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}{CSTYLE "_cstyle298" -1 555 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle281" -1 556 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle355" -1 557 "Times" 1 14 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle85" -1 558 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle283" -1 559 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle37" -1 560 "Times" 1 14 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle71" -1 561 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle183" -1 562 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle363" -1 563 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {PSTYLE "" -1 340 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 218 "" 0 "" {TEXT 406 19 "Lab 7 April 2, 2008" }} {PARA 256 "" 0 "" {TEXT 267 39 "Cylindrical and spherical coordinates, " }}{PARA 256 "" 0 "" {TEXT 267 30 "Vector Fields, Gradient Fields" } }{PARA 256 "" 0 "" {TEXT 267 34 "Line Integrals of scalar functions" } }{PARA 273 "" 0 "" {TEXT 564 0 "" }}}{EXCHG {PARA 273 "> " 0 "" {MPLTEXT 1 414 20 "restart:with(plots):" }}}{SECT 1 {PARA 339 "" 0 "" {TEXT 242 33 "Vector Fields and Gradient Fields" }}{PARA 273 "" 0 "" {TEXT 564 0 "" }}{SECT 1 {PARA 311 "" 0 "" {TEXT 324 26 "Example 1: \+ Vector Fields" }}{PARA 292 "" 0 "" {TEXT 335 69 "MAPLE can plot vector fields for both two and three dimensions using " }{TEXT 447 10 "field plot " }{TEXT 335 50 "commands. You may wish to use the arrows option: " }}{PARA 292 "" 0 "" {TEXT 335 51 "arrows = THICK, SLIM, or LINE. T he default is THIN." }}{PARA 292 "" 0 "" {TEXT 565 0 "" }}{PARA 273 "" 0 "" {TEXT 518 28 "(a) Plot the vector field " }{XPPEDIT 2 0 "F(x, \+ y) = `<,>`(2*x+5*y, 4*x-3*y);" "6#/-I\"FG6\"6$I\"xGF&I\"yGF&-I$<,>GF&6 $,&*&\"\"#\"\"\"F(F0F0*&\"\"&F0F)F0F06#,&*&\"\"%F0F(F0F0*&\"\"$F0F)F0! \"\"" }{TEXT 518 1 " " }{TEXT 518 1 " " }}{PARA 273 "" 0 "" {TEXT 518 28 "(b) Plot the vector field " }{XPPEDIT 2 0 "F(x, y, z) = `<,>`(1, y, 4);" "6#/-I\"FG6\"6%I\"xGF&I\"yGF&I\"zGF&-I$<,>GF&6$\"\"\"6$F)\"\" %" }{TEXT 518 1 " " }{TEXT 518 1 " " }}{PARA 273 "" 0 "" {TEXT 564 0 " " }}{EXCHG {PARA 273 "> " 0 "" {MPLTEXT 1 414 45 "fieldplot([2*x+5*y,4 *x-3*y],x=-5..5,y=-5..5);" }}}{EXCHG {PARA 273 "> " 0 "" {MPLTEXT 1 414 45 "fieldplot3d([1,y,4],x=-2..2,y=-2..2,z=-2..2);" }}}}{SECT 1 {PARA 205 "" 0 "" {TEXT 566 27 "Example 2: Gradient Fields" }}{PARA 326 "" 0 "" {TEXT 567 0 "" }}{PARA 326 "" 0 "" {TEXT 276 5 "Maple" } {TEXT 18 51 " will compute the gradient of a function using the " } {TEXT 276 4 "grad" }{TEXT 18 31 " command, which is part of the " } {TEXT 276 6 "linalg" }{TEXT 18 67 " package. (Don't worry about the w arnings generated when you load " }{TEXT 276 6 "linalg" }{TEXT 18 3 ". ) " }}{PARA 326 "" 0 "" {TEXT 567 0 "" }}{PARA 326 "" 0 "" {TEXT 455 9 "Example: " }{TEXT 567 60 " Find the gradient and the gradient field for the function " }{XPPEDIT 2 0 "f(x, y) = x^2+y;" "6#/-I\"fG6\"6$I \"xGF&I\"yGF&,&*$)F(\"\"#\"\"\"F.F)F." }{TEXT 2 1 " " }{TEXT 2 1 " " } {TEXT 2 1 " " }{TEXT 2 1 " " }{TEXT 2 1 " " }{TEXT 212 1 " " }}{PARA 326 "" 0 "" {TEXT 567 0 "" }}{EXCHG {PARA 326 "> " 0 "" {MPLTEXT 1 289 22 "restart: with(linalg):" }}{PARA 326 "> " 0 "" {MPLTEXT 1 289 19 "grad(x^2+y, [x,y]);" }}}{PARA 326 "" 0 "" {TEXT 567 0 "" }}{PARA 326 "" 0 "" {TEXT 567 13 "The argument " }{TEXT 276 5 "[x,y]" }{TEXT 567 7 " tells " }{TEXT 276 5 "Maple" }{TEXT 567 19 " that you want the " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT 18 1 " " }{TEXT 18 1 " " } {TEXT 18 1 " " }{TEXT 18 1 " " }{TEXT 18 1 " " }{TEXT 18 1 " " }{TEXT 18 23 "-partial derivative of " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 18 1 " " }{TEXT 18 1 " " }{TEXT 18 1 " " }{TEXT 18 1 " " }{TEXT 18 1 " " }{TEXT 18 1 " " }{TEXT 18 33 " in the first component, and the " } {XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT 18 1 " " }{TEXT 18 1 " " }{TEXT 18 1 " " }{TEXT 18 1 " " }{TEXT 18 1 " " }{TEXT 18 1 " " }{TEXT 18 36 "-partial derivative in the second. " }}{PARA 326 "" 0 "" {TEXT 567 0 "" }}{PARA 326 "" 0 "" {TEXT 567 12 "The command " }{TEXT 276 8 "gra dplot" }{TEXT 567 151 " draws the gradient field of the given function . In other words, it plots the gradient vectors of the function at eac h of several points in the plane. " }}{PARA 326 "" 0 "" {TEXT 567 1 " \+ " }}{PARA 326 "" 0 "" {TEXT 567 0 "" }}{EXCHG {PARA 326 "> " 0 "" {MPLTEXT 1 289 12 "with(plots):" }}{PARA 326 "> " 0 "" {MPLTEXT 1 289 36 "setoptions(thickness=2, color=blue):" }}{PARA 326 "> " 0 "" {MPLTEXT 1 289 62 "gradplot(x^2+y, x=-2..2, y=-2..2, grid=[10,10], arr ows=slim); " }}}{PARA 340 "" 0 "" {TEXT 429 16 "It is important " } {TEXT 292 11 "to realize " }{TEXT 292 5 "that " }{TEXT 452 5 "Maple" } {TEXT 292 285 " scales the gradient plot so that the gradient vectors \+ won't overlap each other. For example, the vector at the point (1,1) \+ in the plot above should be [2,1], but it is clearly drawn smaller tha n that. The relative sizes of the vectors (and their directions!) are preserved, however." }}{EXCHG {PARA 320 "" 0 "" {TEXT 568 0 "" }}} {EXCHG {PARA 326 "" 0 "" {TEXT 335 46 "We can also plot the gradient v ector field of " }{TEXT 225 2 "f " }{TEXT 335 34 "together with the co ntour plot of " }{TEXT 225 1 "f" }{TEXT 335 47 " and observe how they \+ are related to each other" }}{PARA 326 "" 0 "" {TEXT 567 0 "" }}} {EXCHG {PARA 326 "> " 0 "" {MPLTEXT 1 289 39 "p1:=contourplot(x^2+y,x= -2..2,y=-2..2):" }}}{EXCHG {PARA 326 "> " 0 "" {MPLTEXT 1 289 36 "p2:= gradplot(x^2+y,x=-2..2,y=-2..2):" }}}{EXCHG {PARA 326 "> " 0 "" {MPLTEXT 1 289 17 "display([p1,p2]);" }}}{EXCHG {PARA 326 "" 0 "" {TEXT 335 128 "Note that the gradient vectors are always perpendicular to the level curves and the vectors are longer when the level curves \+ are" }}{PARA 326 "" 0 "" {TEXT 