi ],\n [cos(t),1+.3*sin(t),t=0..2*Pi]],linestyle=2,color=COLOR(RGB,.4, .5,.9)):\np3 := plot(30*(x-1)^2+1,x=0.7..1.3,color=red):\np4 := plotto ols[arrow]([-.3,0],[1.9,0],0,.08,.04,arrow,color=black):\np5 := plotto ols[arrow]([0,-.3],[0,4],0,.08,.03,arrow,color=black):\np6 := plottool s[arrow]([.1,.1],[-1.5,-1.5],0,.08,.04,arrow,color=black):\nt1 := plot s[textplot]([[1.9,-.2,`y`],[-.15,4,`z`],[-1.4,-1.56,`x`]],color=black) :\nt2 := plots[textplot]([-1.5,1,`z = x + y`],color=blue):\nt3 := plo ts[textplot]([-1.32,1.13,`2 2`],font=[HELVETICA,6],color=blu e):\nt4 := plots[textplot]([[1.6,1,`z = x + 1`],[1.5,1.35,`y = 1`]],c olor=red):\nt5 := plots[textplot]([1.64,1.13,2],font=[HELVETICA,6],col or=red):\nplots[display]([p1,p2,p3,p4,p5,p6,t1,t2,t3,t4,t5],axes=none) ;\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 16 "saddle pictures " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 842 "p1 := plot([seq([t,i/t,t=i/ 5..5],i=1..4),\n seq([t,i/t,t=-5..-i/5],i=1..4)],color=red,\n \+ labels=[``,``],discont=true):\np2 := plot([seq([t,-i/t,t=i/5..5],i=1. .4),\n seq([t,-i/t,t=-5..-i/5],i=1..4)],color=blue,\n labels= [``,``],discont=true):\np3 := plot([[[-5,0],[5,0]],[[0,-5],[0,5]]],col or=COLOR(RGB,.3,.3,.5)):\nt1 := plots[textplot]([[5.5,-.2,`x`],[-.2,5. 5,`y`]]):\nt2 := plots[textplot]([[.74,.6,`x y = 1`],[3,1.9,`x y = 4`] ,\n [-.74,-.6,`x y = 1`],[-3,-1.9,`x y = 4`]],color=COLOR(RGB,.8,0 ,0)):\nt3 := plots[textplot]([[-.74,.6,`x y = -1`],[-3,1.9,`x y = -4`] ,\n [.74,-.6,`x y = -1`],[3,-1.9,`x y = -4`]],color=COLOR(RGB,0,0, .8)):\nt4 := plots[textplot]([2.3,.2,`x y = 0`],color=COLOR(RGB,.1,.1, .3)):\nplots[display]([p1,p2,p3,t1,t2,t3,t4],thickness=2,\n \+ view=[-5.5..5.5,-5.5..5.5],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 554 "p 1 := seq(plots[spacecurve]([t,i/t,i,t=1/5..5],color=red),i=1..4):\np2 \+ := seq(plots[spacecurve]([t,i/t,i,t=-5..-1/5],color=red),i=1..4):\np3 \+ := seq(plots[spacecurve]([t,-i/t,-i,t=1/5..5],color=blue),i=1..4):\np4 := seq(plots[spacecurve]([t,-i/t,-i,t=-5..-1/5],color=blue),i=1..4): \np5 := plots[pointplot3d]([[-5,0,0],[5,0,0]],style=line,color=COLOR(R GB,.3,.3,.5)):\np6 := plots[pointplot3d]([[0,-5,0],[0,5,0]],style=line ,color=COLOR(RGB,.3,.3,.5)):\nplots[display](\{p1,p2,p3,p4,p5,p6\},axe s=boxed,labels=[`x`,`y`,`z`],\n view=[-5..5,-5..5,-5..5],thickness= 2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 351 "p1 := plot3d([x,y,x*y],x=0..5,y=0..5,color=COLOR(RGB ,1,.5,.5)):\np2 := plot3d([x,y,x*y],x=-5..0,y=-5..0,color=COLOR(RGB,1, .5,.5)):\np3 := plot3d([x,y,x*y],x=0..5,y=-5..0,color=COLOR(RGB,.5,.5, 1)):\np4 := plot3d([x,y,x*y],x=-5..0,y=0..5,color=COLOR(RGB,.5,.5,1)): \nplots[display]([p1,p2,p3,p4],view=[-5..5,-5..5,-5..5],axes=boxed,\n \+ labels=[`x`,`y`,`z`]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 405 "p1 := plot3d([x,y,x*y],x=0..5,y=0. .5,color=COLOR(RGB,1,.5,.5)):\np2 := plot3d([x,y,x*y],x=-5..0,y=-5..0, color=COLOR(RGB,1,.5,.5)):\np3 := plot3d([x,y,x*y],x=0..5,y=-5..0,colo r=COLOR(RGB,.5,.5,1)):\np4 := plot3d([x,y,x*y],x=-5..0,y=0..5,color=CO LOR(RGB,.5,.5,1)):\np5 := plot3d([2,y,z],y=-5..5,z=-5..5,color=grey): \nplots[display]([p1,p2,p3,p4,p5],view=[-5..5,-5..5,-5..5],axes=boxed, \n labels=[`x`,`y`,`z`]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 534 "p1 := plot3d([x,y,x*y],x=0. .5,y=0..5,color=COLOR(RGB,1,.5,.5)):\np2 := plot3d([x,y,x*y],x=-5..0,y =-5..0,color=COLOR(RGB,1,.5,.5)):\np3 := plot3d([x,y,x*y],x=0..5,y=-5. .0,color=COLOR(RGB,.5,.5,1)):\np4 := plot3d([x,y,x*y],x=-5..0,y=0..5,c olor=COLOR(RGB,.5,.5,1)):\nfrms := NULL:\n for i from 0 to 10 do\n \+ p5 := plot3d([i/2,y,z],y=-5..5,z=-5..5,color=grey):\n frms := frms, plots[display]([p1,p2,p3,p4,p5],view=[-5..5,-5..5,-5..5],axes=boxed); \nend do:\nplots[display]([frms],insequence=true,orientation=[-26,54], \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 63 "picture for geometrical interpretation of partial de rivatives " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1883 "#p1 := plot([[[0,0],[4,1]],[[0,0],[4,1]]],color=b lack):\np1 := plottools[arrow]([0,0],[0,1.5],0,.1,.05,arrow,color=blac k):\np2 := plottools[arrow]([0,0],[5,1.25],0,.1,.02,arrow,color=black) :\np3 := plottools[arrow]([0,0],[-5,1.25],0,.1,.02,arrow,color=black): \np4 := plot(interp([0,2,4],[1,3/2,5/2],x),x=0..4,color=black):\np5 := plot(interp([0,-2,-4],[1,2,7/2],x),x=-4..0,color=black):\nq6 := plot( interp([-4,-3/2,1],[7/2,9/2,6],x),x=-4..1):\nq7 := plot(interp([1,5/2, 4],[6,4,5/2],x),x=1..4):\np6 := plot(interp([-2,1/4,5/2],[2,8/3,4],x), x=-2..5/2,color=red):\np7 := plot(interp([-3/2,1/4,2],[9/2,8/3,3/2],x) ,x=-3/2..2,color=red):\nrvrs := (_u,_v)->evalb(_u[1]>_v[1]):\npts := [ op(sort(op(1,op(1,p5)),rvrs)),op(op(1,op(1,q6))),op(op(1,op(1,q7))),\n op(sort(op(1,op(1,p4)),rvrs))]:\np8 := plots[polygonplot](pts, color=COLOR(RGB,.94,.94,.94),style=patchnogrid):\np9 := plot([[[1/4,8/ 3],[9/4,32/9]],[[1/4,8/3],[9/4,19/6]],\n [[-5/4,73/24],[1/4,8/3] ],[[-1,157/42],[1/4,8/3]]],\n linestyle=2,color=COLOR(RGB, .7,0,1)):\np10 := plot([1/4+13/10*cos(t),8/3+13/10*sin(t),t=Pi-arctan( 1/4)..Pi-arctan(6/7)],\n numpoints=10,adaptive=false,linestyle =4,color=COLOR(RGB,.7,0,1)):\np11 := plot([1/4+2*cos(t),8/3+2*sin(t),t =arctan(1/4)..arctan(4/9)],\n numpoints=10,adaptive=false,line style=3,color=COLOR(RGB,.7,0,1)):\np12 := plot([[[2,1/2],[2,3/2]],[[-2 ,1/2],[-2,2]]],color=red,linestyle=2):\np13 := plot([[[4,1],[4,5/2]],[ [-4,1],[-4,7/2]]],color=grey,linestyle=2):\nt1 := plots[textplot]([[1. 92,3.27,`a`],[-.71,3.25,`b`]],\n font=[SYMBOL,10],color=CO LOR(RGB,.5,0,.9)):\nt2 := plots[textplot]([[5,1.1,`x`],[-5,1.1,`y`],[. 22,1.4,`z`]],color=black):\nt3 := plots[textplot]([[2.1,.3,`x = a`],[- 2.1,.3,`y = b`]],\n color=COLOR(RGB,.95,0,0)):\n plots[display]([p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,\n p11,p12, p13,t1,t2,t3],axes=none,view=[-5..5,0..6]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{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 }