(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 19878, 651]*) (*NotebookOutlinePosition[ 27807, 828]*) (* CellTagsIndexPosition[ 26076, 791]*) (*WindowFrame->Normal*) Notebook[{ Cell["Differential Properties of Curves and Surfaces", "Title"], Cell[CellGroupData[{ Cell["Examples of Curves", "Subtitle"], Cell["Two examples of closed curves:", "Section"], Cell[TextData[{ "Here are two vectors with components given in terms of the parameter ", StyleBox["t", FontSlant->"Italic"], ":" }], "Subsubsection"], Cell[BoxData[ RowBox[{\(PrettyFlower[t_]\), " ", ":=", " ", RowBox[{\((1\/4\ + \(3\/4\) Cos[3 t])\), RowBox[{ StyleBox["{", FontSize->36], StyleBox[" ", FontSize->24], RowBox[{ StyleBox[\(Cos[t]^3\), FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontSize->72], " ", StyleBox[\(Sin[t]^3\), FontColor->RGBColor[0, 0.682353, 0.0470588]], StyleBox[",", FontSize->72], " ", StyleBox[\(Sin[t]\ Cos[t]^2\), FontColor->RGBColor[1, 0.192157, 0.192157]]}], StyleBox["}", FontSize->36]}]}]}]], "Input", CellTags->"mmtag:12:parametric_curves_example"], Cell["\<\ Both of these functions define a list of 3 positions along each \ axis in terms of a single parameter\ \>", "Text"], Cell[BoxData[ \(Bendy[t_\ ]\ := \ {\ Cos[t], \ Sin[t], \ Sin[t] Cos[t]}\)], "Input"], Cell["Defining Functions to Display Them", "Subsection"], Cell[BoxData[ \(showcurve[VecFunc_\ \ , tl_\ ]\ := \ ParametricPlot3D[Evaluate[VecFunc[tval]], {tval, 0, tl}, Compiled \[Rule] False, \ \[IndentingNewLine]DisplayFunction \[Rule] Identity, PlotRange \[Rule] {{\(-1\), 1}, {\(-1\), 1}, {\(-1\), 1}}, BoxRatios \[Rule] {1, 1, 1}]\)], "Input", CellTags->{ "mmtag:12:parametric_curves__illustrating_with_location_vector", "mmtag:12:parametric_curves__animating_with_location_vector"}], Cell[BoxData[ \(showline[VecFunc_\ , tl_\ ]\ := \ Graphics3D[{Thickness[0.01], Hue[1], Line[{{0, 0, 0}, VecFunc[tl]}]}]\)], "Input", CellTags->{"mmtag:12:Graphics3D[]", "mmtag:12:ParametricPlot3D[]__curve"}], Cell[BoxData[ \(showcurveline[VecFunc_\ , \ tl_\ ]\ := \ Show[{showcurve[VecFunc, tl], showline[VecFunc, tl]}, DisplayFunction \[Rule] $DisplayFunction]\)], "Input"], Cell[BoxData[ \(CurveLineSequence[VecFunc_\ ]\ := Table[showcurveline[VecFunc, i], {i, .1, 3\ Pi, .1}]\)], "Input", CellTags->"mmtag:12:example_animating_function"], Cell["Animating the Curves with Their Parameter", "Subsection"], Cell["\<\ The following command produces a sequence of 3D graphics objects \ that can be animated. To animate: Select all the graphics objects that are \ output; use the \"Cell\" menu to access \"Cell Grouping\" and select \ \"Open/Close Group\" (this will collapse the output to a single plot and make \ your notebook less cluttered); then use the \"Cell\" menu again and select \ \"Animate Selected Graphics.\"\ \>", "Text"], Cell[TextData[StyleBox["If you are viewing this notebook on the web, the \ animations will not appear in pdf format. See the html format for animations", FontColor->RGBColor[1, 0, 0]]], "Text", Background->GrayLevel[1]], Cell[BoxData[ \(\(CurveLineSequence[PrettyFlower];\)\)], "Input"], Cell["\<\ Do the same thing for a different parameterized vector \ function:\ \>", "Text"], Cell[BoxData[ \(\(CurveLineSequence[Bendy];\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Examples of a parameritized surface with embedded curves", "Subtitle", CellTags->"mmtag:12:ParametricPlot3D[]__example_of_parametric_surface"], Cell[BoxData[ RowBox[{\(FlowerPot[u_, \ v_]\), " ", ":=", RowBox[{ StyleBox["{", FontSize->36], RowBox[{ StyleBox[\(\((3\ + \ Cos[v])\) Cos[u]\), FontColor->RGBColor[0, 0, 1]], StyleBox[",", FontSize->72], StyleBox[\(Sin[u]\ + \ \((3\ + \ Cos[v])\) Sin[u]\), FontColor->RGBColor[0, 0.352941, 0.0156863]], StyleBox[",", FontSize->72], " ", StyleBox[\(\((3/2\ + \ Cos[u + v])\) Sin[v]\), FontColor->RGBColor[1, 0, 0]]}], StyleBox["}", FontSize->36]}]}]], "Input", CellTags->"mmtag:12:parametric_surfaces_example"], Cell[BoxData[ \(<< Graphics`ParametricPlot3D`\)], "Input"], Cell[BoxData[ \(Flowerplot\ = \ ParametricPlot3D[FlowerPot[u, v], {u, 0, 2\ Pi}, {v, 0, 2\ Pi}, ViewPoint -> {0.141, \ 1.653, \ 1.117}, PlotPoints -> {120, 40}]\)], "Input"], Cell["A Curve on a parameritized surface", "Section", CellTags->"mmtag:12:curve_on_parameritized_surface"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{\(Vines[t_]\), " ", ":=", " ", RowBox[{"1.025", "*", RowBox[{"FlowerPot", "[", RowBox[{ StyleBox[\(t\ Cos[t]\), FontSize->24, FontColor->RGBColor[1, 0, 0]], ",", StyleBox[\(\(-t^2\) Sin[\ t]\), FontSize->24, FontColor->RGBColor[0, 0, 1]]}], "]"}]}]}], "\[IndentingNewLine]", \(vineplot\ = ParametricPlot3D[Vines[t], {t, 0, \ 2\ Pi}, ViewPoint -> {0.141, \ 1.653, \ 1.117}, PlotPoints -> 500]\)}], "Input"], Cell["\<\ In the above, I want to put the vine outside the surface so I scale \ it out a little bit...\ \>", "Subsubsection"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics3D \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[BoxData[ \(thickvineplot\ = \ Show[{Graphics3D[Thickness[0.02]], Graphics3D[Hue[0.333, 0.5, 0.5]], vineplot}]\)], "Input"], Cell[BoxData[ \(Show[thickvineplot, Flowerplot]\)], "Input"], Cell[BoxData[ \(Show[Flowerplot, thickvineplot]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Arc Length and Re-parameterizing Curves", "Subtitle"], Cell[TextData[{ "Here is a general way to take a function of a general parameter, ", StyleBox["t", FontSlant->"Italic"], ", and compute the arc length traversed as ", StyleBox["t", FontSlant->"Italic"], " varies from one value to another:" }], "Text"], Cell[BoxData[ \(dFlowerDt\ = Simplify[D[PrettyFlower[t], t]]\)], "Input", CellTags->"mmtag:12:parametric_curves__derivative_along"], Cell["This is the arclength up to the parameter t", "Text"], Cell[BoxData[ \(sFlower\ = Integrate[Sqrt[Simplify[dFlowerDt . dFlowerDt]], t]\)], "Input", CellTags->{ "mmtag:12:parametric_curves__integrating_along", "mmtag:12:parametric_curves__computing_arclength"}], Cell[BoxData[ \(In\ other\ words, \ ds\^2 = \ \ \ \ dx\^2\ \ \ + \ \ \ \ dy\^2\ \ \ \ + \ \ \ \ \ \ dz\^2\ \ \ so\ integrating\ the\ square\ root\ of\ this\ is\ the\ \ arclength\)], "Text"], Cell["Applying this to the function Bendy defined above:", "Text"], Cell[BoxData[ \(dBendyDt\ = \ D[Bendy[t], t]\)], "Input"], Cell[BoxData[ \(sBendy\ = Integrate[Sqrt[dBendyDt . dBendyDt], t]\)], "Input"], Cell[TextData[{ "The arc length in this case is given by a tabulated function called an \ elliptic integral and after checking its behavior at ", StyleBox["t", FontSlant->"Italic"], " = 0 we can plot it over the range {t,0,2\[Pi]}:" }], "Text"], Cell[BoxData[ \(sBendy /. t \[Rule] 0\)], "Input"], Cell[BoxData[ \(Plot[sBendy, {t, 0, 2 Pi}]\)], "Input"], Cell["\<\ Alternatively, we can evaluate the expression for arc length \ numberically using the following:\ \>", "Text"], Cell[BoxData[ \(Plot[ Evaluate[NIntegrate[ Sqrt[dFlowerDt . dFlowerDt], {t, 0, uplim}]], {uplim, 0, 6.4}]\)], "Input", CellTags->{"mmtag:12:NIntegrate[]", "mmtag:12:NIntegrate[]_and_plotting"}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Multivariable Calculus: ", StyleBox["Mathematica", FontSlant->"Italic"], " Review" }], "Subtitle"], Cell["AScalarFunction is defined everywhere in (x,y,z)", "Text"], Cell[BoxData[ \(AScalarFunction[x_\ , \ y_\ , \ z_]\ := \ SomeFunction[x, y, z]\)], "Input"], Cell[BoxData[ \(AScalarFunction[x, y, z]\)], "Input"], Cell[TextData[{ "The following lines print ", StyleBox["and", FontVariations->{"Underline"->True}], " they define expressions." }], "Text"], Cell[BoxData[{ \(Print["\"\ ]; \ \ dFuncX\ = \ D[AScalarFunction[x, y, z], x]\), "\[IndentingNewLine]", \(Print["\"\ ]; \ \ dFuncY\ = \ D[AScalarFunction[x, y, z], y]\), "\[IndentingNewLine]", \(Print["\"\ ]; \ \ dFuncZ\ = \ D[AScalarFunction[x, y, z], z]\), "\[IndentingNewLine]", \(\)}], "Input", CellTags->{"mmtag:12:partial_derivatives", "mmtag:12:D[]"}], Cell[CellGroupData[{ Cell[TextData[{ "In the output lines above, partial derivatives are denoted by \ superscripts: e.g., (1,0,0) indicates the first partial derivative with \ respect to the first variable, ", StyleBox["x", FontSlant->"Italic"], ". The second partial with respect to y and z would be denoted by a \ superscript (0,1,1)." }], "Subsubsection"], Cell["\<\ x(w,v), y(w,v), z(w,v) is a restriction of all space to a surface \ parameterized by (w,v), AScalarFunction is now defined on the surface as a function of (w,v)\ \>", \ "Text"], Cell[BoxData[ \(AScalarFunction[x[w, v], \ y[w, v], \ z[w, v]]\)], "Input", CellTags->"mmtag:12:partial_derivatives__restricted_to_surface"], Cell[TextData[{ "Because it is now a function of ", StyleBox["w", FontSlant->"Italic"], " and ", StyleBox["v", FontSlant->"Italic"], ", the derivative with respect to ", StyleBox["x", FontSlant->"Italic"], " will vanish:" }], "Text"], Cell[BoxData[ \(D[AScalarFunction[x[w, v], y[w, v], z[w, v]], x]\)], "Input"], Cell["\<\ Two more flavors of derivatives, these are partial derivatives \ evaluated on the surface\ \>", "Text"], Cell[BoxData[ \(dFuncW\ = \ D[AScalarFunction[x[w, v], \ y[w, v], \ z[w, v]], w]\)], "Input"], Cell[BoxData[ \(dFuncV\ = \ D[AScalarFunction[x[w, v], \ y[w, v], \ z[w, v]], v]\)], "Input"], Cell["\<\ On the surface x(w,v), y(w,v), z(w,v), we can prescribe a curve \ w(t), v(t), now we have AScalarFunction defined on that curve\ \>", "Text"], Cell[BoxData[ \(AScalarFunction[x[w[t], v[t]], \ y[w[t], v[t]], \ z[w[t], v[t]]]\)], "Input"], Cell["\<\ The following is a derivative of the function along the curve \ parameterized by t\ \>", "Text"], Cell[BoxData[ \(dFuncT\ = \ D[AScalarFunction[x[w[t], v[t]], \ y[w[t], v[t]], \ z[w[t], v[t]]], t]\)], "Input", CellTags->"mmtag:12:total_derivative__example_of_curve"] }, Open ]], Cell[TextData[{ "Note on the step immediately above: by specifying ", StyleBox["w", FontSlant->"Italic"], " and ", StyleBox["v", FontSlant->"Italic"], ", values of ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " are specified, and additionally values of ", StyleBox["z", FontSlant->"Italic"], ". The three functions ", StyleBox["x", FontSlant->"Italic"], "(", StyleBox["w,v", FontSlant->"Italic"], "), ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["w,v", FontSlant->"Italic"], "), and ", StyleBox["z", FontSlant->"Italic"], "(", StyleBox["w,v", FontSlant->"Italic"], ") together describe a surface---it specifies that three points can be \ specified by two values (A familiar case is when w=x, and v=y, then z(x,y) is \ a surface that can be specified over the x-y plane. The functions ", StyleBox[" w", FontSlant->"Italic"], "(", StyleBox["t", FontSlant->"Italic"], ")", StyleBox[" ", FontSlant->"Italic"], "and ", StyleBox["v", FontSlant->"Italic"], "(", StyleBox["t", FontSlant->"Italic"], ") trace out a plane curve on the w-v surface and map onto a corresponding \ twisted curve (see Kreyszig p.429 for distinction between \"plane\" and \ \"twisted\" if it is not obvious)." }], "Subsubsection"], Cell["\<\ Finally, we could skip the surface and just define a space curve \ x(t), y(t), z(t) and take the derivative of AScalarFunc along that curve:\ \ \>", "Text"], Cell[BoxData[ \(dFuncT\ = \ D[AScalarFunction[x[t], \ y[t], z[t]], t]\)], "Input", CellTags->"mmtag:12:total_derivative__example_of_thermodynamic_variation"], Cell[BoxData[{ \(This\ last\ equation\ is\ \ the\ \(form : \ dF\/dt\) = \ \(\(\[PartialD]F\/\[PartialD]x\) dx\/dt\ + \ \(\[PartialD]F\/\[PartialD]y\) dy\/dt\ \ + \ \(\[PartialD]F\/\[PartialD]z\) dz\/dt\ \ which\ looks\ like\ a\ thermodynamic\ expression\ if\ \ we\ multiply\ both\ sides\ by\ \(dt : \ \[IndentingNewLine]\[IndentingNewLine]dF\) = \ \(\[PartialD]F\/\[PartialD]x\) dx\ + \ \(\[PartialD]F\/\[PartialD]y\) dy\ \ + \ \(\[PartialD]F\/\[PartialD]z\) dz\)\), "\[IndentingNewLine]", \(Here, \ dF\ is\ interpreted\ as\ any\ change\ in\ the\ scalar\ function\ if\ its\ \ variables\ are\ also\ changed\ dx, \ dy, \ dz\)}], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Visualizing approximations (Taylor Expansions) to surfaces at \ points\ \>", "Subtitle", CellTags->"mmtag:12:Taylor_series__visualizing_approximating_surfaces"], Cell[CellGroupData[{ Cell["\<\ Getting Mathematica to represent a general change in a \ function--multidimensional versions of Taylor expansions\ \>", "Section"], Cell[BoxData[ \(SmallChangeSeries\ = \ Expand[Series[ AScalarFunction[x + \ dx, \ y\ + \ dy, \ z\ + \ dz], {dx, 0, 1}, {dy, 0, 1}, \ {dz, 0, 1}]]\ - \ AScalarFunction[x, y, z]\)], "Input", CellTags->{"mmtag:12:Series[]", "mmtag:12:Expand[]"}], Cell[BoxData[ \(dScalarFunction\ = \ Expand[Normal[SmallChangeSeries]]\)], "Input", CellTags->{ "mmtag:12:Normal[]__turning_a_series_representation_into_an_expression", "mmtag:12:series_representation__converting_to_an_expression"}], Cell["\<\ The next step eliminates second- and third-order terms\[Ellipsis] \ (remember, dx, dy, and dz are small)\ \>", "Text"], Cell[BoxData[ \(dScalarFunction\ = \ \(\(dScalarFunction\)\(/.\)\({dx\ dy\ \[Rule] \ 0, \ dy\ dz \[Rule] 0, \ dx\ dz \[Rule] 0}\)\(\ \)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(The\ above\ form\ is\ like\ the\ thermodynamic\ \(expression : \ \[IndentingNewLine]dF\) = \ \(\[PartialD]F\/\[PartialD]x\) dx\ + \ \(\[PartialD]F\/\[PartialD]y\) dy\ \ + \ \(\[PartialD]F\/\[PartialD]z\) dz\)], "Subsubsection"], Cell["\<\ Example of using Taylor Expansion to turn a function of two \ variables into an approximating function of four variables:\ \>", "Text"], Cell[BoxData[ \(CrazyFun[x_, \ y_]\ := \ \(\(Sin[5 \[Pi]\ x]\) \(Sin[5 \[Pi]\ y]\)\(\ \)\)\/\(x\ \ y\)\ + \(\(\ \)\(Sin[5 \[Pi]\ \((x - 1)\)] Sin[5 \[Pi]\ \((y - 1)\)]\)\)\/\ \(\((x - 1)\) \((\ y - 1)\)\)\)], "Input"], Cell[TextData[{ "Plot this function over suitable range of the variables ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], ":" }], "Text"], Cell[BoxData[ \(theplot\ = \ Plot3D[CrazyFun[x, y], {x, 0.1, .9}, {y, 0.1, .9}, PlotRange \[Rule] All, Mesh \[Rule] False]\)], "Input"], Cell[TextData[{ "Now, approximate the function about a specific point (xo, yo), using ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ", StyleBox["Series", FontWeight->"Bold"], " function:" }], "Text"], Cell[BoxData[ \(Approxfunction[x_, \ y_\ , \ xo_\ , \ yo_]\ := \ Series[CrazyFun[x, y], {x, xo, 2}, {y, yo, 2}] // Normal\)], "Input"], Cell["\<\ and plot the approximate function in the neighborhood of (xo, \ yo):\ \>", "Text"], Cell[BoxData[ \(anapprox\ = Plot3D[Evaluate[ Approxfunction[x, y, .7, .1]], {x, .7 - .1, .7 + .1}, {y, .1 - .1, .1 + \ .1}]\)], "Input", CellTags->"mmtag:12:Plot3D[]"], Cell["\<\ Both the original and the approximate function can be plotted \ simultaneously:\ \>", "Text"], Cell[BoxData[ \(Show[anapprox, \ theplot]\)], "Input"], Cell["\<\ This final little bit selects random points (xo, yo) and fits \ Taylor expansions to ten different points, then displays them individually as \ well as with a superposed plot of the original function.\ \>", "Text"], Cell[BoxData[ \(\(Table[{xo[i] = Random[], yo[i] = Random[]}, {i, 1, 100}];\)\)], "Input"], Cell["\<\ The next function automates the small approximating surface patch\ \ \>", "Text"], Cell[BoxData[ \(ApproxPlot[i_]\ := \ Plot3D[Evaluate[Approxfunction[x, y, xo[i], yo[i]]], {x, xo[i] - .1, xo[i] + .1}, {y, yo[i] - .1, yo[i] + .1}, PlotPoints \[Rule] 6, ColorFunction \[Rule] \((RGBColor[0.9 xo[i], 0.9 yo[i], #] &)\), DisplayFunction \[Rule] \ Identity]\)], "Input", CellTags->{ "mmtag:12:delaying_display", "mmtag:12:DisplayFunction__delaying_graphics"}], Cell["\<\ To build a sequence of graphics, I'll build a stack of ten graphics \ objects by using a recursive method. The next command sets the end of the \ recursion loop.\ \>", "Text", CellTags->"mmtag:12:graphics_stack__recursive_plotting_scheme"], Cell[BoxData[ \(Clear[GraphicsStack]\)], "Input"], Cell[BoxData[ \(GraphicsStack[0]\ = \ Show[ApproxPlot[1], DisplayFunction \[Rule] Identity]\)], "Input"], Cell["Here is the recursive function.", "Text"], Cell[BoxData[ \(GraphicsStack[i_]\ := \ \(GraphicsStack[i]\ = \ Show[GraphicsStack[i - 1], \ ApproxPlot[i + 1]]\)\)], "Input"], Cell[BoxData[ \(Show[ GraphicsArray[{GraphicsStack[10], Show[theplot, GraphicsStack[10]]}], DisplayFunction \[Rule] $DisplayFunction]\)], "Input"] }, Open ]] }, Open ]] }, Closed]] }, FrontEndVersion->"5.2 for Macintosh", ScreenRectangle->{{4, 1280}, {0, 832}}, ScreenStyleEnvironment->"Presentation", PrintingStyleEnvironment->"Printout", CellGrouping->Manual, WindowSize->{1078, 551}, WindowMargins->{{236, Automatic}, {Automatic, 188}}, WindowTitle->"Lecture 12 MIT 3.016 (Fall 2006) \[Copyright] W. 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(******************************************************************* End of Mathematica Notebook file. *******************************************************************)