335 151 "closer to each other and short when they are far apart. This is because the lenght of the gradient f ield is the value of the directional derivative of " }{TEXT 225 1 "f" }{TEXT 335 45 " and close level curves indicate steep graph." }}{PARA 326 "" 0 "" {TEXT 567 0 "" }}{PARA 326 "" 0 "" {TEXT 335 82 "We can al so plot the gradient field together with the level curves and the surf ace" }}{PARA 326 "" 0 "" {TEXT 567 0 "" }}}{EXCHG {PARA 292 "> " 0 "" {MPLTEXT 1 531 28 "with(plots):with(plottools):" }}}{EXCHG {PARA 292 " > " 0 "" {MPLTEXT 1 531 54 "p := plot3d(x^2+y,x=-2..2,y=-2..2,style=pa tchcontour):" }}}{EXCHG {PARA 292 "> " 0 "" {MPLTEXT 1 531 52 "q := co ntourplot(x^2+y,x=-2..2,y=-2..2,thickness=2):" }}}{EXCHG {PARA 292 "> \+ " 0 "" {MPLTEXT 1 531 74 "r := gradplot(x^2+y,x=-2..2,y=-2..2,arrows=s lim,color=black,grid=[10,10]):" }}}{EXCHG {PARA 292 "> " 0 "" {MPLTEXT 1 531 33 "f := transform((x,y) -> [x,y,0]):" }}}{EXCHG {PARA 292 "> " 0 "" {MPLTEXT 1 531 36 "display(\{p,f(q),f(r)\},axes=boxed);" }}} {EXCHG {PARA 292 "" 0 "" {TEXT 335 148 "Note how the gradient vectors \+ are longer at points corresponding to a steepest surface and how they \+ always point in the direction of maximum ascent." }}}}}{SECT 1 {PARA 339 "" 0 "" {TEXT 343 9 "Exercises" }}{SECT 1 {PARA 311 "" 0 "" {TEXT 324 13 "Exercise 1: " }{TEXT 520 22 "Plotting Vector Fields" }}{PARA 299 "" 0 "" {TEXT 432 25 "Plot the vector fields: " }}{PARA 299 "" 0 "" {TEXT 432 27 "(a) Plot the vector field " }{XPPEDIT 2 0 "F(x, y) = `<,>`(x, -y);" "6#/-I\"FG6\"6$I\"xGF&I\"yGF&-I$<,>GF&6$F(6#,$F)!\"\"" }{TEXT 348 1 " " }{TEXT 348 1 " " }{TEXT 432 1 " " }{TEXT 432 2 " ." }{TEXT 432 2 " " }}{PARA 299 "" 0 "" {TEXT 432 26 "(b) Plot the vect or field" }{TEXT 348 1 " " }{XPPEDIT 2 0 "F(x, y) = `<,>`(y^2-2*x*y, 3 *x*y-6*x^2);" "6#/-I\"FG6\"6$I\"xGF&I\"yGF&-I$<,>GF&6$,&*$)F)\"\"#\"\" \"F1*(F0F1F(F1F)F1!\"\"6#,&*(\"\"$F1F(F1F)F1F1*&\"\"'F1*$)F(F0F1F1F3" }{TEXT 348 1 " " }{TEXT 348 1 " " }{TEXT 432 28 " . On which set of po ints is" }{TEXT 311 1 " " }{TEXT 545 1 "F" }{TEXT 348 1 "(" }{TEXT 545 1 "x" }{TEXT 348 1 "," }{TEXT 545 1 "y" }{TEXT 348 9 ") = <0,0>" } {TEXT 348 1 " " }{TEXT 432 1 "?" }}{PARA 299 "" 0 "" {TEXT 569 0 "" }} {PARA 299 "" 0 "" {TEXT 348 1 " " }}{EXCHG {PARA 273 "> " 0 "" {TEXT 564 0 "" }}}{EXCHG {PARA 273 "> " 0 "" {TEXT 564 0 "" }}}{EXCHG {PARA 273 "> " 0 "" {TEXT 564 0 "" }}}{EXCHG {PARA 273 "> " 0 "" {TEXT 564 0 "" }}}{EXCHG {PARA 273 "> " 0 "" {TEXT 564 0 "" }}}{EXCHG {PARA 273 "> " 0 "" {TEXT 564 0 "" }}}{PARA 273 "" 0 "" {TEXT 518 4 " " }}}} {SECT 1 {PARA 313 "" 0 "" {TEXT 339 25 "Line Integrals of Scalars" }} {SECT 0 {PARA 292 "" 0 "" {TEXT 553 8 "Example " }}{PARA 0 "" 0 "" {TEXT 317 21 "Plot the spiral helix" }{TEXT 317 1 " " }{TEXT 317 1 " " }{XPPEDIT 2 0 "r(t) = `<,>`(t*cos(t), t*sin(t), t);" "6#/-I\"rG6\"6#I \"tGF&-I$<,>GF&6$*&F(\"\"\"-I$cosGF&F'F-6$*&F(F--I$sinGF&F'F-F(" } {TEXT 2 1 " " }{TEXT 2 1 " " }{TEXT 305 1 " " }{TEXT 317 16 " for 0 < t < " }{XPPEDIT 2 0 "4*Pi;" "6#*&\"\"%\"\"\"I#PiG%*protectedGF%" } {TEXT 2 1 " " }{TEXT 2 1 " " }{TEXT 305 1 " " }{TEXT 317 48 " and com pute the line integral of the function " }}{PARA 334 "" 0 "" {TEXT 265 1 " " }{XPPEDIT 2 0 "f(x, y, z) = z(2+x^2+y^2);" "6#/-I\"fG6\"6%I \"xGF&I\"yGF&I\"zGF&-F*6#,(\"\"#\"\"\"*$)F(F.F/F/*$)F)F.F/F/" }{TEXT 2 1 " " }{TEXT 388 1 " " }{TEXT 2 1 " " }{TEXT 2 1 " " }{TEXT 2 1 " " }{TEXT 317 56 "over this portion of the spiral helix. We rewrite the " }{XPPEDIT 2 0 "f(x, y, z);" "6#-I\"fG6\"6%I\"xGF%I\"yGF%I\"zGF%" } {TEXT 2 1 " " }{TEXT 388 1 " " }{TEXT 2 1 " " }{TEXT 317 115 " functio n in parametric form and calculate the length of the r'(t) funtion jus t on the side by hand. Confirm that " }{XPPEDIT 2 0 "f(x(t), y(t), z( t)) = 2*t+t^3;" "6#/-I\"fG6\"6%-I\"xGF&6#I\"tGF&-I\"yGF&F*-I\"zGF&F*,& *&\"\"#\"\"\"F+F3F3*$)F+\"\"$F3F3" }{TEXT 2 1 " " }{TEXT 388 1 " " } {TEXT 388 1 " " }{TEXT 317 6 " and " }}{PARA 0 "" 0 "" {TEXT 317 3 "r '(" }{TEXT 496 1 "t" }{TEXT 317 4 ") = " }{XPPEDIT 2 0 "`<,>`(cost-t*s int, sint+t*cost, 1);" "6#-I$<,>G6\"6$,&I%costGF%\"\"\"*&I\"tGF%F)I%si ntGF%F)!\"\"6$,&F,F)*&F+F)F(F)F)F)" }{TEXT 2 1 " " }{TEXT 2 1 " " } {TEXT 305 1 " " }{TEXT 317 17 " , so |r'(t)| = " }{XPPEDIT 2 0 "sqrt( 2+t^2);" "6#-I%sqrtG6\"6#,&\"\"#\"\"\"*$)I\"tGF%F(F)F)" }{TEXT 2 1 " " }{TEXT 2 1 " " }{TEXT 2 1 " " }{TEXT 2 1 " " }{TEXT 2 1 " " }{TEXT 317 1 "." }}{PARA 292 "" 0 "" {TEXT 565 0 "" }}{EXCHG {PARA 292 "> " 0 "" {MPLTEXT 1 531 105 "spacecurve([t*cos(t),t*sin(t),t],t=0..4*Pi, nu mpoints=96, axes=normal, orientation=[30,50], thickness=2);" }}} {EXCHG {PARA 292 "> " 0 "" {MPLTEXT 1 531 46 "Int((2*t+t^3)*sqrt(2+t^2 ),t=0..4*Pi);evalf(%);" }}}}}{SECT 0 {PARA 313 "" 0 "" {TEXT 570 9 "Ex ercise " }{TEXT 339 2 " " }{TEXT 327 25 "Line integrals of scalars" } }{PARA 0 "" 0 "" {TEXT 335 10 "Exercise 2" }}{EXCHG {PARA 320 "" 0 "" {TEXT 335 42 "Compute the line integral of the function " }{XPPEDIT 2 0 "f(x, y, z) = x*exp(yz);" "6#/-I\"fG6\"6%I\"xGF&I\"yGF&I\"zGF&*&F(\" \"\"-I$expGF&6#I#yzGF&F," }{TEXT 2 1 " " }{TEXT 2 1 " " }{TEXT 489 1 " " }{TEXT 335 100 " over the line segment (0,6,-1) to (4,1,5). Write \+ the line segment in parametric form, rewrite the " }{TEXT 438 1 "f" } {TEXT 335 1 "(" }{TEXT 438 1 "x" }{TEXT 335 2 ", " }{TEXT 438 1 "y" } {TEXT 335 2 ", " }{TEXT 438 1 "z" }{TEXT 335 99 ") function in param etric form and calculate the length of the r'(t) funtion just below by hand. " }}}{EXCHG {PARA 292 "" 0 "" {TEXT 565 0 "" }}}{EXCHG {PARA 320 "> " 0 "" {TEXT 568 0 "" }}}{EXCHG {PARA 320 "> " 0 "" {TEXT 568 0 "" }}}{EXCHG {PARA 320 "> " 0 "" {TEXT 568 0 "" }}}}{PARA 268 "" 0 " " {TEXT 571 0 "" }}{PARA 278 "" 0 "" {TEXT 572 0 "" }}{PARA 336 "" 0 " " {TEXT 573 0 "" }}{PARA 222 "" 0 "" {TEXT 574 0 "" }}{PARA 209 "" 0 " " {TEXT 575 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }