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"Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 291 0 "" }{TEXT 292 23 "Calculus IV with Maple\n" }{TEXT 296 32 "Copyright 2002, Dr. J ack Wagner\n" }{TEXT -1 30 "j.wagner@intelligentsearch.com" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 35 "\nLesson 1: Directional Derivatives\n" }} }{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 257 0 "" }{TEXT 258 97 "Topics : The directional derivative and non-linear tran sformations. \nMaple commands introduced: " }{TEXT 259 60 "ScalarMult iply, DirectionalDiff, Jacobian, fieldplot, evalVF" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{TEXT 284 1 "f" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "R^n;" "6#)%\"RG%\"nG" } {TEXT -1 5 " -> " }{XPPEDIT 18 0 "R^n;" "6#)%\"RG%\"nG" }{TEXT -1 7 " , the " }{TEXT 285 27 "directional derivative of f" }{TEXT -1 1 "(" } {TEXT 287 1 "x" }{TEXT -1 1 ")" }{TEXT 286 18 " in the direction " } {TEXT -1 17 "v is defined as: " }{XPPEDIT 18 0 "Limit((f(x+t*v)-f(x))/ t,t = 0);" "6#-%&LimitG6$*&,&-%\"fG6#,&%\"xG\"\"\"*&%\"tGF-%\"vGF-F-F- -F)6#F,!\"\"F-F/F3/F/\"\"!" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 51 "restart: with(LinearAlgebra): with(VectorCalculus): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 299 1 "F" }{TEXT -1 46 " be a vector-valued function o f four variables" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "F := (w , x, y, z) ->:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "F(w, x, y, z);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 297 1 "v" }{TEXT -1 57 " be a direction along which we compu te the derivative of " }{TEXT 298 1 "F" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "v := ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 300 0 "delta" "6#%&deltaG" }{TEXT -1 44 " be the difference q uotient along direction " }{TEXT 301 1 "v" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "delta := (F(w + t * a, x + t * b, y + t * c, z + t * d) - F(w, x, y, z))/t;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The directional derivativ e of " }{TEXT 302 1 "F" }{TEXT -1 7 " along " }{TEXT 303 2 "v " } {TEXT -1 31 "we find by taking the limit as " }{TEXT 304 1 "t" }{TEXT -1 11 " goes to 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "D[v]( F) = limit(delta, t = 0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 50 "This is conveniently written as a matrix \+ equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "D[v](f) = <, , , > * <>;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Observe that each row of the matrix consists of the components which result from operating on a component of " }{TEXT 305 1 "F" }{TEXT -1 37 ", with an operator v ariously called " }{TEXT 306 4 "grad" }{TEXT -1 2 ", " }{TEXT 307 3 "d el" }{TEXT -1 2 ", " }{TEXT 308 5 "nabla" }{TEXT -1 3 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "del = ; " }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "In this notation: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "D[v](f) = <> * <>;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "The matrix whose rows are the gradients of the components of a ve ctor function is known as the Jacobian." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 12 "Example 1.1\011" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Find the derivative of " } {XPPEDIT 18 0 "F(x,y) = [x, y, sin(x)+cos(y)];" "6#/-%\"FG6$%\"xG%\"yG 7%F'F(,&-%$sinG6#F'\"\"\"-%$cosG6#F(F." }{TEXT -1 62 ", at the point X := [2, 2], in the direction, v = [1.5, 1]. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "restart: w ith(LinearAlgebra): with(VectorCalculus):with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "F := (x, y) ->:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "F(x, y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "v := <1.5, 1>;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "X := + t * v;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "delta := (F(X[1], X[2]) - F(x, y))/t;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dF := limit(delta, t = 0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dF := evalf(subs(\{x = 2, y = 2\}, dF));" }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Jacobian" 2 " Jacibian" "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "r := [x, y, z]: J1 := Jacobian(F(x, y), r);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "J2 := evalf(subs(\{x = 2, y = 2\}, \+ J1));\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "Note that we needed to define all our vectors as three d imensional. Now we take the product of this matrix with v." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dF := J2.<1.5, 1, 0>;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "We obtain the same result as with the definition." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "For functions " }{TEXT 288 1 "f" }{TEXT -1 2 ": " } {XPPEDIT 18 0 "R^n;" "6#)%\"RG%\"nG" }{TEXT -1 3 " ->" }{TEXT 289 1 "R " }{TEXT -1 22 " , Maple provides the " }{TEXT 320 15 "DirectionalDiff " }{TEXT -1 10 " function." }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Di rectionalDiff" 2 "" "DirectionalDiff" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "g := (x, y) ->sin(x) + cos(y):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "g(x, y);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 55 "dg := DirectionalDiff(g(x, y), <1.5, 1, 0>, \+ [x, y, z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dg := evalf( subs(\{x = 2, y = 2\}, dg));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "N ote that this differs from the z component of " }{TEXT 309 2 "dF" } {TEXT -1 62 ". This is because the Maple function DirectionalDiff uses the " }{TEXT 290 10 "normalized" }{TEXT -1 8 " vector " }{XPPEDIT 311 0 "v/abs(v);" "6#*&%\"vG\"\"\"-%$absG6#F$!\"\"" }{TEXT -1 14 ", in place of " }{TEXT 310 1 "v" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sqrt(1.5^2 + 1^2) * dg;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "In the same manner as in single variable calculus we write the equation for the tangent lin e to the surface at point " }{TEXT 314 1 "X" }{TEXT -1 13 ", with slop e " }{TEXT 313 2 "dF" }{TEXT -1 83 ". That is, we write the vector eq uation of a line through F(2, 2) and parallel to " }{TEXT 312 2 "dF" } {TEXT -1 3 ". " }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "ScalarMultipl y" 2 "ScalarMultiply" "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 3 "dF;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "F(2 , 2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Tangent vector " }{TEXT 315 1 "T" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "T := F(2, 2) \+ + ScalarMultiply(dF, s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "Now we plot the surface, the direction ve ctor, " }{TEXT 261 1 "v" }{TEXT -1 48 ", and the tangent vector, T. F irst the surface:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "P1 := plot3d(F(x, y), x = 1..4, y = 1..4, color = pink, style = wireframe, axes = framed):\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Next the tangent vector, " }{TEXT 316 1 "T" } {TEXT -1 22 " at the point F(2, 2)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "P2 := spacecurve([T[1], T[2], T[3]], s = - 0.5..1, color = red, thickness = 3):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "Now the vector " }{TEXT 317 1 "v" } {TEXT -1 33 " translated to the point F(2, 2)." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "v1 := F(2, 2) + s.<1.5, 1, 0>;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 318 14 "ScalarMultiply" }{TEXT -1 121 " function is necessary wh en dealing with vectors written out as above. If the vector is defined as a list, then a simple " }{TEXT 319 5 "s * v" }{TEXT -1 10 " will d o. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "P3 := spacecurve([v1 [1], v1[2], v1[3]], s = - 2..2, color = black, thickness = 3):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "display(P1, P2, P3, axes = f ramed, labels = [x, y, z], scaling = constrained);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 320 " If the plot is rotated so that we are looking straight down on the x - y plane, we can see that the direction vecto r and the tangent vector are lined up with each other. Also notice th at the x and y ranges having been defined as equal the grid is square \+ as well as rectilinear.\nSuppose we complicate things just a bit." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 11 "E xample 1.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Define a function, " }{TEXT 270 2 "G " }{TEXT 295 1 "(" } {TEXT 294 7 "x, y, z" }{TEXT 293 6 ") = " }{TEXT -1 1 "[" }{TEXT 271 3 "x, " }{XPPEDIT 18 0 "y^2;" "6#*$%\"yG\"\"#" }{TEXT -1 6 ", sin( " }{TEXT 272 1 "x" }{TEXT -1 8 ") + cos(" }{XPPEDIT 18 0 "y^2;" "6#*$% \"yG\"\"#" }{TEXT -1 67 ")] , keeping everything the same except for the change from y to " }{XPPEDIT 18 0 "y^2;" "6#*$%\"yG\"\"#" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "restart:with(plo ts): with(LinearAlgebra): with(VectorCalculus):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "G := (x, y) ->:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "G(x, y);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 33 "v := <1.5, 1, 0>: r := [x, y, z]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "delta[G] := (G(x + t * v[1], y + t * v[2]) - G(x, y))/t;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "dG := limit(delta[G], t = 0);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dG := evalf(subs(\{x = 2, y = 2\}, dG));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Write the equation for the tangent line." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "T := G(2, 2) + ScalarMultiply(dG, s); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Plot " }{TEXT 263 1 " v" }{TEXT -1 35 ", the tangent line and the surface." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "P1 := plot3d(G(x, y), x = 1..4, y = 1..4, color = pink, style = wireframe):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "P2 := spacecurve([T[1], T[2], T[3]], s = - 0.5..1, c olor = red, thickness = 3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "v1 := G(2, 2) + ScalarMultiply(v, s);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 83 "P3 := spacecurve([v1[1], v1[2], v1[3]], s = - 2..2 , color = black, thickness = 3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "display(P1, P2, P3, axes = framed, labels = [x, y, z] );\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 209 "This time, when we rotat e the plot as before we find that the direction vector is not lined up with the tangent. The projection onto the x - y plane is definitely n ot square and the divisions graduate in size." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "What has actually happened here is that we have effect ed a non - linear coordinate transformation. " }{TEXT 282 1 "H" } {TEXT -1 2 ":[" }{TEXT 283 4 "x, y" }{TEXT -1 8 "] ->[x, " }{XPPEDIT 18 0 "y^2;" "6#*$%\"yG\"\"#" }{TEXT -1 148 "] This can be demonstra ted very neatly in Maple. Remember that if K and H are maps on appropr iately dimensioned spaces, the chain rule yields:\011 " }{TEXT 273 1 "D" }{TEXT -1 1 "(" }{TEXT 274 1 "K" }{TEXT -1 0 "" }{TEXT 276 1 "o" } {TEXT 275 1 "H" }{TEXT -1 6 ") = ((" }{TEXT 277 2 "DK" }{TEXT -1 1 ") " }{TEXT 278 0 "" }{TEXT -1 0 "" }{TEXT 279 1 "o" }{TEXT -1 0 "" } {TEXT 280 1 "H" }{TEXT -1 2 ")(" }{TEXT 281 2 "DH" }{TEXT -1 1 ")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "restart:with(plots): with(Li nearAlgebra):with(VectorCalculus):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "H := (x, y, z) ->(x, y^2, z); # The coordinate \+ change\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "K := (x, y, z) ->:\011# The function from example 1.1 \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "K(x, y, z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "G := K @H;\011\011\011\011\011# Function composition\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 11 "G(x, y, z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "r := [x, y, z]: J1 := Jacobian([H(x, y, z)], r);\n" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "k := K(x, y, z);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "J2 := Jacobian(k, r);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "J2 := subs(y = y^2, J2);\011 \011#compose DK with H\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " J2.J1;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "And now, directly with G, confirming the correctness of t he chain rule formula." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "g := G(x, y, z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "J3 := Ja cobian(g, r);\011\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 248 "It is the non - linear coordinate change, H, that is responsible for the non - alignment of the direction vector and the tangent. If the directiona l derivative happens to be in the direction of one of the coordinate v ectors, say the y coordinate." }}{PARA 0 "" 0 "" {TEXT -1 11 " \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "DirectionalDiff(f(w, x, y, z), <0, 0, 1, 0>, [w, x, y, z]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 266 "This is the \+ partial derivative of F with respect to y. From this we see that the \+ partial derivatives are actually directional derivatives in the direct ion of the coordinate basis vectors. The gradient is the vector whose \+ components are the coordinate derivatives. \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 0 "" }{TEXT 265 8 "Practice" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "1. Find the derivative of the following \+ functions in the direction of v at the point \nindicated. Use both the definition and the Jacobian.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "a. " }{XPPEDIT 18 0 "f(x,y) = [x^2+y^2, x^2-y^2, x*y];" "6#/-%\"fG6$ %\"xG%\"yG7%,&*$F'\"\"#\"\"\"*$F(F,F-,&*$F'F,F-*$F(F,!\"\"*&F'F-F(F-" }{TEXT -1 2 " " }{TEXT 266 1 "v" }{TEXT -1 21 " = [1, 1] p = (2, 2) " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "b. " }{XPPEDIT 18 0 "f(x,y) = [exp(-x+y), exp(x+y), x+y];" "6#/-%\"fG6$%\"xG%\"yG7%-%$expG6#,&F'!\" \"F(\"\"\"-F+6#,&F'F/F(F/,&F'F/F(F/" }{TEXT -1 5 " " }{TEXT 267 1 "v" }{TEXT -1 25 " = [.25, .05] p = (1, 2)" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 4 "c. " }{XPPEDIT 18 0 "f(x,y) = ln(x^2+y^2);" "6#/-%\"fG6 $%\"xG%\"yG-%#lnG6#,&*$F'\"\"#\"\"\"*$F(F.F/" }{TEXT -1 1 " " }{TEXT 268 1 "v" }{TEXT -1 24 " = [2.5.2.0] p = (4, 5)" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 5 "d. " }{XPPEDIT 18 0 "f(x,y) = sin(x+y)+sin(x-y); " "6#/-%\"fG6$%\"xG%\"yG,&-%$sinG6#,&F'\"\"\"F(F.F.-F+6#,&F'F.F(!\"\"F ." }{TEXT -1 3 " " }{TEXT 269 1 "v" }{TEXT -1 22 " = [1, 2] p = (2 , 4)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "0 0 2" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }