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Use options in the Actions Preferences dialog box to control when Unformatted text is generated.\ \>", "\<\ -Graphics3D-\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Norm"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The function", StyleBox[" Norm[v] ", FontFamily->"Courier", FontWeight->"Bold"], "calculates the norm (or the length) of a vector ", StyleBox["v", FontFamily->"Courier", FontWeight->"Bold"], ".\n\nExample:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Norm[{1,2,3}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 14^(1/2)\ \>", "\<\ Sqrt[14]\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "\n", StyleBox["Example:", FontWeight->"Bold"], " Find the speed of a particle whose position function is" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["r[t_] = {t, t^2, t^3};"], "Input", AspectRatioFixed->True], Cell[TextData[{ "And find the distance travelled by the particle from time ", StyleBox["t = 0", FontFamily->"Courier", FontWeight->"Bold"], " to ", StyleBox["t = 1", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Solution:", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["speed[t_] = Norm[r'[t]]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ (1 + 4*t^2 + 9*t^4)^(1/2)\ \>", "\<\ 2 4 Sqrt[1 + 4 t + 9 t ]\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["dist = Integrate[ speed[t], {t, 0, 1} ]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 14^(1/2)/3 + (2*Integrate[(1 + 2*t^2)/ (1 + 4*t^2 + 9*t^4)^(1/2), {t, 0, 1}])/3\ \>", "\<\ 2 1 + 2 t 2 Integrate[---------------------, {t, 0, 1}] 2 4 Sqrt[14] Sqrt[1 + 4 t + 9 t ] -------- + --------------------------------------------- 3 3\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "The decimal approximation is given by the function ", StyleBox["N[]", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["N[dist]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 1.8630229825128\ \>", "\<\ 1.86302\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Pr"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The function ", StyleBox["Pr[v,w] ", FontFamily->"Courier", FontWeight->"Bold"], "computes the projection of the vector ", StyleBox["v", FontFamily->"Courier", FontWeight->"Bold"], " onto the vector ", StyleBox["w", FontFamily->"Courier", FontWeight->"Bold"], ".\n\n", StyleBox["Example:", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Pr[{1,2,3},{4,5,6}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {128/77, 160/77, 192/77}\ \>", "\<\ 128 160 192 {---, ---, ---} 77 77 77\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Cross"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The function ", StyleBox["Cross[v,w]", FontFamily->"Courier", FontWeight->"Bold"], " computes the cross product of the 3-dimensional vectors ", StyleBox["v", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["w", FontFamily->"Courier", FontWeight->"Bold"], ".\n\n", StyleBox["Example:", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Cross[{1,2,3},{4,5,6}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {-3, 6, -3}\ \>", "\<\ {-3, 6, -3}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Grad"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The function", StyleBox[" Grad[f] ", FontFamily->"Courier", FontWeight->"Bold"], "computes the gradient of a function", StyleBox[" f", FontFamily->"Courier", FontWeight->"Bold"], ". The default variables are ", StyleBox["x", FontFamily->"Courier", FontWeight->"Bold"], ", ", StyleBox["y", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["z.\n\nExample:", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Grad[ x^2 + y^2 + z^2 ]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {2*x, 2*y, 2*z}\ \>", "\<\ {2 x, 2 y, 2 z}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[ "To use other variables, just include them as a second argument to the \ function.\n(See the section on Vars for setting the default variables.)\n\n\ Example:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Grad[ u^3 - v^3, {u,v} ]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {3*u^2, -3*v^2}\ \>", "\<\ 2 2 {3 u , -3 v }\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], 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Mathematica uses this format so that the results can \ be substituted into other expressions like ", StyleBox["Discriminant[f[x,y]]", FontFamily->"Courier", FontWeight->"Bold"], ", see the following section." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "To use ", StyleBox["CriticalPoints[]", FontFamily->"Courier", FontWeight->"Bold"], " with functions of other variables, just mention those variables as a \ second argument or reset ", StyleBox["Vars", FontFamily->"Courier", FontWeight->"Bold"], " as usual.\n\n", StyleBox["Example:", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["cp = CriticalPoints[u^2 v - v^3 + v, {u, v} ]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {{u -> 0, v -> 3^(-1/2)}, {u -> 0, v -> -3^(-1/2)}, {u -> I, v -> 0}, {u -> -I, v -> 0}}\ \>", "\<\ 1 1 {{u -> 0, v -> -------}, {u -> 0, v -> -(-------)}, {u -> I, v -> 0}, Sqrt[3] Sqrt[3] {u -> -I, v -> 0}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "Answers can be converted to decimal approximations with the function ", StyleBox["N[]", FontFamily->"Courier", FontWeight->"Bold"], " :" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["N[cp]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {{u -> 0., v -> 0.5773502691896258}, {u -> 0., v -> \ -0.5773502691896258}, {u -> 1.*I, v -> 0.}, {u -> -1.*I, v -> 0.}}\ \>", "\<\ {{u -> 0., v -> 0.57735}, {u -> 0., v -> -0.57735}, {u -> 1. 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I, v -> 0.}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "You should ignore solutions which are complex numbers (those with the \ symbol ", StyleBox["I=Sqrt[-1]", FontFamily->"Courier", FontWeight->"Bold"], " in them)." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Discriminant"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The function ", StyleBox["Discriminant[f]", FontFamily->"Courier", FontWeight->"Bold"], " calculates the quantity", StyleBox[" f", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["xx", FontFamily->"Courier", FontWeight->"Bold", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" f", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["yy", FontFamily->"Courier", FontWeight->"Bold", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" - (f", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["xy", FontFamily->"Courier", FontWeight->"Bold", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[")^2 ", FontFamily->"Courier", FontWeight->"Bold"], "used in determining whether a critical point is a saddle point or a local \ max/min.\n\n", StyleBox["Example:", FontWeight->"Bold"], "" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["f[x_,y_] = x^2 - 3x y + y^3;"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["d = Discriminant[f[x,y]]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ -9 + 12*y\ \>", "\<\ -9 + 12 y\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["cp = CriticalPoints[f[x,y]]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {{x -> 9/4, y -> 3/2}, {x -> 0, y -> 0}}\ \>", "\<\ 9 3 {{x -> -, y -> -}, {x -> 0, y -> 0}} 4 2\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData["Evaluate the discriminant at the critical points:"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["d /. cp"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {9, -9}\ \>", "\<\ {9, -9}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "This shows that the second critical point", StyleBox[" {0,0} ", FontFamily->"Courier", FontWeight->"Bold"], "is a saddle point since", StyleBox[" d < 0", FontFamily->"Courier", FontWeight->"Bold"], ". The first point ", StyleBox["{9/4, 3/2} ", FontFamily->"Courier", FontWeight->"Bold"], "is either a local max or local min since", StyleBox[" d > 0", FontFamily->"Courier", FontWeight->"Bold"], ". We look at the second derivative in any direction to determine which." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["D[f[x,y],{x,2}] /. {x->9/4, y->3/2}"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ 2\ \>", "\<\ 2\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "Positive means curved up, so", StyleBox[" {9/4, 3/2} ", FontFamily->"Courier", FontWeight->"Bold"], "is a local min of ", StyleBox["f", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "To use", StyleBox[" Discriminant[] ", FontFamily->"Courier", FontWeight->"Bold"], "with functions of other variables, just mention those variables as a \ second argument or reset ", StyleBox["Vars", FontFamily->"Courier", FontWeight->"Bold"], " as usual (see the section on", StyleBox[" ", FontFamily->"Courier"], StyleBox["Vars", FontFamily->"Courier", FontWeight->"Bold"], ").\n\n", StyleBox["Example:", FontWeight->"Bold"], StyleBox["", FontFamily->"Courier", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Discriminant[ Cos[u v], {u,v} ]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ u^2*v^2*Cos[u*v]^2 - (-(u*v*Cos[u*v]) - Sin[u*v])^2\ \>", "\<\ 2 2 2 2 u v Cos[u v] - (-(u v Cos[u v]) - Sin[u v])\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Lagrange"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The function", StyleBox[" Lagrange[f, g==c] ", FontFamily->"Courier", FontWeight->"Bold"], "computes the critical points of the function ", StyleBox["f", FontFamily->"Courier", FontWeight->"Bold"], " under the constraint ", StyleBox["g == c", FontFamily->"Courier", FontWeight->"Bold"], ".\n\n", StyleBox["Example:", FontWeight->"Bold"], " Find the maximum and minimum of the function" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["f[x_,y_] = x^2 - 3 x y + y^3;"], "Input", AspectRatioFixed->True], Cell[TextData["on the unit circle."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Solution:", FontWeight->"Bold"], "" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["lp = Lagrange[f[x,y], x^2 + y^2 == 1]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {ToRules[Roots[2*lambda == -70 - 108*y + 178*y^2 + 165*y^3 - 36*y^4 + \ 54*y^5, lambda, Using -> Roots[3*x == 54 + 70*y - 129*y^2 - 106*y^3 + 21*y^4 - 36*y^5, x, Using -> Roots[-40*y^2 + 12*y^3 + 31*y^4 - 12*y^5 + 9*y^6 == -9, y]]]]}\ \ \>", "\<\ 2 3 4 \ 5 {ToRules[Roots[2 lambda == -70 - 108 y + 178 y + 165 y - 36 y + 54 y , lambda, Using -> 2 3 4 5 Roots[3 x == 54 + 70 y - 129 y - 106 y + 21 y - 36 y , x, 2 3 4 5 6 Using -> Roots[-40 y + 12 y + 31 y - 12 y + 9 y == -9, \ y]]]]}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "To get numerical approximations for this answer apply the function", StyleBox[" N[] ", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["N[lp]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ {{lambda -> -2.079213266048797, x -> -0.4379388572612773, y -> -0.899004759331485}, {lambda -> 1.862176943202197, x -> 0.866986980334528, y -> -0.4983307896672842}, {lambda -> 1.07712786382308 - 1.373674370120286*I, x -> -2.221907951905981 - 0.5926295958284982*I, y -> 0.6569674004710488 - 2.004313167689302*I}, {lambda -> 1.07712786382308 + 1.373674370120286*I, x -> -2.221907951905981 + 0.5926295958284982*I, y -> 0.6569674004710488 + 2.004313167689302*I}, {lambda -> -0.4744634494031175, x -> 0.713151197237103, y -> 0.701010249482487}, {lambda -> 2.537244044603341, x -> -0.698383416498307, y -> 0.7157238319075172}}\ \>", "\<\ {{lambda -> -2.07921, x -> -0.437939, y -> -0.899005}, {lambda -> 1.86218, x -> 0.866987, y -> -0.498331}, {lambda -> 1.07713 - 1.37367 I, x -> -2.22191 - 0.59263 I, y -> 0.656967 - 2.00431 I}, {lambda -> 1.07713 + 1.37367 I, x -> -2.22191 + 0.59263 I, y -> 0.656967 + 2.00431 I}, {lambda -> -0.474463, x -> 0.713151, y -> 0.70101}, {lambda -> 2.53724, x -> -0.698383, y -> 0.715724}}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "Evaluate", StyleBox[" f[x,y] ", FontFamily->"Courier", FontWeight->"Bold"], "at these points:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["f[x,y] /. N[lp]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {-1.715921146773415, 1.924053076699357, 4.894176672575971 - 4.101998346128415*I, 4.894176672575971 + 4.101998346128415*I, -0.6467070549222782, 2.353925483545836}\ \>", "\<\ {-1.71592, 1.92405, 4.89418 - 4.102 I, 4.89418 + 4.102 I, -0.646707, \ 2.35393}\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "The maximum value is", StyleBox[" 1.924 ", FontFamily->"Courier", FontWeight->"Bold"], "and the minimum value is", StyleBox[" -1.716 ", FontFamily->"Courier", FontWeight->"Bold"], "(we ignore the other answers since they are complex numbers)." }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[CellGroupData[{ Cell[TextData["Curl"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "The function", StyleBox[" Curl[F] ", FontFamily->"Courier", FontWeight->"Bold"], "calculates the curl of a vector-valued function", StyleBox[" F", FontFamily->"Courier", FontWeight->"Bold"], ".\n\n", StyleBox["Example:", FontWeight->"Bold"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["F[x_,y_,z_] = {x^2 y, z^3 x, z^3};"], "Input", AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Curl[F[x,y,z]]"], "Input", AspectRatioFixed->True], Cell[OutputFormData["\<\ {-3*x*z^2, 0, -x^2 + z^3}\ \>", "\<\ 2 2 3 {-3 x z , 0, -x + z }\ \>"], "Output", Evaluatable->False, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ " ", StyleBox["Curl[F[x,y,z]] ", FontFamily->"Courier", FontWeight->"Bold"], "always returns a 3-dimensional vector. 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