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fluid flow across any closed cur ve--the diference between the total fluid moving out and the total flu id coming in. If there's a sink, total fluid flow should be negative \+ across any closed curve enclosing that sink only, and if there's a sou rce the total fluid flow should be positive across any closed curve en closing that source only." }}{PARA 0 "" 0 "" {TEXT -1 136 "Here we are measuring the divergence of the negative gradient field; for the dive rgence of the gradient vector field, reverse the signs." }}{PARA 0 "" 0 "" {TEXT -1 33 "Note that the curl is still zero." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "curl([-x,-y,0],[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 616 "Level curves are also another way to view potential func tions. In the above paraboloid--the graph of the potential function-- the circles represent points that are at the same height (potential). \+ If we project these down to the plane, we get the level curves, curve s on which the potential is constant. These level curves are orthogon al to the gradient (or the negative gradient) vector field. This is b ecause water will find the shortest/most efficient path to flow from h igher points to lower points; it will not stay at the same height but \+ flow down in the direction perpendicular to points at the same height. " }}{PARA 0 "" 0 "" {TEXT -1 277 "Becuase the level curves are orthogo nal to the gradient vector field, we can easily find them by rotating \+ the gradient (or the negative gradient) vector field by 90 degrees. T his rotated vector field is called the Hamiltonian vector field, and i ts orbits are the level curves." }}{PARA 0 "" 0 "" {TEXT -1 63 "So in \+ the above example the Hamiltonian vector field is ." }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 58 "fieldplot([y,-x],x=-2..2,y=-2..2,arrows=SLIM ,color=black);" }}{PARA 13 "" 1 "" {GLPLOT2D 449 451 451 {PLOTDATA 2 " 6$-%'CURVESG6]dl7'7$$!3V+-D2xcD>!#<$!3M*z\\FHKW2#F*7$F+F(7$$!3@E2P\"4% fB?F*$!3*QPaAw*=R>F*7$$!3)eiXxB531#F*$!3et#H'3fSw>F*F-7'7$$!3wE2Y\"RT] r\"F*$!3C5cc9tfm?F*7$$!3!fKgp(f!R'=F*$!3)**QMao-M$>F*7$$!3:+t7?l-:=F*$ !3(3wl`4KO%>F*7$$!3/0,Tx^K[=F*$!3wg1uT#[3)>F*F>7'7$$!34`7nv]^/:F*$!3q? 9QOBwe?F*7$$!3A_3F*7$$!3ktQ))[*ekg\"F*$!3%y9x%G W2[>F*7$$!3*Reuq6Sej\"F*$!3_Z?&[d!H&)>F*FS7'7$$!3Uz<))f())RH\"F*$!3;Js >et#40#F*7$$!3by8QXL&GW\"F*$!3%)oF!=ks!\\>F*7$$!3MZ/kx8*yR\"F*$!3#[`)e hn^_>F*7$$!3:j!Rn0bLU\"F*$!3sMM'z!Ht*)>F*Fbo7'7$$!3u0B4WCY$3\"F*$!3hTI ,!Q#4V?F*7$$!3)[!>fHqKK7F*$!3;ep)*>w!p&>F*7$$!31@qR1QK*=\"F*$!3!=#**p% 4fp&>F*7$$!34UNS'**p3@\"F*$!3Y@[2T_<%*>F*Fgp7'7$$!3#=KGIGh$H()!#=$!3^_ )G=Sd_.#F*7$$!3?JC!Qr+=-\"F*$!3sZ6<)fUZ'>F*7$$!3!y%f`^Bc2)*Fjq$!3+48\" yU,9'>F*7$$!3M5-og$\\Q)**Fjq$!3m3i=uvh)*>F*F]r7'7$$!34&eL^7)4CmFjq$!3' HmWOUAu-#F*7$$!3@u&H,)Ru7\")Fjq$!3/P`Nwvds>F*7$$!3#Qo,\"Rm)=s(Fjq$!3w& pA4wVe'>F*7$$!3k)*\\Kd()**eyFjq$!3k&f(H2*fI+#F*Fbs7'7$$!3P[)Qs'\\$)=XF jq$!3Ut/YXue>?F*7$$!3[P[BA3[2gFjq$!3OE&RXb7/)>F*7$$!3'4Unm#4@OcFjq$!3u #3MS4'Gq>F*7$$!30)ypR:[Tt&Fjq$!3<#)*3/C-v+#F*Fgt7'7$$!3k6TM4=d8CFjq$!3 K%GwsY_<,#F*7$$!3w+,Mkw@-RFjq$!3!frBF`Z#))>F*7$$!3adJB9_`]NFjq$!3qpa9F %GZ(>F*7$$!3-yXh]vH4OFjq$!3gp._tX%>,#F*F\\v7'7$$!3yZP\\9l3$3$!#>$!3x%4 #4*[F*7$$!37%*))z,&f[Y\"Fjq$!3o coDg2F*7$$!3sn$fs%pW%[\"Fjq$!38cF*7$$\"3hC%e 4cOSS'F_w$!3bVJuR#H3-#F*Fex7'7$$\"3a)4SVm\\1FFjq$!3'3jzkUb!))>F*7$$\"3i_5XfUDlFFjq$!3_IX&Gdr_-#F *Ffy7'7$$\"3EN[BA3[2gFjqFjt7$$\"39Y)Qs'\\$)=XFjqFet7$$\"3e&*Q]Nw;#z%Fj q$!3Q<5ffx\\#*>F*7$$\"3Aji!G'[5!*[Fjq$!30dH,)R u7\")FjqFes7$$\"3(GeL^7)4CmFjqF`s7$$\"3We\"QzMVy(oFjq$!3e/Cq#4Sp*>F*7$ $\"3Dt9;ma&\\,(Fjq$!3./t2Ri:M?F*Fh[l7'7$$\"3)4V-Qr+=-\"F*F`r7$$\"3g>$G IGh$H()FjqF[r7$$\"3I@CPg!>N'*)Fjq$!3y\"z8eU#Q,?F*7$$\"3%Qo;&pg!)R\"*Fj q$!3X\"p)=s&)fQ?F*Fi\\l7'7$$\"3m/>fHqKK7F*Fjp7$$\"3_0B4WCY$3\"F*Fep7$$ \"3Io1Gx%>\\5\"F*$!3Jy^#*eZ#e+#F*7$$\"3L*=(GncYE6F*$!3)z2+`!4/V?F*Fj]l 7'7$$\"35y8QXL&GW\"F*Feo7$$\"3Uz<))f())RH\"F*F`o7$$\"3\"[4C&[q[88F*$!3 Gll.#4n-,#F*7$$\"3h5FiF2&*Q8F*$!3'\\Y6%QK[Z?F*F[_l7'7$$\"3+_3Y0A:F*$!3E_z9D%4Z,#F*7$$\"3AJ#ezyN9 b\"F*$!3Q_G_rb#>0#F*F\\`l7'7$$\"3mD.'p(f!R'=F*FA7$$\"3aE2Y\"RT]r\"F*F< 7$$\"3RZ4,\">A1t\"F*$!3oR$f#e<:>?F*7$$\"3G_PH[3#Rw\"F*$!3ORUj/zOc?F*F] al7'7$$\"3M*z\\FHKW2#F*$!3l+-D2xcD>F*7$$\"3@+-D2xcD>F*F+7$$\"3ntVDi(*= R>F*F/7$$\"3Mt#H'3fSw>F*F4F`bl7'7$FAF?7$FA1t\"F*7$F hal$!3__PH[3#Rw\"F*F[cl7'7$$!3J;\\kpj(Gs\"F*$!3MOhx)*42c=F*7$FfclFdcl7 $$!3&H\"f,(=%e5=F*$!3fMB7CX1Nf?gB[=F*7$$!3oim)HoWbk\"F*$!3j0\"HyM62t\"F*7 $$!3m'[sdh;?g\"F*$!3c@PBdo]RjRy(RJ;F*$!3YEl^9b!Gx\"F*Fidl7' 7$$!3'*of1QP#=I\"F*$!3[dxSU5SS=F*7$$!3+*=(>n$=]V\"F*$!3=&H8gKY&Q\\Ru\"F*7$$!3&fnFOs7*=9F*$!3m8ziZyCx/8QY%RM!)Fjq$!3Ojecg 70i\\#Fjq$!3U5o[^hA,=F*7$$!3U1#)\\#)y'Q#QFj q$!3CUU$p@@xx\"F*7$$!3B(G>J)=61NFjq$!3UV?!f&3;mU([c$Fjq$! 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