{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 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 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 19 "" 0 "" {TEXT -1 16 "Dr. Wei-Chi Yang" }} {PARA 19 "" 0 "" {TEXT -1 18 "Radford University" }}{PARA 19 "" 0 "" {TEXT -1 21 "www.radford.edu/wyang" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "with(plots):with(linalg): with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "S1:=(x,y,z)->x^2+y^2+z^2-1;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 35 "S2:=(x,y,z)->x^2+(y-3)^2+(z-1)^2-1;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "S3:=(x,y,z)->z-(x^2+y^2)-2; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "S4:=(x,y,z)->(4*(x-3)+( y-3)+(z-1))*(x-3)+((x-3)+4*(y-3)+(z-1))*(y-3)+((x-3)+(y-3)+4*(z-1))*(z -1)-3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "F:=(s11,s12,s13,s21,s22,s2 3,s31,s32,s33,s41,s42,s43)->(s11-s21)^2+(s11-s31)^2+(s11-s41)^2+(s12-s 22)^2+(s12-s32)^2+(s12-s42)^2+(s13-s23)^2+(s13-s33)^2+(s13-s43)^2;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "gradF:=grad(F(s11,s12,s13, s21,s22,s23,s31,s32,s33,s41,s42,s43),[s11,s12,s13,s21,s22,s23,s31,s32, s33,s41,s42,s43]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "We check th e condition of Theorem 4 is met below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 732 "diff(F(s11,s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s 43),s11)+diff(F(s11,s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s43),s12)+ diff(F(s11,s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s43),s13)+diff(F(s1 1,s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s43),s21)+diff(F(s11,s12,s13 ,s21,s22,s23,s31,s32,s33,s41,s42,s43),s22)+diff(F(s11,s12,s13,s21,s22, s23,s31,s32,s33,s41,s42,s43),s23)+diff(F(s11,s12,s13,s21,s22,s23,s31,s 32,s33,s41,s42,s43),s31)+diff(F(s11,s12,s13,s21,s22,s23,s31,s32,s33,s4 1,s42,s43),s32)+diff(F(s11,s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s43 ),s33)+diff(F(s11,s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s43),s41)+di ff(F(s11,s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s43),s42)+diff(F(s11, s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s43),s43);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "implicitplot3d(\{S2(x,y,z)=0,S1(x,y,z)=0,S 3(x,y,z)=0,S4(x,y,z)=0\},x=-3..4,y=-2..5,z=-3..5,axes=boxed,scaling=co nstrained,numpoints=5000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "graph1:=implicitplot3d(S1(x,y,z)=0,x=-3..4,y=-2..5,z=-3..5,axes=b oxed,scaling=constrained,style=patchnogrid,lightmodel=light2,glossines s=1,transparency=0.1,numpoints=5000,color=yellow):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "graph2:=implicitplot3d(S2(x,y,z)=0,x=-3..4 ,y=-2..5,z=-3..5,axes=boxed,scaling=constrained,style=patchnogrid,ligh tmodel=light2,glossiness=1,transparency=0.1,numpoints=5000,color=blue) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "graph3:=implicitplot3 d(S3(x,y,z)=0,x=-3..4,y=-2..5,z=-3..5,axes=boxed,scaling=constrained,s tyle=patchnogrid,lightmodel=light2,glossiness=1,transparency=0.1,numpo ints=5000,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "g raph4:=implicitplot3d(S4(x,y,z)=0,x=-3..4,y=-2..5,z=-3..5,axes=boxed,s caling=constrained,style=patchnogrid,lightmodel=light2,glossiness=1,tr ansparency=0.1,numpoints=5000,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "display(\{graph1,graph2,graph3,graph4\});" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 26 "U sing Lagrange Multipliers" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "L:=(s11,s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s43,L1,L2,L3,L4)- >(s11-s21)^2+(s11-s31)^2+(s11-s41)^2+(s12-s22)^2+(s12-s32)^2+(s12-s42) ^2+(s13-s23)^2+(s13-s33)^2+(s13-s43)^2+L1*S1(s11,s12,s13)+L2*S2(s21,s2 2,s23)+L3*S3(s31,s32,s33)+L4*S4(s41,s42,s43,s44);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "diff(L(s11,s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s43,L1,L2,L3,L4 ),s11);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eq1:=diff(L(s11, s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s43,L1,L2,L3,L4),s11)=0;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eq2:=diff(L(s11,s12,s13,s21, s22,s23,s31,s32,s33,s41,s42,s43,L1,L2,L3,L4),s12)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eq3:=diff(L(s11,s12,s13,s21,s22,s23,s31,s 32,s33,s41,s42,s43,L1,L2,L3,L4),s13)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eq4:=diff(L(s11,s12,s13,s21,s22,s23,s31,s32,s33,s41,s 42,s43,L1,L2,L3,L4),s21)=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "eq5:=diff(L(s11,s12,s13,s21, s22,s23,s31,s32,s33,s41,s42,s43,L1,L2,L3,L4),s22)=0;;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eq6:=diff(L(s11,s12,s13,s21,s22,s23 ,s31,s32,s33,s41,s42,s43,L1,L2,L3,L4),s23)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eq7:=diff(L(s11,s12,s13,s21,s22,s23,s31,s32,s33, s41,s42,s43,L1,L2,L3,L4),s31)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eq8:=diff(L(s11,s12,s13,s21,s22,s23,s31,s32,s33,s41,s 42,s43,L1,L2,L3,L4),s32)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eq9:=diff(L(s11,s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s43,L1,L2, L3,L4),s33)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "eq10:=dif f(L(s11,s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s43,L1,L2,L3,L4),s41)= 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "eq11:=diff(L(s11,s12, s13,s21,s22,s23,s31,s32,s33,s41,s42,s43,L1,L2,L3,L4),s42)=0;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "eq12:=diff(L(s11,s12,s13,s21 ,s22,s23,s31,s32,s33,s41,s42,s43,L1,L2,L3,L4),s43)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eq13:=diff(L(s11,s12,s13,s21,s22,s2 3,s31,s32,s33,s41,s42,s43,L1,L2,L3,L4),L1)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eq14:=diff(L(s11,s12,s13,s21,s22,s23,s31,s32,s33 ,s41,s42,s43,L1,L2,L3,L4),L2)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eq15:=diff(L(s11,s12,s13,s21,s22,s23,s31,s32,s33,s41, s42,s43,L1,L2,L3,L4),L3)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "eq16:=diff(L(s11,s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s43,L1,L2 ,L3,L4),L4)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "fsolve( \{eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8,eq9,eq10,eq11,eq12,eq13,eq14,eq15,eq 16\},\{s11,s12,s13,s21,s22,s23,s31,s32,s33,s41,s42,s43,L1,L2,L3,L4\}); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "fsolve(\{eq1,eq2,eq3,e q4,eq5,eq6,eq7,eq8,eq9,eq10,eq11,eq12,eq13,eq14,eq15,eq16\},\{s11,s12, s13,s21,s22,s23,s31,s32,s33,s41,s42,s43,L1,L2,L3,L4\},s12=0..2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 257 6 "Case 1" }{TEXT -1 203 ". This is t he case that shows Lagrange Multiplier Method gives only a necessary c ondition when optimizing a function. In this case, we get the solution which does not point to the minimum total distance." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "with(linalg): with(student):with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "L1:=-10.98816726;L2:=-5 .092601153;L3:=-5.121411430;L4:=1.107790413;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "s13:= -.5418543061;s22:= 3.923034632;s11:= -.31 89240652;s12:= -.7776125973;s21:=.7792698416e-1;s23:= 1.376741893;s31: = -.5209975982e-1;s32:= -.1270315852;s33:= 2.018851409;s41:= 2.5217915 52;s42:= 2.415696446;s43:=.9328295271;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "gradS1:=grad(S1(x,y,z),[x,y,z]);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 32 "gradS2:=grad(S2(x,y,z),[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "gradS3:=grad(S3(x,y,z),[x,y,z]);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "gradS4:=grad(S4(x,y,z),[x, y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "S1S2:=vector([s21 -s11,s22-s12,s23-s13]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " S1S3:=vector([s31-s11,s32-s12,s33-s13]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "S1S4:=vector([s41-s11,s42-s12,s43-s13]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "N1:=subs(x=s11,y=s12,z=s13,op(gradS 1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "N2:=subs(x=s21,y=s2 2,z=s23,op(gradS2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "N3: =subs(x=s31,y=s32,z=s33,op(gradS3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "N4:=subs(x=s41,y=s42,z=s43,op(gradS4));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalm(L1*N1+L2*N2+L3*N3+L4*N4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 189 "6*s11-2*s21-2*s31-2*s41;6*s12-2*s2 2-2*s32-2*s42;6*s13-2*s23-2*s33-2*s43;-2*s11+2*s21;-2*s12+2*s22;-2*s13 +2*s23;-2*s11+2*s31;-2*s12+2*s32;-2*s13+2*s33;-2*s11+2*s41;-2*s12+2*s4 2;-2*s13+2*s43;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "S1S2vec: =spacecurve(evalm(X0+t*S1S2),t=0..0.8,color=red,thickness=3):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "S1S3vec:=spacecurve(evalm(X0 +t*S1S3),t=0..1.2,color=red,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 69 "S1S4vec:=spacecurve(evalm(X0+t*S1S4),t=0..1.2,color =red,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with( plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "display(graph1,graph2,graph3,graph4,S1S2vec,S1S3 vec,S1S4vec,axes=boxed,scaling=constrained);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 7 "Case 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "with(linalg): with(student):with(plots):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 208 "s33:= 2.042652198;s41:= 2.387696012;s42:=2.4863128 30;s43:=1.099333105;s11:=.3966581137;s12:= .7009795128;s13:=.592697278 1;s21:=.1674881704;s22:= 2.029242761;s23:=.8280171630;s31:=.1017095600 ;s32:=.1797424919;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "gradS 1:=grad(S1(x,y,z),[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "S1S2:=vector([s21-s11,s22-s12,s23-s13]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "S1S3:=vector([s31-s11,s32-s12,s33-s13]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "S1S4:=vector([s41-s11,s42-s1 2,s43-s13]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "gradS2:=gra d(S2(x,y,z),[x,y,z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "x0 :=s11;y0:=s12;z0:=s13;X0:=vector([x0,y0,z0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "N:=subs(x=x0,y=y0,z=z0,op(gradS1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Nvec:=spacecurve(evalm(X0+t*N),t=-2..1,color=red,thic kness=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "S1S2vec:=space curve(evalm(X0+t*S1S2),t=0..1,color=red,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "S1S3vec:=spacecurve(evalm(X0+t*S1S3 ),t=0..1.2,color=red,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "S1S4vec:=spacecurve(evalm(X0+t*S1S4),t=0..1.2,color=r ed,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Sum:=sp acecurve(evalm(X0+t*(S1S2+S1S3+S1S4)),t=0..1.2,color=black,thickness=3 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "display(graph1,graph2,graph3,graph4,S1S2vec,S1S3vec,S 1S4vec,Sum,axes=boxed,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "display(graph1,graph2,graph3,graph4,S1S2vec,S1S3v ec,S1S4vec,Sum,Nvec,axes=boxed,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "87" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }