{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 "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {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 "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "#In this lab we wil l learn how to do several integrations in 1, 2, and 3 manifolds using \+ Maple. First, observe that the integration command is int();" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "int(1/(x^2+1), x=Pi/2..infin ity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%'arctanG6#,$*&\"\"#!\"\"% #PiG\"\"\"F,F**&F)F*F+F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "#Maple makes a difference between \"Path Integrals\" (Integrals o f scalar fields) and \"Line Integrals\" (Integrals of vector fields." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "#Let us take problem #3 \+ from our last exam, to integrate the differential form (2xy+2)dx+ (x^2 +3)dy along the curvey =x(x^2+1)(3-x)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "with(Student[VectorCalculus]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7^o%#&xG%\"*G%\"+G%\"-G%\".G%$<,>G%$<|gr>G%&AboutG%*Arc LengthG%,BasisFormatG%)BinormalG%.ConvertVectorG%-CrossProductG%%CurlG %*CurvatureG%\"DG%$DelG%0DirectionalDiffG%+DivergenceG%+DotProductG%)F lowLineG%%FluxG%/GetCoordinatesG%1GetPVDescriptionG%-GetRootPointG%)Ge tSpaceG%)GradientG%(HessianG%1IsPositionVectorG%/IsRootedVectorG%.IsVe ctorFieldG%)JacobianG%*LaplacianG%(LineIntG%+MapToBasisG%&NablaG%%Norm G%*NormalizeG%(PathIntG%3PlotPositionVectorG%+PlotVectorG%/PositionVec torG%0PrincipalNormalG%2RadiusOfCurvatureG%-RootedVectorG%0ScalarPoten tialG%/SetCoordinatesG%+SpaceCurveG%0SpaceCurveTutorG%+SurfaceIntG%)TN BFrameG%(TangentG%,TangentLineG%-TangentPlaneG%.TangentVectorG%(Torsio nG%'VectorG%,VectorFieldG%1VectorFieldTutorG%0VectorPotentialG%,Vector SpaceG%%diffG%'evalVFG%$intG%&limitG%'seriesG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "LineInt(VectorField(<2*x*y+2,x^2+3>),Path(,t=0..2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#u" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "#Let us now do problem #9" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "PathInt( x*y^2, [x,y] = Pa th( , t=0..2 ) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"$+\" \"\"\"\"#E#F&\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "#B oth commands are much more versatile. By typing ?LineInt or ?PathInt w e can see examples of other possible paths. Let us do a few more probl ems." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "#Let us compute th e line integral of the one-form (y/(x^2*y^2+1)-1)*(dx) +(x/(x^2*y^2+1 )-1)*(dy) along the line segment going from (1,1) to (2,2) to (3,1) to (4,3)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "LineInt( Vector Field( <(y/(x^2*y^2+1)-1),(x/(x^2*y^2+1)-1)> ), LineSegments( <1,1>,<2 ,2>, <3,1>, <4,3>),'output' = 'integral' );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$IntG6$,&*(\"\"#\"\"\",&F*F*%\"tGF*F*,&*$)F+\"\"%F* F*F*F*!\"\"F*F)F1/F,;\"\"!F*F*-F%6$,&*&,&F)F*F,F1F*,&*&),&F)F*F,F*F)F* )F9F)F*F*F*F*F1F**&F=F*F:F1F1F2F*-F%6$,(*&,&F*F**&F)F*F,F*F*F*,&*&),& \"\"$F*F,F*F)F*)FDF)F*F*F*F*F1F*FJF1*(F)F*FIF*FFF1F*F2F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 223 "#Let us now compute the line integ ral of the one-form x*(dx) +y*(dy) along the arc of a circle of radiu s 4 centred at the origin going from Pi/6 to Pi/5.\nLineInt( VectorFie ld( ), Arc(Circle( <0,0>, 4), Pi/6, Pi/5));\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 " #Let us now calculate the path integral of x||dx|| along the above lin e segment and the arc of the circle.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "PathInt( x, [x,y]=Line( <0,0>, <2*sqrt(3),2> )); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%\"\"\"\"\"$#F&\"\"#F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "PathInt( x, [x,y]=Arc(Circle ( <0,0>, 4), Pi/6, Pi/5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\")! \"\"*&\"#;\"\"\"-%$cosG6#,$*(\"\"$F(\"#5F%%#PiGF(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "#Let us integrate f(x,y)=xy inside \+ the triangle with vertices (0,0), (3,1), (4,4)." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "int(x*y, [x,y]=Triangle(<0,0>,<3,1>,<4,4>));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "#Suppose this time that you wanted to know how to set up the integration limits. Then you may use\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 68 "int(x*y, [x,y]=Triangle(<0,0>,<3,1>,<4,4>),'ou tput' = 'integral' );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$IntG6$ -F%6$*&%\"xG\"\"\"%\"yGF+/F,;,$*&\"\"$!\"\"F*F+F+F*/F*;\"\"!F1F+-F%6$- F%6$F)/F,;,&\"\")F2*&F1F+F*F+F+F*/F*;F1\"\"%F+" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 321 "#Let us do problem #6 from the last exam. We \+ will integrate this using Cartesian coordinates. By symmetry, it is en ough to integrate in the first quadrant and multiply by 4. Notice that we are using the order dx dy. (I use the command simplify(), because \+ otherwise, Maple produces a very complicated expression as output." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "simplify(4*int(x^2+y^2, [y ,x] = Region(0..1,sqrt(1-y^2)..sqrt(4-y^2)) )+ 4*int(x^2+y^2, [y,x] = \+ Region(1..2,0..sqrt(4-y^2)) ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$* (\"#:\"\"\"\"\"#!\"\"%#PiGF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "#Here is problem #8 from the exam, using both orders of integr ation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "int(x+y, [y,x]=Reg ion(1..2,1..(y+1)/2)) + int(x+y, [y,x]=Region(2..3,y-1..(y+1)/2));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "int(x+y, [x,y]=Region(1..2,2*x-1..x+1));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"& \"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "#Is Maple smart en ough to switch limits of integration? Let us try problem #10 from the \+ exam." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "int(exp(x^2/2), [y ,x]=Region(0..2,y..2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"!\" \"-%$expG6#\"\"#F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "#Let \+ us consider now a few triple integrals." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "int( sin(x)*cos(y)*tan(z), [x,y,z] = Parallelepiped( \+ 0..Pi, 0..Pi/2, 0..Pi/4 ), 'inert' );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-F$6$*(-%$sinG6#%\"xG\"\"\"-%$cosG6#%\"yGF/-%$tanG6#% \"zGF//F.;\"\"!%#PiG/F3;F:,$*&\"\"#!\"\"F;F/F//F7;F:,$*&\"\"%FAF;F/F/ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "int( sin(x)*cos(y)*tan( z), [x,y,z] = Parallelepiped( 0..Pi, 0..Pi/2, 0..Pi/4 ) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#Here is an integral over a tetrahedron." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "int(x,[x,y,z]=Tetrahedron(<0, 0, 0> ,<3, 2, 0>,<0, 3, 0>,<0, 0, 2>));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6## \"\"*\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "#Maple can b e used to give the limits of integration, but it is sometimes too clev er to give the answer we really want." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "int(x,[x,y,z]=Tetrahedron(<0, 0, 0>,<3, 2, 0>,<0, 3, \+ 0>,<0, 0, 2>),'inert');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\" \"\"-%$IntG6$-F(6$-F(6$%\"xG/%\"yG;,(F%F&*&F%F&%\"tGF&!\"\"*(,&F%F&*&F %F&F4F&F5F&,&\"\"$F&*&F:F&F4F&F5F5,(F.F&F:F5*&F:F&F4F&F&F&F&,(F%F&*&F% F&F4F&F5*(,&F&F5F4F&F&F9F5F " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "33 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }