{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 32 "Vector Fields and Line In tegrals" }}{PARA 0 "" 0 "" {TEXT -1 28 "Dale Winter and Oliver Knill" }}{PARA 0 "" 0 "" {TEXT -1 7 "6/14/01" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Part 1: \+ Vector Valued Functions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f:=(x,y)->vector(2,[x+y,x-y]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 204 "T he next set of commands allow you to evaluate the vector field and vis ualize the output at the point (1,1). Adapt these commands to evaluat e and visualize the output of the vector field at other points." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "x1:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "y1:=1:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "f(x1,y1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "x2:=t->(1-t)*x1+t*(x1+f(x1,y1)[1]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "y2:=t->(1-t)*y1+t*(y1+f(x1,y 1)[2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot([x2(t),y2(t ),t=0..1],thickness=3,view=[-10..10,-10..10]);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "What are the diagrams of vector fields actually showing you?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "fieldplot([x+y,x-y],x=-2..2,y=-2..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "fieldplot3d([(x,y,z)->2*x,(x,y,z)->2*y,(x,y,z)->1] ,-1..1,-1..1,-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "List your matchings betw een graphical and symbolic representations of vector fields here. Des cribe how you could use your computer algebra system to check your ans wers." }}{PARA 0 "" 0 "" {TEXT -1 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 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Part 2: The Concept of Work in Physics" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "w ith(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "alpha:=fie ldplot3d([(x,y,z)->2,(x,y,z)->3,(x,y,z)->1],0..10,0..10,0..10):beta:=p olygonplot3d([[1,3,4],[2,7,9]],axes=boxed):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "display3d(\{alpha,beta\});" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 68 "How much work is done when the object mov es from (1,3,4) to (2,7,9)?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "F:=vector(3,[2,3,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "d:=vector(3,[?,?,?]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Work:=dotprod(F,d);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Ho w much work would be done moving along the path shown?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "alpha:=fieldplot([1,1],x=-2. .2,y=-2..2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "beta:=poly gonplot([[-1.5,1.5],[1.5,-1.5]],thickness=3):delta:=polygonplot([[1.5, -1.5],[1.5,-1.25]],thickness=3):epsilon:=polygonplot([[1.5,-1.5],[1.25 ,-1.5]],thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "di splay(\{alpha,beta\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Along which paths would the force F do no work?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "fieldplot([2,-1],x=-2..2,y=-2..2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 43 "Calculate the work along each line segmen t." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "alpha:= fieldplot([1,1],x=0..5,y=0..4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "beta:=polygonplot([[1,1],[4,1],[4,2],[3,3],[1,3],[1,1]],thick ness=3):delta:=polygonplot([[4,1],[3.8,1.2],[3.8,0.8],[4,1]],thickness =3):epsilon:=polygonplot([[1,3],[1.2,3.2],[1.2,2.8],[1,3]],thickness=3 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "display(\{alpha,beta, delta,epsilon\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dotprod([1,1],[3,0]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dotprod([1,1],[0,1]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dotprod([1,1],[-1,1]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dotprod([1,1],[-2,0]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dotprod([1,1],[0,-2]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Calculate the total work. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "dotprod ([1,1],[3,0]+[0,1]+[-1,1]+[-2,0]+[0,-2]);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Re cord and test your conjecture here." }}{PARA 0 "" 0 "" {TEXT -1 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 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Part 3: Experiments with Work and Vector Fields" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Under what conditions is it pos sible for an object to move and no work be done?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "Describe the curves (if any) along which it would be possible for an object to move and no work be done." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Vector Field 1: " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "Vector Field 2: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Vector Field 3: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Vector Fi eld 4: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Vector Field 5: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Vector Field 6: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 17 "Vector Field 7: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Vector Fi eld Used: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Describe (in words) closed curves that will give:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Large, positive wo rk:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Sm all, positive work:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 21 "Small, negative work:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Large, negative work:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Ex planations on the shape of the curve and the amount of work done." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Vector Field Used: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "Describe (in words) closed curves that wi ll give:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Large, positive work:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 21 "Small, positive work:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Small, negative work:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Large, negative wo rk:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Pa rt 4: Approximating the Work Done Along a Curved Path" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "nalpha:=plot(sqrt(1-t^2),t= 0..1,color=green,thickness=3):eta:=fieldplot([x+y,x-y],x=0..1,y=0..1): nu:=polygonplot([[0,1],[0.03,0.97],[0.03,1.03],[0,1]],thickness=3,colo r=green):display(\{nalpha,eta,nu\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 91 "Calculate the work done when the semicircular arc \+ is approximated by a single line segment." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 27 "Work:=dotprod([1,1],[?,?]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 87 "Calculate the work done when the semicircular arc \+ is apprximated by four line segments." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 5 "n:=4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "delta_x:=1/n:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=x ->sqrt(1-x^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "Work:=su m(dotprod([delta_x*k+f(delta_x*k),delta_x*k-f(delta_x*k)],[delta_x*k,f (delta_x*k)]-[delta_x*(k+1),f(delta_x*(k+1))]),k=0..n-1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(evalf(Work));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Calculate the work done when th e semi-circular arc is approximated by more than four line segments." }}{PARA 0 "" 0 "" {TEXT -1 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 "" {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 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Part 5: Line Integrals" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Evaluate the following integrals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gamma1:=t->1-t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "gamma2:=t->sqrt(1-(1-t)^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "int((gamma1(t)+gamma2(t))*diff(gamma1(t),t),t=0..1); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 52 "int((gamma1(t)-gamma2(t))*diff(gamma2(t),t),t= 0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Set up an aprroximation for the amount \+ of work done using sums." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gamma3:=t->t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "gamma4:=t->1-t^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "alpha:=fieldplot([x+y^2,x^2-y],x=0..1,y=0..1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "beta:=plot([gamma3(t),gamma4(t),t=0..1],col or=red,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "del ta:=polygonplot([[1,0],[1.04,0.08],[0.92,0.06],[1,0]],thickness=3):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "display(\{alpha,beta,delta \});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "delta_x:=1/n;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "Work:=sum((gamma3(k*delt a_x)+gamma4(k*delta_x)^2)*delta_x+(gamma3(k*delta_x)^2-gamma4(k*delta_ x))*(gamma4((k+1)*delta_x)-gamma4(k*delta_x)),k=0..n-1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(evalf(Work));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 47 "Formulate integrals to calculate the work done." } }{PARA 0 "" 0 "" {TEXT -1 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 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Part 6: Summar y" }}{PARA 0 "" 0 "" {TEXT -1 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 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "fieldplot([x/sqrt(x^2+y^2),y /sqrt(x^2+y^2)],x=-2..2,y=-2..2);" }}}{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 "" }}}}{MARK "1 0 0" 12 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }