{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 261 "" 0 "" {TEXT -1 34 "A Potential for Being Conservat ive" }}{PARA 262 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA 263 "" 0 "" {TEXT -1 21 "School of Mathematics" }}{PARA 264 "" 0 "" {TEXT -1 12 "G eorgia Tech" }}{PARA 265 "" 0 "" {TEXT -1 21 "herod@math.gatech.edu" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 "This wo rksheet presents the criteria for a function F to be conservative, as \+ well as implications of this condition. First we define these terms. " }}{PARA 0 "" 0 "" {TEXT -1 6 "If F: " }{XPPEDIT 18 0 "R^3" "6#*$%\"R G\"\"$" }{TEXT -1 4 " -> " }{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" } {TEXT -1 27 " is defined on the region " }{XPPEDIT 18 0 "Omega" "6#%& OmegaG" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" } {TEXT -1 29 " and there is a function g: " }{XPPEDIT 18 0 "R^3" "6#*$ %\"RG\"\"$" }{TEXT -1 17 " -> R such that " }{XPPEDIT 18 0 "grad(g)" "6#-%%gradG6#%\"gG" }{TEXT -1 30 " = F, then we say that F is a " } {TEXT 256 18 "conservative field" }{TEXT -1 17 " and that g is a " } {TEXT 257 15 "potential for F" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 " In the computation of gradients, we use the linear algebr a package. As preparation for understanding this worksheet, recall the definition of the " }{XPPEDIT 18 0 "gradient" "6#%)gradientG" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "'grad(g,[x,y,z])'=[Diff(g, x),Diff(g,y),Diff(g,z)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 10 "Exa mple 1:" }{TEXT -1 68 " Let F(x,y,z) = [ y z, x z, x y ]. This is a co nservative field for " }{XPPEDIT 18 0 "grad(g)" "6#-%%gradG6#%\"gG" } {TEXT -1 28 " = F where g(x,y,z) = x y z." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "grad(x*y*z, [x,y,z]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 191 "Having F to be conservat ive -- to be the gradient of g -- is important in computing the line i ntegral. Here is an explanation of the connection between line integra ls and conservative fields." }}{PARA 0 "" 0 "" {TEXT -1 20 " Recal l that if " }{XPPEDIT 18 0 "grad(g)" "6#-%%gradG6#%\"gG" }{TEXT -1 20 " = F, and if r(t): " }{XPPEDIT 18 0 "R" "6#%\"RG" }{TEXT -1 2 "->" } {XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" }{TEXT -1 66 " is a differentiabl e function defined on the interval [a, b], then" }}{PARA 0 "" 0 "" {TEXT -1 19 " " }{XPPEDIT 18 0 "diff(g(r(t)),t)" "6# -%%diffG6$-%\"gG6#-%\"rG6#%\"tGF," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "g rad(g)(r(t))" "6#--%%gradG6#%\"gG6#-%\"rG6#%\"tG" }{TEXT -1 2 " " } {XPPEDIT 18 0 "diff(r(t),t)" "6#-%%diffG6$-%\"rG6#%\"tGF)" }{TEXT -1 20 " = F(r(t)) r '(t)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 45 "Thus, if we integrate from t = a to t = b," }} {PARA 0 "" 0 "" {TEXT -1 13 " " }{XPPEDIT 18 0 "int(F(r(t) ),r)" "6#-%$intG6$-%\"FG6#-%\"rG6#%\"tGF*" }{TEXT -1 22 " = g(r(b)) - g(r(a))." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 80 " A connection between conservative fields and line integrals ov er closed curves:" }}{PARA 0 "" 0 "" {TEXT -1 87 "Hypothesis: Suppose \+ that F is a conservative field and r(t) describes a closed curve C." } }{PARA 0 "" 0 "" {TEXT -1 28 "Conclusion: If follows that " }{XPPEDIT 18 0 "int(F,r)" "6#-%$intG6$%\"FG%\"rG" }{TEXT -1 55 " = 0, where the \+ integal is taken over the closed curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 10 "Example 2:" }{TEXT -1 149 " The fun ction F(x,y,z) = [y, -x, 0] is not a conservative field for take the c urve to be the circle with radius 1 in the x-y plane. We will see that " }{XPPEDIT 18 0 "int(F,r)" "6#-%$intG6$%\"FG%\"rG" }{TEXT -1 3 " = \+ " }{XPPEDIT 18 0 "int(y,x)" "6#-%$intG6$%\"yG%\"xG" }{TEXT -1 3 " - " }{XPPEDIT 18 0 "int(x,y)" "6#-%$intG6$%\"xG%\"yG" }{XPPEDIT 18 0 "` ` \+ <> ` `;" "6#0%\"~GF$" }{TEXT -1 0 "" }{TEXT -1 19 " 0 over that curve. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "Int(dotprod([sin(t),-co s(t)],[diff(cos(t),t),diff(sin(t),t)]),t=0..2*Pi)\n= int(dotprod([sin( t),-cos(t)],[diff(cos(t),t),diff(sin(t),t)]),t=0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Summary." }}{PARA 0 "" 0 "" {TEXT -1 102 "(1) The definition for F being a conservative field is t hat there is a function g so that grad(g) = f." }}{PARA 0 "" 0 "" {TEXT -1 54 "(2) If F is conservative and C is a closed path, then " } {XPPEDIT 18 0 "int(F,r)" "6#-%$intG6$%\"FG%\"rG" }{TEXT -1 41 " = 0, w here the integral is taken over C." }}{PARA 0 "" 0 "" {TEXT -1 76 "We \+ next give two criteria for determining that a function F is conservati ve." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Is F conservative? Resul t 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 8 "T heorem:" }{TEXT -1 34 " Let F be continuous in a region " }{XPPEDIT 18 0 "Omega" "6#%&OmegaG" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "R^3" "6#* $%\"RG\"\"$" }{TEXT -1 24 ". These are equivalent:" }}{PARA 0 "" 0 " " {TEXT -1 31 "(a) F is a conservative field." }}{PARA 0 "" 0 "" {TEXT -1 22 "(b) The line integral " }{XPPEDIT 18 0 "int(F(x),x)" "6#- %$intG6$-%\"FG6#%\"xGF)" }{TEXT -1 28 " is independent of the path." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 246 " Th is is a nice result. We have argued that if F is a conservative field \+ then the integral around a closed path is zero. That would show that s tatement (a) implies statement (b). To get from statement (b) to state ment (a) is more interesting!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 260 21 "Example 1 revisited: " }{TEXT -1 140 "We have seen that with F(x,y,z) = [ y z, x z, x y ], we have a conservat ive field. It should be that the integral around any closed path in " }{XPPEDIT 18 0 "R^3" "6#*$%\"RG\"\"$" }{TEXT -1 67 " is zero. Take, fo r this example, the unit circle in the x-y plane." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 380 "F:=(x,y,z)->[y*z,x*z,x*y];\nx:=t->cos(t); y: =t->sin(t); z:=t->0;\nInt('F(x(t),y(t),z(t))[1]'*'D(x)(t)',t=0..2*Pi) \n + Int('F(x(t),y(t),z(t))[2]'*'D(y)(t)',t=0..2*Pi)\n + \+ Int('F(x(t),y(t),z(t))[3]'*'D(z)(t)',t=0..2*Pi) =\nint(F(x(t),y(t),z(t ))[1]*D(x)(t),t=0..2*Pi)\n + int(F(x(t),y(t),z(t))[2]*D(y)(t),t=0. .2*Pi)\n + int(F(x(t),y(t),z(t))[3]*D(z)(t),t=0..2*Pi);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 21 "Example 2 revisited: " }{TEXT -1 162 "We have seen that the integr al over a closed curve would not necessarily be zero for this example. We show, another way, that there could be no function g so that" }} {PARA 0 "" 0 "" {TEXT -1 12 " " }{XPPEDIT 18 0 "diff(g,x) = y" "6#/-%%diffG6$%\"gG%\"xG%\"yG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "diff(g,y" "6#-%%diffG6$%\"gG%\"yG" }{TEXT -1 7 " = -x. " }}{PARA 0 "" 0 "" {TEXT -1 91 "If there were such a g, then because mixed partia l would be equal, we have a contradiction:" }}{PARA 0 "" 0 "" {TEXT -1 27 " -1 = " }{XPPEDIT 18 0 "diff(g,y,x" "6#-%% diffG6%%\"gG%\"yG%\"xG" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "diff(g,x,y) " "6#-%%diffG6%%\"gG%\"xG%\"yG" }{TEXT -1 5 " = 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "Is F conservative ? Result 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 9 "Theorem: " }{TEXT -1 158 "Let F(x,y,z) = [P(x,y,z), Q(x,y,z), R (x,y,z)] and suppose that P, Q, and R, together with their first parti al derivatives are continuous in any simple region " }{XPPEDIT 18 0 "O mega" "6#%&OmegaG" }{TEXT -1 23 ". These are equivalent:" }}{PARA 0 " " 0 "" {TEXT -1 33 "(a) F is a conservative field in " }{XPPEDIT 18 0 "Omega" "6#%&OmegaG" }{TEXT -1 5 ", and" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "diff(P,y)" "6#-%%diffG6$%\"PG%\"yG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "diff(Q,x)" "6#-%%diffG6$%\"QG%\"xG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "diff(P,z)" "6#-%%diffG6$%\"PG%\"zG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "diff(R,x)" "6#-%%diffG6$%\"RG%\"xG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "diff(Q,z)" "6#-%%diffG6$%\"QG%\"zG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "diff(R,y)" "6#-%%diffG6$%\"RG%\"yG" }{TEXT -1 5 " in " }{XPPEDIT 18 0 "Omega" "6#%&OmegaG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 " Thi s is a good result, for it provides a simple way to check a function t o see if it is conservative." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 21 "Example 1, revisited:" }{TEXT -1 68 " With F(x ,y,z) = [ y z, x z, x y ], we check all the mixed partials." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "x:='x': y:='y': z:='z':\nF:=(x,y,z )->[y*z,x*z,x*y];\ndiff(F(x,y,z)[1],y),diff(F(x,y,z)[2],x);\ndiff(F(x, y,z)[1],z),diff(F(x,y,z)[3],x);\ndiff(F(x,y,z)[3],y),diff(F(x,y,z)[2], z);" }}}{PARA 259 "" 0 "" {TEXT -1 21 "Example 2, revisited:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "F:=(x,y,z)->[y,-x,0];\ndiff (F(x,y,z)[1],y),diff(F(x,y,z)[2],x);\ndiff(F(x,y,z)[1],z),diff(F(x,y,z )[3],x);\ndiff(F(x,y,z)[3],y),diff(F(x,y,z)[2],z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Finding the Potential." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 324 " It is important \+ to know how to find the potential for a conservative field. The techni ques are standard. In fact, they are so straight forward, they can be \+ programmed! The student will find a way to derive the potential and re concile the result with these programmed results. We use the linear al gebra package in Maple." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "w ith(linalg):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 29 "Example 1, yet one more time:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 62 "F:=(x,y,z)->[y*z,x*z,x*y];\npotential(F(x,y,z),[x,y ,z],'g');\ng;" }}}{PARA 0 "" 0 "" {TEXT 262 28 "Example 2, yet one mor e time" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "F: =(x,y,z)->[y,-x,0];\npotential(F(x,y,z),[x,y,z],'g');" }}}{PARA 257 " " 0 "" {TEXT -1 10 "Example 3:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "F:=(x,y,z)->[exp(y+2*z),x*exp(y+2*z),2*x*exp(y+2*z)];\npotential (F(x,y,z),[x,y,z],'g');\ng;" }}}{PARA 258 "" 0 "" {TEXT -1 10 "Example 4:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "F:=(x,y,z)->[exp(x)* sin(y)+2*y,exp(x)*cos(y)+2*x-2*y,0];\npotential(F(x,y),[x,y,z],'g');\n g;\ngrad(exp(x)*sin(y)+2*x*y-y^2,[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Exercise for the student" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Determine which of the f ollowing functions F is a conservative field. If possible, find a pote ntial g for F." }}{PARA 0 "" 0 "" {TEXT -1 4 "(1) " }{XPPEDIT 18 0 "F( x,y)=[x-y*cos(x),-sin(x)]" "6#/-%\"FG6$%\"xG%\"yG7$,&F'\"\"\"*&F(F+-%$ cosG6#F'F+!\"\",$-%$sinG6#F'F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "(2) " }{XPPEDIT 18 0 "F(x,y,z)=[x^2+y^2+1,-x*y+y]" "6#/-% \"FG6%%\"xG%\"yG%\"zG7$,(*$F'\"\"#\"\"\"*$F(F-F.F.F.,&*&F'F.F(F.!\"\"F (F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "(3) " }{XPPEDIT 18 0 "F(x,y)=[x,x-y]" "6#/-%\"FG6$%\"xG%\"yG7$F',&F'\"\"\"F(!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "(4) " }{XPPEDIT 18 0 "F(x ,y,z)=e^(y+2*z)*[1,x,2*x]" "6#/-%\"FG6%%\"xG%\"yG%\"zG*&)%\"eG,&F(\"\" \"*&\"\"#F.F)F.F.F.7%F.F'*&F0F.F'F.F." }{TEXT -1 1 "." }}}}{MARK "0 0 " 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }