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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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 74 "The gradient of functions of 2 an d 3 variables - directional derivatives " }}{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 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "The gradi ent of a function of two variables " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 27 "The gradient of a f unction " }{XPPEDIT 18 0 "g(x, y);" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 32 ", is the vector valued function " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "grad(g)(x,z) = Diff(g(x,y),x);" "6#/--%%gradG6#%\"g G6$%\"xG%\"zG-%%DiffG6$-F(6$F*%\"yGF*" }{TEXT -1 1 " " }{TEXT 415 1 "i " }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Diff(g(x, y),y);" "6#-%%DiffG6$-% \"gG6$%\"xG%\"yGF*" }{TEXT -1 1 " " }{TEXT 416 1 "j" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "grad(g)(x,y) = g[x](x, y);" "6#/--%%gradG6#%\"gG6$%\"xG %\"yG-&F(6#F*6$F*F+" }{TEXT -1 1 " " }{TEXT 417 1 "i" }{TEXT -1 2 " + " }{XPPEDIT 18 0 "g[y](x, y);" "6#-&%\"gG6#%\"yG6$%\"xGF'" }{TEXT -1 1 " " }{TEXT 420 1 "j" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "w here " }{TEXT 418 1 "i" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([[1], [0]]) = [1, 0];" "6#/-%'matrixG6#7$7#\"\"\"7#\"\"!7$F)F+" }{TEXT -1 6 " and " }{TEXT 419 1 "j" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([ [0], [1]]) = [0, 1];" "6#/-%'matrixG6#7$7#\"\"!7#\"\"\"7$F)F+" }{TEXT -1 55 " are the unit vectors in the coordinate directions, or " } {TEXT 259 22 "standard basis vectors" }{TEXT -1 19 ", in 2 dimensions. " }}{PARA 0 "" 0 "" {TEXT -1 52 "Again we assume that the partial der ivatives exist. " }}{PARA 0 "" 0 "" {TEXT -1 11 "Evaluating " } {XPPEDIT 18 0 "grad(g)" "6#-%%gradG6#%\"gG" }{TEXT -1 12 " at a point \+ " }{XPPEDIT 18 0 "``(x[0],y[0])" "6#-%!G6$&%\"xG6#\"\"!&%\"yG6#F)" } {TEXT -1 84 " gives a 2 - dimensional vector, which we may picture as \+ originating from the point " }{XPPEDIT 18 0 "``(x[0],y[0])" "6#-%!G6$& %\"xG6#\"\"!&%\"yG6#F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 "The resulting picture is a " }{TEXT 259 14 "gradient field" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 8 "gradplot" }{TEXT -1 8 " in the \+ " }{TEXT 0 5 "plots" }{TEXT -1 60 " package can be used to illustrate \+ a gradient vector field. " }}{PARA 0 "" 0 "" {TEXT -1 84 "The length o f the arrows is proportional to the magnitude of the associated vector s." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "with(plots):\ng := (x,y) -> x*y:\n'g(x,y)'=g(x,y);\n gradplot(g(x,y),x=-3..3,y=-3..3,arrows=slim,\n color=COLOR(RGB ,.8,0,1),grid=[30,30]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6$% \"xG%\"yG*&F'\"\"\"F(F*" }}{PARA 13 "" 1 "" {GLPLOT2D 430 354 354 {PLOTDATA 2 "6$-%'CURVESG6acn7'7$$!3eFF*$!3`aNpzd* =$HF*7$$!3_W?kDCDmGF*$!3[XkI?U5oIF*7$$!3Ks')QWSg^GF*$!3#Gu>N)pK?IF*7$$ !3\")\\aBM>bz08DF*$!3.\"H`n`Sp$HF*7$$ !3i1z\"R(eNfEF*$!3'*3nCj%fI1$F*7$$!3_+'*R.jWVEF*$!3uK[MqmYh#F*$!3%R#z-f6/aIF*FO7'7$$!3)QShuOhhI#F*$!3)z-8QH&)>%HF*7$$!3soP >A$fCX#F*$!3-sp=1Z,eIF*7$$!3FG0Ti&)GNCF*$!3?A*prN1Y,$F*7$$!3EUqJ47G1CF *$!3U8I&e%3=^IF*F^o7'7$$!3'fEPd\"[E*4#F*$!3[kF(30Iq%HF*7$$!3#3jp/xibC# F*$!3_Ns7\\*pH0$F*7$$!3Yc9U@38FAF*$!3c7]*R/Y<,$F*7$$!3rQy&o%ek+AF*$!3K 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unapply(convert(evalm(1/nm*gd),list),x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gdG-%'vectorG6#7$%\"yG%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nmG*$-%%sqrtG6#,&*$)-%$absG6#%\"yG\"\"#\"\"\"F1*$)-F -6#%\"xGF0F1F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6$%\"xG%\" yG6\"6$%)operatorG%&arrowGF)7$*&,&*$)-%$absG6#9%\"\"#\"\"\"F7*$)-F36#9 $F6F7F7#!\"\"F6F5F7*&F/F=F " 0 "" {MPLTEXT 1 0 334 "g := (x,y) -> x*y:\n'g(x,y) '=g(x,y);\np1:=contourplot(g(x,y),x=-3..3,y=-3..3,grid=[35,35],colorin g=[red,yellow],\n contours=[-7,-6,-5,-4,-3,-2,-1.5,-1,-.5,-.15,\n \+ .15,.5,1,1.5,2,3,4,5,6,7]):\np2:=fieldplot( h(x,y),x=-3..3,y=-3..3,arrows=slim,\n color=COLOR(RGB,.8,0,1) ,grid=[30,30]):\ndisplay([p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"gG6$%\"xG%\"yG*&F'\"\"\"F(F*" }}{PARA 13 "" 1 "" {GLPLOT2D 550 401 401 {PLOTDATA 2 "68-%'CURVESG657$7$$!321Zw6%HN#G!#<$!3_mmmmm;zCF*7 $$!3fKLLLLLLGF*$!3?=THN#)eqCF*7$F-7$$!\"$\"\"!$!3[LLLLLLLBF*7$F'7$$!3A 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XwzYZl`GF*Fecjn7'7$FjbclFidel7$FgbclF\\eel7$FfeelFdeel7$FaeelF_eelFbdj n7'7$FicclF_ggl7$FdcclFbggl7$F\\hglFjggl7$FggglFegglFgdjn7'7$F^eclFahi l7$FidclFdhil7$F^iilF\\iil7$FihilFghilF\\ejn7'7$FcfclFch[m7$F^fclFfh[m 7$F`i[mF^i[m7$F[i[mFih[mFaejn7'7$FhgclFfg]m7$FcgclFig]m7$Fch]mFah]m7$F ^h]mF\\h]mFfejn7'7$F]iclFje_m7$FhhclF]f_m7$Fgf_mFef_m7$Fbf_mF`f_mF[fjn 7'7$FbjclFdcam7$F]jclFgcam7$FadamF_dam7$F\\damFjcamF`fjn7'7$Fg[dlFj_cm 7$Fb[dlF]`cm7$Fg`cmFe`cm7$Fb`cmF``cmFefjn7'7$F\\]dlFd[em7$Fg\\dlFg[em7 $Fa\\emF_\\em7$F\\\\emFj[emFjfjn7'7$Fa^dlF^ffm7$F\\^dlFaffm7$F[gfmFiff m7$FfffmFdffmF_gjn7'7$Ff_dlF\\`hm7$Fa_dlF_`hm7$Fi`hmFg`hm7$Fd`hmFb`hmF dgjn7'7$F[adlF[iim7$Ff`dlF^iim7$FhiimFfiim7$FciimFaiimFigjn7'7$F`bdlF] a[n7$F[bdlF`a[n7$Fja[nFha[n7$Fea[nFca[nF^hjn7'7$FecdlFih\\n7$F`cdlF\\i \\n7$Ffi\\nFdi\\n7$Fai\\nF_i\\nFchjn7'7$F]`^nF[`^n7$Fb`^nF``^n7$F\\a^n Fj`^n7$Fg`^nFe`^nFhhjn7'7$Fgc_nF``^n7$F[fdlF[`^n7$Fbd_nF`d_n7$F]d_nF[d _nF]ijn7'7$F_gdlF\\i\\n7$F\\gdlFih\\n7$Fdf`nFbf`n7$F_f`nF]f`nFbijn7'7$ F`hdlF`a[n7$F]hdlF]a[n7$F\\ganFjfan7$FgfanFefanFgijn7'7$FaidlF^iim7$F^ idlF[iim7$FhfbnFffbn7$FcfbnFafbnF\\jjn7'7$F_ecnF_`hm7$F_jdlF\\`hm7$Fje cnFhecn7$FeecnFcecnFajjn7'7$Fc[elFaffm7$F`[elF^ffm7$F`ddnF^ddn7$F[ddnF icdnFfjjn7'7$Fd\\elFg[em7$Fa\\elFd[em7$FjaenFhaen7$FeaenFcaenF[[[o7'7$ Fe]elF]`cm7$Fb]elFj_cm7$F\\_fnFj^fn7$Fg^fnFe^fnF`[[o7'7$Ff^elFgcam7$Fc ^elFdcam7$Ff[gnFd[gn7$Fa[gnF_[gnFe[[o7'7$Fg_elF]f_m7$F`ggnFje_m7$Fjggn Fhggn7$FeggnFcggnFj[[o7'7$Fh`elFig]m7$Fe`elFfg]m7$F\\dhnFjchn7$FgchnFe chnF_\\[o7'7$FiaelFfh[m7$FfaelFch[m7$F^_inF\\_in7$Fi^inFg^inFd\\[o7'7$ FjbelFdhil7$FgbelFahil7$FhiinFfiin7$FciinFaiinFi\\[o7'7$F[delFbggl7$Fh celF_ggl7$F^djnF\\djn7$FicjnFgcjnF^][o7'7$F\\eelF\\eel7$FidelFidel7$$ \"3qjgP&yh(fIF*$\"3b+Sp'H(=BIF*7$Fg][oFe][oFc][o-%&COLORG6&F_q$\"\")F_ ^mFfqF`[_l-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curv e 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16 " "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "The gradient of a function of th ree varables " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 27 "The gradient of a function " }{XPPEDIT 18 0 "g( x,y,z)" "6#-%\"gG6%%\"xG%\"yG%\"zG" }{TEXT -1 32 ", is the vector valu ed function " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "grad( g)(x, y, z) = Diff(g(x,y,z),x);" "6#/--%%gradG6#%\"gG6%%\"xG%\"yG%\"zG -%%DiffG6$-F(6%F*F+F,F*" }{TEXT -1 1 " " }{TEXT 263 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "Diff(g(x,y,z),y)" "6#-%%DiffG6$-%\"gG6%%\"xG%\"y G%\"zGF*" }{TEXT -1 1 " " }{TEXT 264 1 "j" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "Diff(g(x,y,z),z)" "6#-%%DiffG6$-%\"gG6%%\"xG%\"yG%\"zGF +" }{TEXT -1 1 " " }{TEXT 265 1 "k" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " grad(g)(x,y,z) = g[x](x,y,z);" "6#/--%%gradG6#%\"gG6%%\"xG%\"yG%\"zG-& F(6#F*6%F*F+F," }{TEXT -1 1 " " }{TEXT 266 1 "i" }{TEXT -1 2 " +" } {XPPEDIT 18 0 "g[y](x,y,z)" "6#-&%\"gG6#%\"yG6%%\"xGF'%\"zG" }{TEXT -1 1 " " }{TEXT 279 1 "j" }{TEXT -1 2 " +" }{XPPEDIT 18 0 "g[z](x,y,z) " "6#-&%\"gG6#%\"zG6%%\"xG%\"yGF'" }{TEXT -1 1 " " }{TEXT 267 1 "k" } {TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 268 1 "i " }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([[1],[0],[0]]) = [1,0,0]" " 6#/-%'matrixG6#7%7#\"\"\"7#\"\"!7#F+7%F)F+F+" }{TEXT -1 3 ", " } {TEXT 269 1 "j" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([[0],[1],[0]] ) = [0,1,0]" "6#/-%'matrixG6#7%7#\"\"!7#\"\"\"7#F)7%F)F+F)" }{TEXT -1 5 " and " }{TEXT 270 1 "k" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([[ 0],[0],[1]]) = [0,0,1]" "6#/-%'matrixG6#7%7#\"\"!7#F)7#\"\"\"7%F)F)F, " }{TEXT -1 55 " are the unit vectors in the coordinate directions, or " }{TEXT 259 22 "standard basis vectors" }{TEXT -1 19 ", in 3 dimensi ons. " }}{PARA 0 "" 0 "" {TEXT -1 51 "We assume here that the partial \+ derivatives exist. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT 274 9 "Example I" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "g(x,y,z) = x^2+y^3+z-x*y^2*z^3;" "6#/-%\"gG6% %\"xG%\"yG%\"zG,**$F'\"\"#\"\"\"*$F(\"\"$F-F)F-*(F'F-*$F(F,F-F)F/!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " grad(g)(x,y,z) = ``(2*x-y^2*z^3) " "6#/--%%gradG6#%\"gG6%%\"xG%\"yG%\"zG-%!G6#,&*&\"\"#\"\"\"F*F3F3*&F+ F2F,\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 271 1 "i" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "``(3*y^2-2*x*y*z^3)" "6#-%!G6#,&*&\"\"$\"\"\"*$%\"yG\" \"#F)F)**F,F)%\"xGF)F+F)%\"zGF(!\"\"" }{TEXT -1 1 " " }{TEXT 272 1 "j " }{TEXT -1 3 " + " }{XPPEDIT 18 0 "``(1-3*x*y^2*z^2)" "6#-%!G6#,&\"\" \"F'**\"\"$F'%\"xGF'%\"yG\"\"#%\"zGF,!\"\"" }{TEXT -1 1 " " }{TEXT 273 1 "k" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``= [2*x-y^2*z^3, 3*y^2-2*x*y*z^3, 1-3*x*y^2*z^2]" "6#/ %!G7%,&*&\"\"#\"\"\"%\"xGF)F)*&%\"yGF(%\"zG\"\"$!\"\",&*&F.F)*$F,F(F)F )**F(F)F*F)F,F)F-F.F/,&F)F)**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 68 "with(linalg):\ng := x -> x^2+y^3+z-x*y^2*z^3;\ngrad(g(x,y,z),[x,y, z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operator G%&arrowGF(,**$)9$\"\"#\"\"\"F1*$)%\"yG\"\"$F1F1%\"zGF1*(F/F1)F4F0F1)F 6F5F1!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,&% \"xG\"\"#*&)%\"yGF)\"\"\")%\"zG\"\"$F-!\"\",&*$F+F-F0**F)F-F(F-F,F-F.F -F1,&F-F-**F0F-F(F-F+F-)F/F)F-F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 278 10 "Example II" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "g(x,y,z) = x^3*y/z+sin(x*y*z);" "6#/-%\"gG6%%\"xG%\"yG%\"zG,&*(F '\"\"$F(\"\"\"F)!\"\"F--%$sinG6#*(F'F-F(F-F)F-F-" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "grad(g)(x,y,z) = ``(3*x^2*y/z+y*z*cos(x*y*z));" "6#/-- %%gradG6#%\"gG6%%\"xG%\"yG%\"zG-%!G6#,&**\"\"$\"\"\"*$F*\"\"#F3F+F3F,! \"\"F3*(F+F3F,F3-%$cosG6#*(F*F3F+F3F,F3F3F3" }{TEXT -1 1 " " }{TEXT 275 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "``(x^3/z+x*z*cos(x*y*z))" "6#-%!G6#,&*&%\"xG\"\"$%\"zG!\"\"\"\"\"*(F(F,F*F,-%$cosG6#*(F(F,%\"yGF ,F*F,F,F," }{TEXT -1 1 " " }{TEXT 276 1 "j" }{TEXT -1 2 " +" } {XPPEDIT 18 0 "``(-x^3*y/(z^2)+x*y*cos(x*y*z))" "6#-%!G6#,&*(%\"xG\"\" $%\"yG\"\"\"*$%\"zG\"\"#!\"\"F/*(F(F+F*F+-%$cosG6#*(F(F+F*F+F-F+F+F+" }{TEXT -1 1 " " }{TEXT 277 1 "k" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= [ 3*x^2*y/z+y*z*cos(x*y*z),x^3/z+x* z*cos(x*y*z),-x^3*y/(z^2)+x*y*cos(x*y*z)]" "6#/%!G7%,&**\"\"$\"\"\"*$% \"xG\"\"#F)%\"yGF)%\"zG!\"\"F)*(F-F)F.F)-%$cosG6#*(F+F)F-F)F.F)F)F),&* &F+F(F.F/F)*(F+F)F.F)-F26#*(F+F)F-F)F.F)F)F),&*(F+F(F-F)*$F.F,F/F/*(F+ F)F-F)-F26#*(F+F)F-F)F.F)F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "with(linalg) :\ng := x -> x^3*y/z+sin(x*y*z);\ngrad(g(x,y,z),[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*(9$ \"\"$%\"yG\"\"\"%\"zG!\"\"F1-%$sinG6#*(F.F1F0F1F2F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,&*(%\"xG\"\"#%\"yG\"\"\"%\"zG !\"\"\"\"$*(-%$cosG6#*(F)F,F+F,F-F,F,F+F,F-F,F,,&*&F)F/F-F.F,*(F1F,F)F ,F-F,F,,&*(F)F/F+F,F-!\"#F.*(F1F,F)F,F+F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Graphical examples " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examp le 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 29 "The gradient of the function " }{XPPEDIT 18 0 "g(x,y,z) =sqrt(x^2+y^2+z^2)" "6#/-%\"gG6%%\"xG%\"yG%\"zG-%%sqrtG6#,(*$F'\"\"#\" \"\"*$F(F/F0*$F)F/F0" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "grad(g)(x,y,z) = x/sqrt(x^2+y^2+z^2);" "6#/--%%g radG6#%\"gG6%%\"xG%\"yG%\"zG*&F*\"\"\"-%%sqrtG6#,(*$F*\"\"#F.*$F+F4F.* $F,F4F.!\"\"" }{TEXT -1 1 " " }{TEXT 424 1 "i" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "y/sqrt(x^2+y^2+z^2);" "6#*&%\"yG\"\"\"-%%sqrtG6#,(*$%\" xG\"\"#F%*$F$F,F%*$%\"zGF,F%!\"\"" }{TEXT -1 1 " " }{TEXT 425 1 "j" } {TEXT -1 3 " + " }{XPPEDIT 18 0 "z/sqrt(x^2+y^2+z^2);" "6#*&%\"zG\"\" \"-%%sqrtG6#,(*$%\"xG\"\"#F%*$%\"yGF,F%*$F$F,F%!\"\"" }{TEXT -1 1 " " }{TEXT 426 1 "k" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "``=[ x/sqrt(x^2+y^2+z^2), y/sqrt(x^2+y^2+z^2), z/sqrt(x ^2+y^2+z^2)]" "6#/%!G7%*&%\"xG\"\"\"-%%sqrtG6#,(*$F'\"\"#F(*$%\"yGF.F( *$%\"zGF.F(!\"\"*&F0F(-F*6#,(*$F'F.F(*$F0F.F(*$F2F.F(F3*&F2F(-F*6#,(*$ F'F.F(*$F0F.F(*$F2F.F(F3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "For example, the gradient of " } {XPPEDIT 18 0 "g(x,y,z)" "6#-%\"gG6%%\"xG%\"yG%\"zG" }{TEXT -1 14 " at the point " }{XPPEDIT 18 0 "P(2,1,-2)" "6#-%\"PG6%\"\"#\"\"\",$F&!\" \"" }{TEXT -1 64 " is a unit vector in the direction of the position v ector of P. " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "grad (g)" "6#-%%gradG6#%\"gG" }{TEXT -1 1 " " }{TEXT 427 1 "|" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``[``(2,1,-2)];" "6#&%!G6#-F$6%\"\"#\"\"\",$F(!\" \"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\" " }{TEXT -1 1 " " }{TEXT 428 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "1 /3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 430 1 "j" }{TEXT -1 1 " " }{XPPEDIT 18 0 "- 2/3" "6#,$*&\"\"#\"\"\"\"\"$!\"\"F(" } {TEXT -1 1 " " }{TEXT 429 1 "k" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``= [2 /3,1/3,-2/3]" "6#/%!G7%*&\"\"#\"\"\"\"\"$!\"\"*&F(F(F)F*,$*&F'F(F)F*F* " }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 100 "with(linalg):\ng := x -> sqrt(x^2+y^2+z^2); \ngrad(g(x,y,z),[x,y,z]);\nsimplify(eval(%,\{x=2,y=1,z=-2\}));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(-%%sqrtG6#,(*$)9$\"\"#\"\"\"F4*$)%\"yGF3F4F4*$)%\"zGF3F4F4F(F(F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%*&,(*$)%\"xG\"\"#\"\"\" F-*$)%\"yGF,F-F-*$)%\"zGF,F-F-#!\"\"F,F+F-*&F(F4F0F-*&F(F4F3F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%#\"\"#\"\"$#\"\"\"F)#!\" #F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "W e can draw a picture of this gradient field using the procedure " } {TEXT 0 10 "gradplot3d" }{TEXT -1 2 ". 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0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 29 "The gradient of the function " }{XPPEDIT 18 0 "g(x,y,z) = x*y*z;" "6#/-%\"gG6%%\"xG%\"yG%\"zG*(F'\"\"\"F(F+F)F+" }{TEXT -1 4 " \+ is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "grad(g)(x,y,z) = y*z;" "6#/--%%gradG6#%\"gG6%%\"xG%\"yG%\"zG*&F+\"\"\"F,F." }{TEXT -1 1 " " }{TEXT 431 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "x*z;" "6#* &%\"xG\"\"\"%\"zGF%" }{TEXT -1 1 " " }{TEXT 432 1 "j" }{TEXT -1 3 " + \+ " }{XPPEDIT 18 0 "x*y;" "6#*&%\"xG\"\"\"%\"yGF%" }{TEXT -1 1 " " } {TEXT 433 1 "k" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "`` = [y*z, x*z, x*y];" "6#/%!G7%*&%\"yG\"\"\"%\"zGF(*&% \"xGF(F)F(*&F+F(F'F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "The vector field of unit vectors havin g the same direction as the vectors in the gradient field can be plott ed as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "with(linalg):\ng := x -> x*y*z;\ngd := grad(g(x ,y,z),[x,y,z]);\nnm := norm(%,2);\nh := unapply(convert(evalm(1/nm*gd) ,list),x,y,z);\nfieldplot3d(h(x,y,z),x=-3..3,y=-3..3,z=-3..3,\n \+ arrows=SLIM,shading=ZHUE,axes=boxed);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*(9$\"\"\"%\"y GF.%\"zGF.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gdG-%'vectorG6# 7%*&%\"yG\"\"\"%\"zGF+*&%\"xGF+F,F+*&F.F+F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nmG*$-%%sqrtG6#,(*$)-%$absG6#*&%\"yG\"\"\"%\"zGF1\" \"#F1F1*$)-F-6#*&%\"xGF1F2F1F3F1F1*$)-F-6#*&F9F1F0F1F3F1F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&a rrowGF*7%*(,(*$)-%$absG6#*&9%\"\"\"9&F8\"\"#F8F8*$)-F46#*&9$F8F9F8F:F8 F8*$)-F46#*&F@F8F7F8F:F8F8#!\"\"F:F7F8F9F8*(F0FFF@F8F9F8*(F0FFF@F8F7F8 F*F*F*" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6&-%'CUR VESG6\\[m7'7%$!3i'p_'HeVZK!#4WrGF1=>F*$!3o!fOrZoK 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1 "k" }{XPPEDIT 18 0 "`` = [2*x, 4*y, 1];" "6#/%!G7%*&\" \"#\"\"\"%\"xGF(*&\"\"%F(%\"yGF(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "with(linal g):\ng := x -> x^2+2*y^2+z+1;\ngrad(g(x,y,z),[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,**$)9$\" \"#\"\"\"F1*&F0F1)%\"yGF0F1F1%\"zGF1F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,$%\"xG\"\"#,$%\"yG\"\"%\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "The gradient fi eld is independent of z. 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"" 0 "" {TEXT -1 60 "The directional derivative of a function \+ of three variables " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 22 "directional derivativ e" }{TEXT -1 34 " of a function of three variables " }{XPPEDIT 18 0 "g (x,y,z)" "6#-%\"gG6%%\"xG%\"yG%\"zG" }{TEXT -1 11 " at a point" } {XPPEDIT 18 0 "``(x[0],y[0],z[0])" "6#-%!G6%&%\"xG6#\"\"!&%\"yG6#F)&% \"zG6#F)" }{TEXT -1 37 " in the direction of the unit vector " }{TEXT 287 1 "u" }{TEXT -1 8 " is the " }{TEXT 259 11 "dot product" }{TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "D[\"u\"](g )(x[0],y[0],z[0]) = grad(g)(x[0],y[0],z[0]);" "6#/--&%\"DG6#Q\"u6\"6#% \"gG6%&%\"xG6#\"\"!&%\"yG6#F1&%\"zG6#F1--%%gradG6#F,6%&F/6#F1&F36#F1&F 66#F1" }{TEXT -1 1 " " }{TEXT 281 1 "." }{TEXT -1 1 " " }{TEXT 280 1 " u" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 282 17 " _________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 34 "This directional derivative gives " } {TEXT 259 34 "the rate of change of the function" }{TEXT -1 1 " " } {XPPEDIT 18 0 "w = g(x,y,z);" "6#/%\"wG-%\"gG6%%\"xG%\"yG%\"zG" } {TEXT -1 28 " (with respect to distance) " }{TEXT 259 35 "in the direc tion of the unit vector" }{TEXT -1 1 " " }{TEXT 286 1 "u" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Since" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "grad(g)(x[0], y[0], z[0]) = Diff(g(x,y,z),x) ;" "6#/--%%gradG6#%\"gG6%&%\"xG6#\"\"!&%\"yG6#F-&%\"zG6#F--%%DiffG6$-F (6%F+F/F2F+" }{TEXT -1 1 " " }{TEXT 283 1 "i" }{TEXT -1 3 " + " } {XPPEDIT 18 0 "Diff(g(x,y,z),y)" "6#-%%DiffG6$-%\"gG6%%\"xG%\"yG%\"zGF *" }{TEXT -1 1 " " }{TEXT 284 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 " Diff(g(x,y,z),z)" "6#-%%DiffG6$-%\"gG6%%\"xG%\"yG%\"zGF+" }{TEXT -1 1 " " }{TEXT 285 1 "k" }{TEXT -1 1 " " }{XPPEDIT 18 0 "eval(``,(x, y, z) = (x[0], y[0], z[0]));" "6#-%%evalG6$%!G/6%%\"xG%\"yG%\"zG6%&F)6#\"\" !&F*6#F/&F+6#F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = g[x](x[0],y[0],z [0]);" "6#/%!G-&%\"gG6#%\"xG6%&F)6#\"\"!&%\"yG6#F-&%\"zG6#F-" }{TEXT -1 1 " " }{TEXT 294 1 "i" }{TEXT -1 2 " +" }{XPPEDIT 18 0 "g[y](x[0], \+ y[0], z[0]);" "6#-&%\"gG6#%\"yG6%&%\"xG6#\"\"!&F'6#F,&%\"zG6#F," } {TEXT -1 1 " " }{TEXT 296 1 "j" }{TEXT -1 2 " +" }{XPPEDIT 18 0 "g[z]( x[0], y[0], z[0]);" "6#-&%\"gG6#%\"zG6%&%\"xG6#\"\"!&%\"yG6#F,&F'6#F, " }{TEXT -1 1 " " }{TEXT 295 1 "k" }{TEXT -1 3 ", " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "the special cases " } {TEXT 297 1 "u" }{TEXT -1 3 " = " }{TEXT 298 1 "i" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "[1,0,0]" "6#7%\"\"\"\"\"!F%" }{TEXT -1 2 ", " }{TEXT 299 1 "u" }{TEXT -1 3 " = " }{TEXT 300 1 "j" }{TEXT -1 4 " = " } {XPPEDIT 18 0 "[0,1,0]" "6#7%\"\"!\"\"\"F$" }{TEXT -1 5 " and " } {TEXT 301 1 "u" }{TEXT -1 3 " = " }{TEXT 302 1 "k" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "[0,0,1]" "6#7%\"\"!F$\"\"\"" }{TEXT -1 79 " reduce to t he respective partial derivatives which give the rate of change of " } {XPPEDIT 18 0 "w = g(x,y,z);" "6#/%\"wG-%\"gG6%%\"xG%\"yG%\"zG" } {TEXT -1 31 " in the coordinate directions: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "D[\"i\"](g)(x[0],y[0],z[0]) = grad(g)( x[0],y[0],z[0])" "6#/--&%\"DG6#Q\"i6\"6#%\"gG6%&%\"xG6#\"\"!&%\"yG6#F1 &%\"zG6#F1--%%gradG6#F,6%&F/6#F1&F36#F1&F66#F1" }{TEXT -1 1 " " } {TEXT 291 1 "." }{TEXT -1 1 " " }{TEXT 288 1 "i" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "g[x](x[0],y[0],z[0]);" "6#-&%\"gG6#%\"xG6%&F'6#\"\"!&% \"yG6#F+&%\"zG6#F+" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "D[\"j\"](g)(x[0],y[0],z[0]) = grad(g)(x[0],y[0],z[0 ])" "6#/--&%\"DG6#Q\"j6\"6#%\"gG6%&%\"xG6#\"\"!&%\"yG6#F1&%\"zG6#F1--% %gradG6#F,6%&F/6#F1&F36#F1&F66#F1" }{TEXT -1 1 " " }{TEXT 292 1 "." } {TEXT -1 1 " " }{TEXT 289 1 "j" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "g[y] (x[0],y[0],z[0]);" "6#-&%\"gG6#%\"yG6%&%\"xG6#\"\"!&F'6#F,&%\"zG6#F," }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 " D[\"k\"](g)(x[0],y[0],z[0]) = grad(g)(x[0],y[0],z[0])" "6#/--&%\"DG6#Q \"k6\"6#%\"gG6%&%\"xG6#\"\"!&%\"yG6#F1&%\"zG6#F1--%%gradG6#F,6%&F/6#F1 &F36#F1&F66#F1" }{TEXT -1 1 " " }{TEXT 293 1 "." }{TEXT -1 1 " " } {TEXT 290 1 "k" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "g[z](x[0],y[0],z[0]) " "6#-&%\"gG6#%\"zG6%&%\"xG6#\"\"!&%\"yG6#F,&F'6#F," }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Explanation of the g eneral case" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 23 "Let the unit vector be " }{TEXT 303 1 "u" }{TEXT -1 3 " = " }{TEXT 307 1 "a" }{TEXT -1 1 " " }{TEXT 304 1 "i" }{TEXT -1 3 " + " }{TEXT 308 1 "b" }{TEXT -1 1 " " }{TEXT 305 1 "j" }{TEXT -1 3 " + " }{TEXT 309 1 "c" }{TEXT -1 1 " " }{TEXT 306 1 "k" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "[a,b,c]" "6#7%%\"aG%\"bG%\"cG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "Then the line through the point \+ " }{XPPEDIT 18 0 "P(x[0],y[0],z[0]);" "6#-%\"PG6%&%\"xG6#\"\"!&%\"yG6# F)&%\"zG6#F)" }{TEXT -1 37 " in the direction of the unit vector " } {TEXT 310 1 "u" }{TEXT -1 31 " has the parametric equations: " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([x=x[0]+a* s,``],[y=y[0]+b*s,``],[z=z[0]+c*s,``])" "6#-%*PIECEWISEG6%7$/%\"xG,&&F (6#\"\"!\"\"\"*&%\"aGF-%\"sGF-F-%!G7$/%\"yG,&&F46#F,F-*&%\"bGF-F0F-F-F 17$/%\"zG,&&F<6#F,F-*&%\"cGF-F0F-F-F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "dx/ds = a" "6#/*&%#dxG\" \"\"%#dsG!\"\"%\"aG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "dy/ds = b" "6#/* &%#dyG\"\"\"%#dsG!\"\"%\"bG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dz/dt = c" "6#/*&%#dzG\"\"\"%#dtG!\"\"%\"cG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "The point on this line determined by the parameter v alue " }{TEXT 311 1 "s" }{TEXT -1 20 " lies at a distance " }{TEXT 312 1 "s" }{TEXT -1 16 " from the point " }{XPPEDIT 18 0 "P(x[0],y[0], z[0])" "6#-%\"PG6%&%\"xG6#\"\"!&%\"yG6#F)&%\"zG6#F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{TEXT 259 22 "the rate of chang e of " }{XPPEDIT 18 0 "w=g(x,y,z)" "6#/%\"wG-%\"gG6%%\"xG%\"yG%\"zG" } {TEXT -1 1 " " }{TEXT 259 19 "in the direction of" }{TEXT -1 1 " " } {TEXT 313 1 "u" }{TEXT -1 46 " (with respect to distance along the lin e) is " }{XPPEDIT 18 0 "dw/ds" "6#*&%#dwG\"\"\"%#dsG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "By the chain rule " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dw/ds=``(Diff(w,y))*``(dx/ds)+`` (Diff(w,y))*``(dy/ds)+``(Diff(w,z))*``(dz/ds)" "6#/*&%#dwG\"\"\"%#dsG! \"\",(*&-%!G6#-%%DiffG6$%\"wG%\"yGF&-F,6#*&%#dxGF&F'F(F&F&*&-F,6#-F/6$ F1F2F&-F,6#*&%#dyGF&F'F(F&F&*&-F,6#-F/6$F1%\"zGF&-F,6#*&%#dzGF&F'F(F&F &" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``= ``(Diff(w,y))*a+``(Diff(w,y))*b+``(Diff(w,z))*c" "6#/%!G,(*&-F$6#- %%DiffG6$%\"wG%\"yG\"\"\"%\"aGF.F.*&-F$6#-F*6$F,F-F.%\"bGF.F.*&-F$6#-F *6$F,%\"zGF.%\"cGF.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "Evaluating at the point" }{XPPEDIT 18 0 "``(x[0],y[0],z[0])" "6#-%!G6 %&%\"xG6#\"\"!&%\"yG6#F)&%\"zG6#F)" }{TEXT -1 7 " gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "eval(dw/ds,(x, y, z) = (x[0], y [0], z[0])) = g[x](x[0],y[0],z[0])*a+g[x](x[0],y[0],z[0])*b+g[x](x[0], y[0],z[0])*c;" "6#/-%%evalG6$*&%#dwG\"\"\"%#dsG!\"\"/6%%\"xG%\"yG%\"zG 6%&F.6#\"\"!&F/6#F4&F06#F4,(*&-&%\"gG6#F.6%&F.6#F4&F/6#F4&F06#F4F)%\"a GF)F)*&-&F=6#F.6%&F.6#F4&F/6#F4&F06#F4F)%\"bGF)F)*&-&F=6#F.6%&F.6#F4&F /6#F4&F06#F4F)%\"cGF)F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = grad(g)(x[0],y[0],z[0]);" "6#/%!G--%%gradG6 #%\"gG6%&%\"xG6#\"\"!&%\"yG6#F.&%\"zG6#F." }{TEXT -1 1 " " }{TEXT 315 1 "." }{TEXT -1 1 " " }{TEXT 314 1 "u" }{TEXT -1 5 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Example " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "g(x,y,z)=x*exp(y)+y*z" "6#/-%\"gG6%%\"xG%\"yG%\"zG,&*&F '\"\"\"-%$expG6#F(F,F,*&F(F,F)F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 38 "We find the directional derivative of " }{XPPEDIT 18 0 "g(x,y,z)" "6#-%\"gG6%%\"xG%\"yG%\"zG" }{TEXT -1 14 " at the point " } {XPPEDIT 18 0 "P(2,0,0)" "6#-%\"PG6%\"\"#\"\"!F'" }{TEXT -1 53 " in th e direction of the line joining P to the point " }{XPPEDIT 18 0 "Q(4,1 ,-2)" "6#-%\"QG6%\"\"%\"\"\",$\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 17 " The gradient of " }{XPPEDIT 18 0 "g(x,y,z)" "6#-% \"gG6%%\"xG%\"yG%\"zG" }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "grad(g)(x,y,z) = exp(y)" "6#/--%%gradG6#%\"g G6%%\"xG%\"yG%\"zG-%$expG6#F+" }{TEXT -1 1 " " }{TEXT 316 1 "i" } {TEXT -1 3 " + " }{XPPEDIT 18 0 "``(x*exp(y)+z)" "6#-%!G6#,&*&%\"xG\" \"\"-%$expG6#%\"yGF)F)%\"zGF)" }{TEXT -1 1 " " }{TEXT 317 1 "j" } {TEXT -1 5 " + y " }{TEXT 318 1 "k" }{TEXT -1 3 " = " }{XPPEDIT 18 0 " [exp(y),x*exp(y)+z,y]" "6#7%-%$expG6#%\"yG,&*&%\"xG\"\"\"-F%6#F'F+F+% \"zGF+F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 24 "Evaluating at the point " }{XPPEDIT 18 0 "``(2,0,0 )" "6#-%!G6%\"\"#\"\"!F'" }{TEXT -1 18 " gives the vector " }{TEXT 332 1 "w" }{TEXT -1 3 " = " }{TEXT 319 1 "i" }{TEXT -1 5 " + 2 " } {TEXT 320 1 "j" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "[1,2,0]" "6#7%\"\"\" \"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " } {TEXT 321 2 "PQ" }{TEXT -1 3 " = " }{TEXT 322 1 "v" }{TEXT -1 5 " = 2 \+ " }{TEXT 323 1 "i" }{TEXT -1 3 " + " }{TEXT 324 1 "j" }{TEXT -1 5 " - \+ 2 " }{TEXT 325 1 "k" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "[2,1,-2]" "6#7% \"\"#\"\"\",$F$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "Si nce ||" }{TEXT 331 1 "v" }{TEXT -1 5 "|| = " }{XPPEDIT 18 0 "sqrt(4+1+ 4)=3" "6#/-%%sqrtG6#,(\"\"%\"\"\"F)F)F(F)\"\"$" }{TEXT -1 33 ", a unit vector in the direction " }{TEXT 327 1 "v" }{TEXT -1 4 " is " }{TEXT 326 1 "u" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$ !\"\"" }{TEXT -1 1 " " }{TEXT 328 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 329 1 "j" } {TEXT -1 3 " - " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" } {TEXT -1 1 " " }{TEXT 330 1 "k" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "[2/3 ,1/3,-2/3]" "6#7%*&\"\"#\"\"\"\"\"$!\"\"*&F&F&F'F(,$*&F%F&F'F(F(" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 22 "directional derivative" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "g(x,y,z)" "6#-%\"gG6%%\"xG%\"yG%\"zG" }{TEXT -1 14 " at the point " }{XPPEDIT 18 0 "P(2,0,0)" "6#-%\"PG6%\"\"#\"\"!F' " }{TEXT -1 37 " in the direction of the unit vector " }{TEXT 333 1 "u " }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 340 1 "w" }{TEXT -1 1 " " }{TEXT 342 1 "." }{TEXT -1 1 " " }{TEXT 341 1 "u" }{TEXT -1 4 " = (" }{TEXT 334 1 "i" }{TEXT -1 5 " + 2 " }{TEXT 335 1 " j" }{TEXT -1 2 ") " }{TEXT 337 1 "." }{TEXT -1 3 " ( " }{XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 336 1 "i" } {TEXT -1 3 " + " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 338 1 "j" }{TEXT -1 3 " - " }{XPPEDIT 18 0 "2/3" "6#*& \"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 339 1 "k" }{TEXT -1 5 " \+ ) = " }{XPPEDIT 18 0 "2/3+2/3+0 = 4/3;" "6#/,(*&\"\"#\"\"\"\"\"$!\"\"F '*&F&F'F(F)F'\"\"!F'*&\"\"%F'F(F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "with(linalg ):\ng := (x,y,z) -> x*exp(y)+y*z;\ngrad(g(x,y,z),[x,y,z]);\nw := eval( %,\{x=2,y=0,z=0\});\nv := vector([2,1,-2]);\nu := 1/sqrt(dotprod(v,v)) *v;\ndotprod(w,u); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6%%\"xG %\"yG%\"zG6\"6$%)operatorG%&arrowGF*,&*&9$\"\"\"-%$expG6#9%F1F1*&F5F19 &F1F1F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%-%$expG6# %\"yG,&*&%\"xG\"\"\"F'F.F.%\"zGF.F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"wG-%'vectorG6#7%\"\"\"\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'vectorG6#7%\"\"#\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG,$%\"vG#\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\" \"%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "We now use the dir ectional derivative to " }{TEXT 259 32 "estimate the change in the val ue" }{TEXT -1 17 " of the function " }{XPPEDIT 18 0 "w = g(x,y,z);" "6 #/%\"wG-%\"gG6%%\"xG%\"yG%\"zG" }{TEXT -1 25 " in moving a distance of " }{XPPEDIT 18 0 "1/100" "6#*&\"\"\"F$\"$+\"!\"\"" }{TEXT -1 31 " alo ng the line from the point " }{XPPEDIT 18 0 "P(2,0,0)" "6#-%\"PG6%\"\" #\"\"!F'" }{TEXT -1 19 " towards the point " }{XPPEDIT 18 0 "Q(4,1,-2) " "6#-%\"QG6%\"\"%\"\"\",$\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "For a small change " }{XPPEDIT 18 0 "Delta*s" "6#*&%&D eltaG\"\"\"%\"sGF%" }{TEXT -1 18 " in the distance " }{TEXT 344 1 "s " }{TEXT -1 25 " along PQ from P we have " }{XPPEDIT 18 0 "Delta*w/(De lta*s)" "6#*(%&DeltaG\"\"\"%\"wGF%*&F$F%%\"sGF%!\"\"" }{TEXT -1 1 " " }{TEXT 343 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4/3" "6#*&\"\"%\"\"\" \"\"$!\"\"" }{TEXT -1 10 ". Hence " }{XPPEDIT 18 0 "Delta*w" "6#*&%& DeltaG\"\"\"%\"wGF%" }{TEXT -1 1 " " }{TEXT 352 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4/3" "6#*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Delta*s" "6#*&%&DeltaG\"\"\"%\"sGF%" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "Delta*s = 1/100;" " 6#/*&%&DeltaG\"\"\"%\"sGF&*&F&F&\"$+\"!\"\"" }{TEXT -1 18 ", which giv es the " }{TEXT 259 33 "estimate for the change in value" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "Delta*w" "6#*&%&DeltaG\"\"\"%\"wGF%" }{TEXT -1 1 " " }{TEXT 345 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(4/3)*``(1 /100) = 4/300;" "6#/*&-%!G6#*&\"\"%\"\"\"\"\"$!\"\"F*-F&6#*&F*F*\"$+\" F,F**&F)F*\"$+$F," }{TEXT -1 1 " " }{TEXT 346 1 "~" }{TEXT -1 16 " 0.0 1333333333. " }}{PARA 0 "" 0 "" {TEXT -1 21 "Moving a distance of " } {XPPEDIT 18 0 "1/100" "6#*&\"\"\"F$\"$+\"!\"\"" }{TEXT -1 6 " from " } {XPPEDIT 18 0 "P(2,0,0)" "6#-%\"PG6%\"\"#\"\"!F'" }{TEXT -1 37 " in th e direction of the unit vector " }{TEXT 348 1 "u" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " } {TEXT 349 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\" \"$!\"\"" }{TEXT -1 1 " " }{TEXT 350 1 "j" }{TEXT -1 3 " - " } {XPPEDIT 18 0 "2/3" "6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " } {TEXT 351 1 "k" }{TEXT -1 24 " brings us to the point " }{XPPEDIT 18 0 "``(2+2/300, 1/300, -2/300)" "6#-%!G6%,&\"\"#\"\"\"*&F'F(\"$+$!\"\"F (*&F(F(F*F+,$*&F'F(F*F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Hence the " }{TEXT 259 13 "actual change" }{TEXT -1 5 " is " } {XPPEDIT 18 0 "g(2+2/300,1/300,-2/300)-g(2,0,0)" "6#,&-%\"gG6%,&\"\"# \"\"\"*&F(F)\"$+$!\"\"F)*&F)F)F+F,,$*&F(F)F+F,F,F)-F%6%F(\"\"!F2F," } {TEXT -1 3 " = " }{XPPEDIT 18 0 "301/150*exp(1/300)-90001/45000" "6#,& *(\"$,$\"\"\"\"$]\"!\"\"-%$expG6#*&F&F&\"$+$F(F&F&*&\"&,+*F&\"&+]%F(F( " }{TEXT -1 1 " " }{TEXT 347 1 "~" }{TEXT -1 14 " 0.013344494. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "w1 := g(2,0,0);\nvector([2,0,0])+1/100*evalm(u);\nevalm(%);\nw2 : = g(op(convert(%,list)));\nw2-w1;\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#w1G\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'vec torG6#7%\"\"#\"\"!F)\"\"\"*&#F*\"$+\"F*-F%6#7%#F(\"\"$#F*F2#!\"#F2F*F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%#\"$,$\"$]\"#\"\"\" \"$+$#!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#w2G,&-%$expG6##\" \"\"\"$+$#\"$,$\"$]\"#F*\"&+]%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,&-%$expG6##\"\"\"\"$+$#\"$,$\"$]\"#\"&,+*\"&+]%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")%\\WL\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 355 6 "Remark" }}{PARA 0 "" 0 "" {TEXT -1 90 "We can also obtain the previous estimate by appealing to the linear approximation formul a " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta*w" "6#*&% &DeltaG\"\"\"%\"wGF%" }{TEXT -1 1 " " }{TEXT 353 1 "~" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Diff(w,x)*Delta*x+Diff(w,y)*Delta*y+Diff(w,z)*Delta* z;" "6#,(*(-%%DiffG6$%\"wG%\"xG\"\"\"%&DeltaGF*F)F*F**(-F&6$F(%\"yGF*F +F*F/F*F**(-F&6$F(%\"zGF*F+F*F3F*F*" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Delta*w" "6#*&%&DeltaG\"\"\"%\"wGF%" }{TEXT -1 1 " " } {TEXT 354 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1)*``(2/300)+``(2)* ``(1/300) = 4/300" "6#/,&*&-%!G6#\"\"\"F)-F'6#*&\"\"#F)\"$+$!\"\"F)F)* &-F'6#F-F)-F'6#*&F)F)F.F/F)F)*&\"\"%F)F.F/" }{TEXT -1 1 " " }{TEXT 356 1 "~" }{TEXT -1 16 " 0.01333333333. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "The direct ion of the gradient vector at a point " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 41 "The directional der ivative of a function " }{XPPEDIT 18 0 "g(x,y,z)" "6#-%\"gG6%%\"xG%\"y G%\"zG" }{TEXT -1 11 " at a point" }{XPPEDIT 18 0 "``(x[0],y[0],z[0]) " "6#-%!G6%&%\"xG6#\"\"!&%\"yG6#F)&%\"zG6#F)" }{TEXT -1 37 " in the di rection of the unit vector " }{TEXT 359 1 "u" }{TEXT -1 4 ": is" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "D[\"u\"](g)(x[0],y[0] ,z[0]) = grad(g)(x[0],y[0],z[0]);" "6#/--&%\"DG6#Q\"u6\"6#%\"gG6%&%\"x G6#\"\"!&%\"yG6#F1&%\"zG6#F1--%%gradG6#F,6%&F/6#F1&F36#F1&F66#F1" } {TEXT -1 1 " " }{TEXT 358 1 "." }{TEXT -1 1 " " }{TEXT 357 1 "u" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "If the angle between " }{XPPEDIT 18 0 "grad(g)(x[0],y[0],z[0])" "6#--%%gradG6#%\"gG6%&%\"xG6# \"\"!&%\"yG6#F,&%\"zG6#F," }{TEXT -1 5 " and " }{TEXT 363 1 "u" } {TEXT -1 4 " is " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 7 ", t hen " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "D[\"u\"](g)(x [0],y[0],z[0])" "6#--&%\"DG6#Q\"u6\"6#%\"gG6%&%\"xG6#\"\"!&%\"yG6#F0&% \"zG6#F0" }{TEXT -1 6 " = || " }{XPPEDIT 18 0 "grad(g)(x[0],y[0],z[0]) " "6#--%%gradG6#%\"gG6%&%\"xG6#\"\"!&%\"yG6#F,&%\"zG6#F," }{TEXT -1 4 " || " }{TEXT 362 1 "." }{TEXT -1 4 " || " }{TEXT 361 1 "u" }{TEXT -1 4 " || " }{XPPEDIT 18 0 "cos(theta)" "6#-%$cosG6#%&thetaG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "D[\"u\"](g)(x[0],y[0],z[0])" "6#--&%\"D G6#Q\"u6\"6#%\"gG6%&%\"xG6#\"\"!&%\"yG6#F0&%\"zG6#F0" }{TEXT -1 6 " = \+ || " }{XPPEDIT 18 0 "grad(g)(x[0],y[0],z[0])" "6#--%%gradG6#%\"gG6%&% \"xG6#\"\"!&%\"yG6#F,&%\"zG6#F," }{TEXT -1 4 " || " }{XPPEDIT 18 0 "co s(theta)" "6#-%$cosG6#%&thetaG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 360 21 "_____________________" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "If the d irection of the unit vector " }{TEXT 364 1 "u" }{TEXT -1 12 " is varie d, " }{XPPEDIT 18 0 "cos(theta)" "6#-%$cosG6#%&thetaG" }{TEXT -1 18 " \+ can vary between " }{XPPEDIT 18 0 "-``" "6#,$%!G!\"\"" }{TEXT -1 35 "1 and 1. From this we see that the " }{TEXT 259 13 "maximum value" } {TEXT -1 31 " of the directional derivative " }{XPPEDIT 18 0 "D[\"u\"] (g)(x[0],y[0],z[0])" "6#--&%\"DG6#Q\"u6\"6#%\"gG6%&%\"xG6#\"\"!&%\"yG6 #F0&%\"zG6#F0" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "g(x,y,z)" "6#-%\"gG6 %%\"xG%\"yG%\"zG" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(x[0] ,y[0],z[0])" "6#-%!G6%&%\"xG6#\"\"!&%\"yG6#F)&%\"zG6#F)" }{TEXT -1 7 " is || " }{XPPEDIT 18 0 "grad(g)(x[0],y[0],z[0])" "6#--%%gradG6#%\"gG6 %&%\"xG6#\"\"!&%\"yG6#F,&%\"zG6#F," }{TEXT -1 26 " ||, and this occurs when " }{XPPEDIT 18 0 "theta=0" "6#/%&thetaG\"\"!" }{TEXT -1 64 ". He nce the maximum rate of increase in the value of a function " } {XPPEDIT 18 0 "g(x,y,z)" "6#-%\"gG6%%\"xG%\"yG%\"zG" }{TEXT -1 43 " at any point is given by the magnitude || " }{XPPEDIT 18 0 "grad(g)(x[0] ,y[0],z[0])" "6#--%%gradG6#%\"gG6%&%\"xG6#\"\"!&%\"yG6#F,&%\"zG6#F," } {TEXT -1 93 " || of the gradient vector, and occurs in the direction o f the gradient vector at that point." }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 259 13 "minimum value" } {TEXT -1 31 " of the directional derivative " }{XPPEDIT 18 0 "D[\"u\"] (g)(x[0],y[0],z[0])" "6#--&%\"DG6#Q\"u6\"6#%\"gG6%&%\"xG6#\"\"!&%\"yG6 #F0&%\"zG6#F0" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "g(x,y,z)" "6#-%\"gG6 %%\"xG%\"yG%\"zG" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(x[0] ,y[0],z[0])" "6#-%!G6%&%\"xG6#\"\"!&%\"yG6#F)&%\"zG6#F)" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "-``" "6#,$%!G!\"\"" }{TEXT -1 3 "|| " } {XPPEDIT 18 0 "grad(g)(x[0],y[0],z[0])" "6#--%%gradG6#%\"gG6%&%\"xG6# \"\"!&%\"yG6#F,&%\"zG6#F," }{TEXT -1 23 " ||, which occurs when " } {XPPEDIT 18 0 "theta=Pi" "6#/%&thetaG%#PiG" }{TEXT -1 64 ". Hence the \+ maximum rate of decrease in the value of a function " }{XPPEDIT 18 0 " g(x,y,z)" "6#-%\"gG6%%\"xG%\"yG%\"zG" }{TEXT -1 84 " at any point occu rs in the opposite direction to the gradient vector at that point." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 259 82 "value of a function increases most rapidly in the direct ion of the gradient vector" }{TEXT -1 186 ", that is, the gradient vec tor points in the direction in which the function increases most rapid ly, and its length is the rate of increase (with respect to distance) \+ in that direction. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Example " }}{PARA 0 "" 0 "" {TEXT 365 8 "Questio n" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 38 "(a) Find the gradi ent of the function " }{XPPEDIT 18 0 "g(x,y,z) = (x^2+y^2)/(z^2);" "6# /-%\"gG6%%\"xG%\"yG%\"zG*&,&*$F'\"\"#\"\"\"*$F(F-F.F.*$F)F-!\"\"" } {TEXT -1 14 " at the point " }{XPPEDIT 18 0 "``(2,-1,1)" "6#-%!G6%\"\" #,$\"\"\"!\"\"F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 68 "(b) F ind the directional derivative of the function in (a) the point" } {XPPEDIT 18 0 "``(2,-1,1);" "6#-%!G6%\"\"#,$\"\"\"!\"\"F(" }{TEXT -1 37 ", and in the direction of the vector " }{TEXT 413 1 "i" }{TEXT -1 3 " + " }{TEXT 412 1 "j" }{TEXT -1 3 " + " }{TEXT 411 1 "k" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 87 "(c) Give the direction in which t he function in (a) increases most rapidly at the point" }{XPPEDIT 18 0 "``(2,-1,1);" "6#-%!G6%\"\"#,$\"\"\"!\"\"F(" }{TEXT -1 31 ", in the \+ form of a unit vector." }}{PARA 0 "" 0 "" {TEXT -1 74 "(d) Find the ma ximum rate of increase of the function in (a) at the point " } {XPPEDIT 18 0 "``(2,-1,1);" "6#-%!G6%\"\"#,$\"\"\"!\"\"F(" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 366 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 21 "(a) The gr adient of " }{XPPEDIT 18 0 "g(x,y,z)" "6#-%\"gG6%%\"xG%\"yG%\"zG" } {TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 12 " grad(g ) = " } {XPPEDIT 18 0 "diff(g(x,y,z),x);" "6#-%%diffG6$-%\"gG6%%\"xG%\"yG%\"zG F)" }{TEXT -1 1 " " }{TEXT 408 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "diff(g(x,y,z),y);" "6#-%%diffG6$-%\"gG6%%\"xG%\"yG%\"zGF*" }{TEXT -1 1 " " }{TEXT 409 1 "j" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "diff(g(x,y,z) ,z);" "6#-%%diffG6$-%\"gG6%%\"xG%\"yG%\"zGF+" }{TEXT -1 1 " " }{TEXT 410 1 "k" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = 2*x/(z^2);" "6#/%!G*(\"\"#\"\"\"%\"xGF'*$%\"zGF&! \"\"" }{TEXT -1 1 " " }{TEXT 367 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "2*y/(z^2)" "6#*(\"\"#\"\"\"%\"yGF%*$%\"zGF$!\"\"" }{TEXT -1 1 " " } {TEXT 368 1 "j" }{TEXT -1 2 " " }{XPPEDIT 18 0 "-2*(x^2+y^2)/(z^3)" " 6#,$*(\"\"#\"\"\",&*$%\"xGF%F&*$%\"yGF%F&F&*$%\"zG\"\"$!\"\"F/" } {TEXT -1 1 " " }{TEXT 369 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The gradient vector of f \+ at " }{XPPEDIT 18 0 "``(2,-1,1)" "6#-%!G6%\"\"#,$\"\"\"!\"\"F(" } {TEXT -1 7 " is: " }{XPPEDIT 18 0 "grad(g)(2,-1,1);" "6#--%%gradG6#% \"gG6%\"\"#,$\"\"\"!\"\"F+" }{TEXT -1 5 " = 4 " }{TEXT 377 1 "i" } {TEXT -1 1 " " }{XPPEDIT 18 0 "- 2" "6#,$\"\"#!\"\"" }{TEXT -1 1 " " } {TEXT 378 1 "j" }{TEXT -1 1 " " }{XPPEDIT 18 0 "- 10" "6#,$\"#5!\"\"" }{TEXT -1 1 " " }{TEXT 379 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "(b) A unit vector in the direction of " }{TEXT 370 2 " i " }{TEXT -1 3 " + " }{TEXT 371 1 "j" }{TEXT -1 3 " + " }{TEXT 372 1 "k " }{TEXT -1 4 " is " }{TEXT 373 1 "u" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/sqrt(3)" "6#*&\"\"\"F$-%%sqrtG6#\"\"$!\"\"" }{TEXT -1 3 " ( " } {TEXT 374 1 "i" }{TEXT -1 3 " + " }{TEXT 375 1 "j" }{TEXT -1 3 " + " } {TEXT 376 1 "k" }{TEXT -1 4 " ). " }}{PARA 0 "" 0 "" {TEXT -1 36 "Henc e the directional derivative of " }{XPPEDIT 18 0 "g(x,y,z)" "6#-%\"gG6 %%\"xG%\"yG%\"zG" }{TEXT -1 3 " at" }{XPPEDIT 18 0 "``(2,-1,1);" "6#-% !G6%\"\"#,$\"\"\"!\"\"F(" }{TEXT -1 21 " in the direction of " }{TEXT 390 1 "u" }{TEXT -1 6 " is: " }}{PARA 0 "" 0 "" {TEXT -1 21 " \+ " }{XPPEDIT 18 0 "grad(g)(2,-1,1);" "6#--%%gradG6#%\"gG6% \"\"#,$\"\"\"!\"\"F+" }{TEXT -1 1 " " }{TEXT 389 3 ". u" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "grad(g)(2,-1,1);" "6#--%%gradG6#%\"gG6%\"\"#,$\" \"\"!\"\"F+" }{TEXT -1 1 " " }{TEXT 434 1 "." }{TEXT -1 1 " " } {XPPEDIT 18 0 "1/sqrt(3)" "6#*&\"\"\"F$-%%sqrtG6#\"\"$!\"\"" }{TEXT -1 3 " ( " }{TEXT 380 1 "i" }{TEXT -1 3 " + " }{TEXT 381 1 "j" }{TEXT -1 3 " + " }{TEXT 382 1 "k" }{TEXT -1 11 " ) = ( 4 " }{TEXT 383 1 "i " }{TEXT -1 1 " " }{XPPEDIT 18 0 "- 2" "6#,$\"\"#!\"\"" }{TEXT -1 1 " \+ " }{TEXT 384 1 "j" }{TEXT -1 1 " " }{XPPEDIT 18 0 "- 10" "6#,$\"#5!\" \"" }{TEXT -1 1 " " }{TEXT 385 1 "k" }{TEXT -1 3 " ) " }{TEXT 398 1 ". " }{TEXT -1 2 " " }{XPPEDIT 18 0 "1/sqrt(3)" "6#*&\"\"\"F$-%%sqrtG6# \"\"$!\"\"" }{TEXT -1 3 " ( " }{TEXT 386 1 "i" }{TEXT -1 3 " + " } {TEXT 387 1 "j" }{TEXT -1 3 " + " }{TEXT 388 1 "k" }{TEXT -1 7 " ) = \+ " }{XPPEDIT 18 0 "-8/sqrt(3)" "6#,$*&\"\")\"\"\"-%%sqrtG6#\"\"$!\"\"F +" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "(c) At the point " }{XPPEDIT 18 0 "``(2,-1,1)" "6#-%!G6% \"\"#,$\"\"\"!\"\"F(" }{TEXT -1 14 " the function " }{XPPEDIT 18 0 "g( x,y,z)" "6#-%\"gG6%%\"xG%\"yG%\"zG" }{TEXT -1 66 " increases most rapi dly in the direction of the gradient vector 4 " }{TEXT 391 1 "i" } {TEXT -1 1 " " }{XPPEDIT 18 0 "- 2" "6#,$\"\"#!\"\"" }{TEXT -1 1 " " } {TEXT 392 1 "j" }{TEXT -1 1 " " }{XPPEDIT 18 0 "- 10" "6#,$\"#5!\"\"" }{TEXT -1 1 " " }{TEXT 393 1 "k" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 68 " A unit vector in the direction of the gradient vector at the point " }{XPPEDIT 18 0 "``(2,-1,1)" "6#-%!G6%\"\"#,$\"\"\"!\"\"F( " }{TEXT -1 6 " is " }{TEXT 414 1 "v" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/sqrt(30)" "6#*&\"\"\"F$-%%sqrtG6#\"#I!\"\"" }{TEXT -1 5 " ( 2 \+ " }{TEXT 394 1 "i" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-``" "6#,$%!G!\"\" " }{TEXT -1 1 " " }{TEXT 395 1 "j" }{TEXT -1 1 " " }{XPPEDIT 18 0 "- 5 " "6#,$\"\"&!\"\"" }{TEXT -1 1 " " }{TEXT 396 1 "k" }{TEXT -1 4 " ). \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "(d) \+ The maximum rate of change of the function occurs in the direction of \+ the gradient vector, and is given by the directional derivative in thi s direction." }}{PARA 0 "" 0 "" {TEXT -1 10 " This is " }{XPPEDIT 18 0 "grad(g)(2,-1,1);" "6#--%%gradG6#%\"gG6%\"\"#,$\"\"\"!\"\"F+" } {TEXT -1 1 " " }{TEXT 397 3 ". v" }{TEXT -1 8 " = ( 4 " }{TEXT 402 1 "i" }{TEXT -1 1 " " }{XPPEDIT 18 0 "- 2" "6#,$\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 403 1 "j" }{TEXT -1 1 " " }{XPPEDIT 18 0 "- 10" "6#,$\"#5! \"\"" }{TEXT -1 1 " " }{TEXT 404 1 "k" }{TEXT -1 3 " ) " }{TEXT 435 1 "." }{TEXT -1 2 " " }{XPPEDIT 18 0 "1/sqrt(30)" "6#*&\"\"\"F$-%%sqrtG 6#\"#I!\"\"" }{TEXT -1 5 " ( 2 " }{TEXT 399 1 "i" }{TEXT -1 1 " " } {XPPEDIT 18 0 "-``" "6#,$%!G!\"\"" }{TEXT -1 1 " " }{TEXT 400 1 "j" } {TEXT -1 1 " " }{XPPEDIT 18 0 "- 5" "6#,$\"\"&!\"\"" }{TEXT -1 1 " " } {TEXT 401 1 "k" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "(8 + 2 + 50)/sqrt( 30) = 60/sqrt(30)" "6#/*&,(\"\")\"\"\"\"\"#F'\"#]F'F'-%%sqrtG6#\"#I!\" \"*&\"#gF'-F+6#F-F." }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*sqrt(30)" "6# *&\"\"#\"\"\"-%%sqrtG6#\"#IF%" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 " More simply, this is jus t the magnitude of the gradient vector, namely || 4 " }{TEXT 405 1 "i " }{TEXT -1 1 " " }{XPPEDIT 18 0 "- 2" "6#,$\"\"#!\"\"" }{TEXT -1 1 " \+ " }{TEXT 406 1 "j" }{TEXT -1 1 " " }{XPPEDIT 18 0 "- 10" "6#,$\"#5!\" \"" }{TEXT -1 1 " " }{TEXT 407 1 "k" }{TEXT -1 6 " || = " }{XPPEDIT 18 0 "sqrt(16+4+100) = 2*sqrt(30);" "6#/-%%sqrtG6#,(\"#;\"\"\"\"\"%F) \"$+\"F)*&\"\"#F)-F%6#\"#IF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 35 "Find the directional derivative of " }{XPPEDIT 18 0 "g(x,y,z) = x^ 2+2*y^2+3*z^2;" "6#/-%\"gG6%%\"xG%\"yG%\"zG,(*$F'\"\"#\"\"\"*&F,F-*$F( F,F-F-*&\"\"$F-*$F)F,F-F-" }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(1, 1, 1);" "6#-%!G6%\"\"\"F&F&" }{TEXT -1 22 " in the direction \+ of " }{TEXT 446 1 "v" }{TEXT -1 3 " = " }{TEXT 447 1 "i" }{TEXT -1 3 " + " }{TEXT 448 1 "j" }{TEXT -1 3 " + " }{TEXT 449 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 35 "Find the directional derivative of " }{XPPEDIT 18 0 "g(x,y,z)=x*y+y*z+z*x" "6#/-%\"gG6%%\" xG%\"yG%\"zG,(*&F'\"\"\"F(F,F,*&F(F,F)F,F,*&F)F,F'F,F," }{TEXT -1 13 " at the point" }{XPPEDIT 18 0 "``(1,-1,2)" "6#-%!G6%\"\"\",$F&!\"\"\" \"#" }{TEXT -1 21 " in the direction of " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 450 1 "v" }{TEXT -1 6 " = 10 " }{TEXT 451 1 "i" }{TEXT -1 6 " + 11 " }{TEXT 452 1 "j" }{TEXT -1 5 " - 2 " }{TEXT 453 1 "k" } {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 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 35 "Find the direct ional derivative of " }{XPPEDIT 18 0 "g(x,y,z) = cos(x*y)+exp(y*z)+ln( z*x);" "6#/-%\"gG6%%\"xG%\"yG%\"zG,(-%$cosG6#*&F'\"\"\"F(F/F/-%$expG6# *&F(F/F)F/F/-%#lnG6#*&F)F/F'F/F/" }{TEXT -1 13 " at the point" } {XPPEDIT 18 0 "``(1,0,1/2);" "6#-%!G6%\"\"\"\"\"!*&F&F&\"\"#!\"\"" } {TEXT -1 21 " in the direction of " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {TEXT 442 1 "v" }{TEXT -1 3 " = " }{TEXT 443 1 "i" }{TEXT -1 5 " + 2 \+ " }{TEXT 444 1 "j" }{TEXT -1 5 " + 2 " }{TEXT 445 1 "k" }{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 40 "(a) Find the directi onal derivative of " }{XPPEDIT 18 0 "g(x,y,z) = ln(sqrt(x^2+y^2+z^2)) ;" "6#/-%\"gG6%%\"xG%\"yG%\"zG-%#lnG6#-%%sqrtG6#,(*$F'\"\"#\"\"\"*$F(F 2F3*$F)F2F3" }{TEXT -1 14 " at the point " }{XPPEDIT 18 0 "``(3,4,12) " "6#-%!G6%\"\"$\"\"%\"#7" }{TEXT -1 34 " in the direction of the vect or 3 " }{TEXT 439 1 "i" }{TEXT -1 5 " + 6 " }{TEXT 440 1 "j" }{TEXT -1 5 " - 2 " }{TEXT 441 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 67 "(b) Use partial derivatives to estimate the change in the value of " }{XPPEDIT 18 0 "g(x,y,z) = ln(sqrt(x^2+y^2+z^2));" "6#/-% \"gG6%%\"xG%\"yG%\"zG-%#lnG6#-%%sqrtG6#,(*$F'\"\"#\"\"\"*$F(F2F3*$F)F2 F3" }{TEXT -1 26 " in moving from the point " }{XPPEDIT 18 0 "``(3,4,1 2)" "6#-%!G6%\"\"$\"\"%\"#7" }{TEXT -1 23 " through a distance of " } {XPPEDIT 18 0 "1/100" "6#*&\"\"\"F$\"$+\"!\"\"" }{TEXT -1 34 " in the \+ direction of the vector 3 " }{TEXT 436 1 "i" }{TEXT -1 5 " + 6 " } {TEXT 437 1 "j" }{TEXT -1 5 " - 2 " }{TEXT 438 1 "k" }{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 32 "________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 40 "(a) Find the directi onal derivative of " }{XPPEDIT 18 0 "g(x,y,z)=exp(x)*cos(y*z)" "6#/-% \"gG6%%\"xG%\"yG%\"zG*&-%$expG6#F'\"\"\"-%$cosG6#*&F(F.F)F.F." }{TEXT -1 48 " at the origin in the direction of the vector 2 " }{TEXT 457 1 "i" }{TEXT -1 3 " + " }{TEXT 458 1 "j" }{TEXT -1 5 " - 2 " }{TEXT 459 1 "k" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 67 "(b) Use partial \+ derivatives to estimate the change in the value of " }{XPPEDIT 18 0 "g (x,y,z)=exp(x)*cos(y*z)" "6#/-%\"gG6%%\"xG%\"yG%\"zG*&-%$expG6#F'\"\" \"-%$cosG6#*&F(F.F)F.F." }{TEXT -1 49 " in moving from the origin thro ugh a distance of " }{XPPEDIT 18 0 "1/100" "6#*&\"\"\"F$\"$+\"!\"\"" } {TEXT -1 34 " in the direction of the vector 2 " }{TEXT 454 1 "i" } {TEXT -1 3 " + " }{TEXT 455 1 "j" }{TEXT -1 5 " - 2 " }{TEXT 456 1 "k " }{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 32 "_______________________________ _" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 41 "Find the direct ion in which the function " }{XPPEDIT 18 0 "g(x,y,z) = exp(x*y)+z^2;" "6#/-%\"gG6%%\"xG%\"yG%\"zG,&-%$expG6#*&F'\"\"\"F(F/F/*$F)\"\"#F/" } {TEXT -1 36 " increases most rapidly at the point" }{XPPEDIT 18 0 "``( 0,2,3)" "6#-%!G6%\"\"!\"\"#\"\"$" }{TEXT -1 24 ", and the rate at whic h " }{XPPEDIT 18 0 "g(x,y,z)" "6#-%\"gG6%%\"xG%\"yG%\"zG" }{TEXT -1 55 " changes (with respect to distance) in that direction. " }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 41 "Find the direction in \+ which the function " }{XPPEDIT 18 0 "g(x,y,z) = z*ln(x^2+y^2);" "6#/-% \"gG6%%\"xG%\"yG%\"zG*&F)\"\"\"-%#lnG6#,&*$F'\"\"#F+*$F(F1F+F+" } {TEXT -1 36 " increases most rapidly at the point" }{XPPEDIT 18 0 "``( 1,1,1);" "6#-%!G6%\"\"\"F&F&" }{TEXT -1 24 ", and the rate at which " }{XPPEDIT 18 0 "g(x,y,z)" "6#-%\"gG6%%\"xG%\"yG%\"zG" }{TEXT -1 55 " c hanges (with respect to distance) in that direction. " }}{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 pi ctures" }}{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 }