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end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plugin: =proc(g,r) simplify(eval(g,\{x=r[1],y=r[2],z=r[3]\})) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Type I surface integral" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 31 "f:=(x-1/2)*(y-1/2)^2*(z-1/2)^3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "r:=[sin(v)*cos(u),sin(v)*sin(u),cos (v)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%*&-%$sinG6#%\"vG\"\" \"-%$cosG6#%\"uGF+*&F'F+-F(F.F+-F-F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "dA:=cross(diff(r,u),diff(r,v)); # pointing inward!! 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!\"\"%#PiG\"\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Total Area:" }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A:=int(int(mag(dA),u=0..2*P i),v=0..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,$*&\"\"#\"\" \"%#PiGF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Average value of \+ f:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "%% /%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"$?$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Center of mass:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "[int(int(r[1]*mag(dA),u=0..2*Pi),v =0..Pi/2)/A,int(int(r[2]*mag(dA),u=0..2*Pi),v=0..Pi/2)/A,\nint(int(r[3 ]*mag(dA),u=0..2*Pi),v=0..Pi/2)/A];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$#\"\"\"\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Moment of inertia with respect to x:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "M/A*int(int(plugin(y^2+z^ 2,r)*mag(dA),u=0..2*Pi),v=0..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*(\"\"#\"\"\"\"\"$!\"\"%\"MGF&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Type II surface integrals (switch to full sphere):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "g:=[x*(y^2+z^2),y*(x^2+z^2), z*(x^2+y^2)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG7%*&%\"xG\"\"\" ,&*$)%\"yG\"\"#F(F(*$)%\"zGF-F(F(F(*&F,F(,&*$)F'F-F(F(F.F(F(*&F0F(,&F3 F(F*F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "r:=[sin(v)*cos( u),sin(v)*sin(u),cos(v)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%* &-%$sinG6#%\"vG\"\"\"-%$cosG6#%\"uGF+*&F'F+-F(F.F+-F-F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "dA:=cross(diff(r,u),diff(r,v)); # pointing inward!!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dAG7%*&,&\"\" \"!\"\"*$)-%$cosG6#%\"vG\"\"#F(F(F(-F-6#%\"uGF(*&F'F(-%$sinGF2F(,$*&-F 6F.F(F,F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "int(int(dot( plugin(g,r),dA),u=0..2*Pi),v=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$*(\"\")\"\"\"\"\"&!\"\"%#PiGF&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Demonstrating Gauss Theorem:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "int(int(int(div(g),z=-sqrt(1-x^2-y^2)..sqrt(1-x^2-y^2 )),y=-sqrt(1-x^2)..sqrt(1-x^2)),x=-1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\")\"\"\"\"\"&!\"\"%#PiGF&F&" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 54 "To demonstrate Stokes' theorem, go back to half-sp here" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " g:=[y-z,z-x,x-y];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG7%,&%\"yG\" \"\"%\"zG!\"\",&F)F(%\"xGF*,&F,F(F'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "int(int(dot(plugin(curl(g),r),dA),u=0..2*Pi),v=0..Pi/ 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"%#PiGF&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r:=[cos(t),sin(t),0]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dr:=diff(r,t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "int(dot(plugin(g,r),dr),t=0..2*Pi); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"%#PiGF&!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "REVIEW EXERCISES:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "1)" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "r:=[u,v,u^2+v^2]: #area \+ of paraboloid" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "plot3d(r,v =-sqrt(1-u^2)..sqrt(1-u^2),u=-1..1,numpoints=1000,scaling=constrained, axes=NORMAL);" }}{PARA 13 "" 1 "" {GLPLOT3D 496 496 496 {PLOTDATA 3 "6 &-%%MESHG6#X.%)anythingG6\"6\"[gl'!%\"!!$^gw\"B\"B\"$BFF00000000000000 0000000000000003FF0000000000000BFEE000000000000BFD645640568C1C33FF0000 000000000BFEC000000000000BFDEFBDEB14F4EDA3FF0000000000000BFEA000000000 000BFE2A79E3A2CD2E63FF0000000000000BFE8000000000000BFE52A7FA9D2F8EA3FF 0000000000000BFE6000000000000BFE73CE704FB7B233FF0000000000000BFE400000 0000000BFE8FAE0C15AD38A3FF0000000000000BFE2000000000000BFEA751F9447B72 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"6#>%\"AG,$**\"\"%\"\"\"%#a|irGF()%#PiG\"\"#F(%#b|irGF(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "int(int(plugin(x^2+y^2, r)*mag(cross(diff(r,u),diff(r,v))),u=0..2*Pi),v=0..2*Pi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simplify(M*%/A);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*(\"\"#!\"\"%\"MG\"\"\",&*&F%F()%#a|irGF%F(F(*&\"\" $F()%#b|irGF%F(F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "6) Vo lume Integrals" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plugin:=proc(g,r);eval(g, \{x=r[1],y=r[2],z=r[3]\})end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "r:=[(a+rho*cos(u))*cos(v),(a+rho*cos(u))*sin(v),rho*sin(u)]:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(linalg): " }}{PARA 7 " " 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have b een redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "det(augment(diff(r,u),diff(r,v),diff(r,rho))): simplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%$rhoG\"\"\",&%\"aGF&*&F%F&-%$ cosG6#%\"uGF&F&F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "i nt(int(int(plugin(x^2+y^2,r)*rho*(a+rho*cos(u)),u=0..2*Pi),v=0..2*Pi), rho=0..b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&**\"\"#\"\"\")%\"aG\" \"$F&)%#PiGF%F&)%\"bGF%F&F&*&#F)F%F&*(F(F&F*F&)F-\"\"%F&F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "simplify(M*%/int(int(int(rho *(a+rho*cos(u)),u=0..2*Pi),v=0..2*Pi),rho=0..b));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"%!\"\"%\"MG\"\"\",&*&F%F()%\"aG\"\"#F(F(*&\"\"$ F()%\"bGF-F(F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "7) C of \+ M:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "int(int(int(x,x=0..1- y-z),y=0..1-z),z=0..1)/int(int(int(1,x=0..1-y-z),y=0..1-z),z=0..1);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "M of I w/r [1,1,1]:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "6*M*int(int(int(x^2+y^2+z^2-(x+y+z) ^2/3,x=0..1-y-z),y=0..1-z),z=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*&\"#5!\"\"%\"MG\"\"\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "8) " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r:=[ u,v,-sqrt(1-u^2-v^2)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "c ross:=proc(r,q) simplify([r[2]*q[3]-r[3]*q[2],r[3]*q[1]-r[1]*q[3],r[1] *q[2]-r[2]*q[1]]) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "m ag:=proc(q) simplify(sqrt(add(q[i]^2,i=1..3))) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "mag(cross(diff(r,u),diff(r,v))): simpli fy(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,$*&\"\"\"F&,(F&!\"\"*$)% \"uG\"\"#F&F&*$)%\"vGF,F&F&F(F(#F&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(h<1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "2*Pi*int(1/sqrt(1-rho^2)*rho,rho=0..sqrt(h*(2-h)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&%#h|irGF&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Volume: z-thickness is sqrt(1 -x^2-y^2)-(1-h)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "2*Pi*int ((sqrt(1-rho^2)-1+h)*rho,rho=0..sqrt(h*(2-h)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\"%#PiGF&,&*&#F&F%F&*$)%#h|irGF%F&F&F&*&#F &\"\"'F&*$)F-\"\"$F&F&!\"\"F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "9)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g :=[x+y,2*x-z,y+z]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Using Stoke s:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "r: =[u,v,6-3*u-2*v]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "curl: =proc(g) simplify([diff(g[3],y)-diff(g[2],z),diff(g[1],z)-diff(g[3],x) ,diff(g[2],x)-diff(g[1],y)]) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "dot:=proc(r,q) simplify(add(r[i]*q[i],i=1..3)) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "int(int(dot(plugin(curl(g),r),cross(diff(r,u),di ff(r,v))),u=0..2-2/3*v),v=0..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" #@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Directly:" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "r:=evalm([2,0,0]+[-2,3, 0]*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG-%'vectorG6#7%,&\"\"# \"\"\"*&F*F+%\"tGF+!\"\",$*&\"\"$F+F-F+F+\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "int(dot(plugin(g,r),[-2,3,0]),t=0..1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "r:=evalm([0,3,0]+[0,-3,6]*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG-%'vectorG6#7%\"\"!,&\"\"$\"\"\"*&F+F,%\"tGF,!\" \",$*&\"\"'F,F.F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "int( dot(plugin(g,r),[0,-3,6]),t=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#O" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "r:=evalm([0,0,6]+[2 ,0,-6]*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG-%'vectorG6#7%,$*& \"\"#\"\"\"%\"tGF,F,\"\"!,&\"\"'F,*&F0F,F-F,!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 38 "int(dot(plugin(g,r),[2,0,-6]),t=0..1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#!#;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "10)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "g:=[y*z,x*z,x*y]: curl(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7%\"\"!F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "r:=[3*cos( t),sin(t),0]: simplify(plugin(x^2+9*y^2-9,r));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "r:= [3*cos(t),sin(t),1-sin(t)^2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "int(dot(plugin(g,r),diff(r,t)),t=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}{MARK "117" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }