{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 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 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 4 257 1 {CSTYLE "" -1 -1 "" 1 18 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 256 "" 0 "" {TEXT -1 23 "Vector Valued Functions " }}{PARA 256 "" 0 "" {TEXT -1 12 "January 2009" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "In t his tutorial we experiment with the Vector Calculus package. We also \+ include the Plots package for 3d graphs." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "with(VectorCalculus);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "The list above shows all the commands in the VectorCalcu lus Package. Along with (hopefully) familiar maple commands we discus s\n the " }{TEXT 266 16 "bracket notation" }{TEXT -1 37 " to define v ector-valued functions\n " }{TEXT 267 10 "SpaceCurve" }{TEXT -1 14 " \+ for graphs\n " }{TEXT 268 5 "diff " }{TEXT -1 18 "for derivatives\n \+ " }{TEXT 269 10 "DotProduct" }{TEXT -1 4 " \n " }{TEXT 270 13 "CrossP roduct " }{TEXT -1 3 "\n " }{TEXT 271 4 "Norm" }{TEXT -1 4 " \n " } {TEXT 272 13 "TangentVector" }{TEXT -1 4 " \n " }{TEXT 273 15 "Princi palNormal" }{TEXT -1 3 "\n " }{TEXT 274 8 "Binormal" }{TEXT -1 5 " \+ \n " }{TEXT 275 9 "ArcLength" }{TEXT -1 3 "\n " }{TEXT 276 9 "Curvat ure" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 6 "Basics" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "We use the curve " }{XPPEDIT 18 0 "r = `<,>`(sin(t), cos(t), exp(t)/(1+exp(t)));" "6#/%\"rG-%$<,>G6%-%$sinG6#%\"tG-%$cosG6# F+*&-%$expG6#F+\"\"\",&F3F3-F16#F+F3!\"\"" }{TEXT -1 132 " as an \+ example throughout this worksheet and define it once and for all.. Us ing a bracket-notation, as we did in class, we set" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 42 "r := < cos(t), sin(t), exp(t)/(1+exp(t))>;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Let's look at a graph first." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "SpaceCurve(r,t=-20..20,axes=normal, numpoints=500);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Derivatives ca n be calculated using " }{TEXT 256 4 "diff" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "v := diff(r,t); # velocity\nv := simplify(v );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a := diff(r,t,t); # \+ acceleration\na := simplify(a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 257 "" 0 "" {TEXT -1 21 "Two Dimesional Curv es" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "R := ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "SpaceCurve(R,t=0..2* Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "diff(R,t);\ndiff(R, t,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Tangent Lines" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "The TangentLine command allows us to obtain a tangent at \+ a desired location." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "TL := Tangen tLine(r,t=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Let's look at a \+ graph:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "with(plots):\nG := space curve(r,t=-20..20,axes=normal,numpoints=2000):\nTANLINE := spacecurve( TL,t=-1..1,axes=normal,numpoints=2000,color=blue, thickness=2):\ndispl ay(G,TANLINE);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "We already calc ulated the velocity (derivative) vector" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "v;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The same vector, in \+ different notation, can be compute with the " }{TEXT 257 13 "TangentVe ctor" }{TEXT -1 9 " command." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "TV \+ := TangentVector(r,t): TV := simplify(TV);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "We use the norm command to calculate the length of a vect or. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Norm(TV);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "We see that the TangentVector is " }{TEXT 263 18 "not a unit vector." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "Tangent, Normal and Binormal " }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Unit Tangent, Unit Normal" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "The unit tangent vector is messy ( due to the contribution from the thrid component)" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "T := v/Norm(v); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Norm(T); # just checking" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 118 "Let's construct the normal vector, using the definitio n from the text. We try to suppress a lot of messy output here!" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "dT := simplify( diff(T,t) ): # ta ke derivative\nN := simplify( dT/Norm(dT) ); # normalize to unit vect or\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Okay, it's still a mess. \+ But the norm is correct:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Norm(N) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Directly from maple we can c alculate the normal vector like this" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "PN := PrincipalNormal(r,t): PN := simplify(PN);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 46 "BUT, this vector is not normalized to le ngth 1" }{TEXT -1 41 ". Let's test the norm of the last vector" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "PNN := simplify(Norm(PN));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot(PNN,t=-20..20); # it's practically one, but not near zero." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Vectors at a specific point" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 182 "At this point we calculated the unit tangent vectors and the unit normal vectors. These quantities are VERY elaborate quantit ies changing with t. Let's study the scenario at t=0. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "r0 := subs(t=0,r); # position\nv0 \+ := subs(t=0,v); # velocity vector\nNorm(v0); # speed \na0 := subs(t=0,a); # acceleration vector\nT0 := subs(t=0,T); # unit tangent \nN0 := subs(t=0,N); # unit normal " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Looks like maple has trouble taking the square root of 16. Let 's try again:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "T0 := simplify(T0) ;" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "Components of Acceleration " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "For the " }{TEXT 265 27 "compon ents of acceleration " }{TEXT -1 7 "we find" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "AT := DotProduct(a,T); # Tangential Component of acce leration" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "AN := simplify( DotProduct(a,N) ); # Normal Component of accelaration" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Let's put it to the test:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "simplify( AT*T + AN*N);\na;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "It works. At t=0 we have" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "AT0 := DotProduct(a0,T0); AT0 := simplify(AT0);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "AN0 := DotProduct(a0,N0); AN 0 := simplify(AN0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a0; N 0;" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Binormals" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 264 8 "Binormal" }{TEXT -1 47 " fro m the definition is constructed as follows:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "B := CrossProduct(T,N): \nB := simplify(B);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Let's check orthogonalities and no rms with the dot product:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "simpl ify(DotProduct(T,T));\nsimplify(DotProduct(T,N));\nsimplify(DotProduct (T,B));\nsimplify(DotProduct(N,N));\nsimplify(DotProduct(N,B));\nsimpl ify(DotProduct(B,B));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 16 "Maple's Binormal" }{TEXT -1 17 " vector is again " }{TEXT 261 21 "not set to \+ length one" }{TEXT -1 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "BN := Binormal(r,t):\nBnn := Norm(BN):\nBnn := simplify(Bnn);\nplot(Bnn,t=- 20..20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Arc Length" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Now we calculate the length of a curve. " }{TEXT 258 9 " ArcLength" }{TEXT -1 20 " will do the trick. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ArcLength(r,t=0..2*Pi);\nvalue(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The integral is too hard. We use a numerical e stimate:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 " The curve is stretched a little bit, and the result is slightly bigger than " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\" #\"\"\"%#PiGF%" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Curvature" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Finally, we look at " }{TEXT 262 9 "curvature" }{TEXT -1 37 ". \+ The calculation is straightforward" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Curvature(r,t): kappa := simplify(%);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "plot(kappa,t=-3..3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 238 "For large |t| we are almost on a circle of radius one, h ence the curvature is about one as well. Near t=0 there is a bit of a stretch and the curvature drops a little. \nUsing the definitions in \+ the text, we can confirm the results abouve" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "kappa2 := Norm( CrossProduct(v,a) )/Norm(v)^3:\nkappa 2 := simplify(kappa2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s implify(kappa - kappa2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(%, t=-20..20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 1 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }