{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 0 2 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 "" 3 256 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 "" 3 257 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 258 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 259 1 {CSTYLE "" -1 -1 "" 1 12 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 256 "" 0 "" {TEXT -1 20 "Flux Across Surfaces" }}{PARA 257 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA 258 "" 0 "" {TEXT -1 21 "Sc hool of Mathematics" }}{PARA 259 "" 0 "" {TEXT 256 21 "herod@math.gate ch.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 " Let S be an oriented surface embedded in a three dimensional vector field F. The flux of the field across S in the direction of th e normal " }{XPPEDIT 18 0 "eta" "6#%$etaG" }{TEXT -1 24 " is given by \+ the formula" }}{PARA 0 "" 0 "" {TEXT -1 55 " \+ Flux = DoubleIntegral < F, " }{XPPEDIT 18 0 "eta" "6#%$etaG" }{TEXT -1 5 " > d " }{XPPEDIT 18 0 "sigma" "6#%&sigmaG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 178 "In this \+ worksheet, we take one field and compute the flux across three surface s all having a common boundary. The reader is asked to repeat the comp utation for a fourth surface. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 73 "This worksheet will use the plots package and the linear algebra package." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(plots): with(linalg):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "Four Surfaces." }}{PARA 0 "" 0 " " {TEXT -1 67 "First we draw all four surfaces. Each is drawn separat ely.\nA disk:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "plot3d([r,t heta,0],r=0..1,theta=0..2*Pi,coords=cylindrical,\n axes=NORMAL,c olor=BLUE);" }}}{PARA 0 "" 0 "" {TEXT -1 24 "The northern hemisphere. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "plot3d([r,theta,sqrt(1- r^2)],r=0..1,theta=0..2*Pi,coords=cylindrical,\n axes=NORMAL,col or=RED);" }}}{PARA 0 "" 0 "" {TEXT -1 17 "Hills and Valleys" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "plot3d([r,theta,cos(theta)*sin(Pi* r)],r=0..1,theta=0..2*Pi,\n coords=cylindrical,axes=NORMAL,color= GREEN);" }}}{PARA 0 "" 0 "" {TEXT -1 35 "A Revere bowl: (turned upside down)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "plot3d([r,theta,1- r^2],r=0..1,theta=0..2*Pi,coords=cylindrical,\n axes=NORMAL,colo r=GREY);" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "The Field." }}{PARA 0 "" 0 "" {TEXT -1 28 "We present a u niform field ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "fieldplot3 d([-1,3,2],x=-1..1,y=-1..1,z=-1..1,grid=[3,3,3],\n axes=NORMAL,col or=BLACK);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "Flux over the disk" }}{PARA 0 "" 0 "" {TEXT -1 73 "Convince yourself that a uni t normal to the surface is [0, 0, 1] and that" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "`flux`=Int(Int(dotprod([F1,F2,F3],[0,0,1]),\n \+ y=-sqrt(1-x^2)..sqrt(1-x^2)),x=-1..1);" }}}{PARA 0 "" 0 "" {TEXT -1 40 "In this case, as in those which follow, " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "F:=(x,y,z)->[-1,3,2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "Int(Int(dotprod(F(x,y,z),[0,0,1]),\n y=- sqrt(1-x^2)..sqrt(1-x^2)),x=-1..1)=\n int(int(dotprod(F(x,y,z),[0,0 ,1]),\n y=-sqrt(1-x^2)..sqrt(1-x^2)),x=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 "Flux over the Northern Hemisphe re" }}{PARA 0 "" 0 "" {TEXT -1 54 " Convince yourself that a norma l to the surface is" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "S:=(x ,y)->[x,y,sqrt(1-x^2-y^2)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Sx:=map(diff,S(x,y,z),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Sy:=map(diff,S(x,y,z),y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "N:=crossprod(Sx,Sy);" }}}{PARA 0 "" 0 "" {TEXT -1 171 "Maple's computation of the dotproduct of the Field with N was so complicated \+ that I summed the product of components myself. Here is the sum of the three components added." }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "F(x,y,z)[1]*N[1]+F(x,y,z)[2]*N[2]+F(x,y,z)[3]*N[3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "int(int(%,\n y=-s qrt(1-x^2)..sqrt(1-x^2)),x=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Flux over Hills and Valleys" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 35 "assume(r,real); assume(theta,real);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "S:=(r,theta)->[r*cos(theta), r*sin(theta),cos(theta)*sin(Pi*r)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Sr:=map(diff,S(r,theta),r);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "St:=map(diff,S(r,theta),theta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "N:=crossprod(Sr,St);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dotprod([-1,3,2],N);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "int(int(dotprod([-1,3,2],N),r=0..1) ,theta=0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "E xercise for the Student" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "This is your problem: find the flux over the Revere \+ Bowl drawn above." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }