(************** 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 66652, 2028]*) (*NotebookOutlinePosition[ 67402, 2054]*) (* CellTagsIndexPosition[ 67358, 2050]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "An Introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], " for Multivariable Calculus " }], "Title", Evaluatable->False, TextAlignment->Center, AspectRatioFixed->True, Background->RGBColor[0, 1, 1]], Cell[CellGroupData[{ Cell["Eliminating some unnecessary warning messages.", "Subsection"], Cell[BoxData[{ \(\(Off[General::"\"];\)\), "\n", \(\(Off[General::"\"];\)\)}], "Input", InitializationCell->True] }, Closed]] }, Closed]], Cell["\<\ This notebook is by Steven Amgott. Please send any questions or comments to \ samgott1@swarthmore.edu. Feel free to use and distribute this notebook, but \ keep this author information in any copy you use or distribute.\ \>", "SmallText"], Cell[TextData[{ "What you are reading is called a ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook. It consists of text, ", StyleBox["Mathematica", FontSlant->"Italic"], " commands, graphics, and other types of cells. A cell is a ", StyleBox["Mathematica", FontSlant->"Italic"], " \"unit,\" bordered by a bracket on the right side of the screen. The \ words \"An Introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], " for Multivariable Calculus\" above are in one cell, while the rest of the \ text you are currently reading is in another cell. What you are currently \ reading is a text cell." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Perhaps the most important type of cell is an input cell, such as the one \ below.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(2 + 2\)], "Input", AspectRatioFixed->True], Cell["\<\ You might notice that the brackets at the right bordering the two cells above \ are slightly different. The type of bracket is one way to see what type the \ cell is.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Input cells are your way to tell ", StyleBox["Mathematica", FontSlant->"Italic"], " to do something. If you click anywhere in the input cell containing 2+2 \ above, and press the ", StyleBox["enter", FontColor->RGBColor[0, 0, 1]], " key on the number pad at the lower right side of the keyboard (NOT the ", StyleBox["return", FontColor->RGBColor[0, 0, 1]], " key on a Macintosh or the ", StyleBox["Enter", FontColor->RGBColor[0, 0, 1]], " key next to the single and double quote marks on a Windows machine!), \ something should happen. ", StyleBox["Do so now, answering Yes to the initialization prompt. ", FontColor->RGBColor[1, 0, 1]], "(Note that pressing just the ", StyleBox["return", FontColor->RGBColor[0, 0, 1]], " key on a Macintosh or the ", StyleBox["Enter", FontColor->RGBColor[0, 0, 1]], " key next to the single and double quote marks on a Windows machine only \ produces a line break. You can, however, use the ", StyleBox["Shift", FontColor->RGBColor[0, 0, 1]], " and ", StyleBox["Enter", FontColor->RGBColor[0, 0, 1]], " keys on a Windows machine, or the ", StyleBox["shift", FontColor->RGBColor[0, 0, 1]], " and ", StyleBox["return", FontColor->RGBColor[0, 0, 1]], " keys on a Mac, simultaneously to evaluate a cell. In particular, that is \ how you evaluate a cell on a notebook computer without a number pad.)" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "What (hopefully) happened is that ", StyleBox["Mathematica", FontSlant->"Italic"], " started its Kernel, which is the portion of the program used to perform \ calculations. After a short(?) while, the arithmetic you requested should \ have been done, and the sum of 2 and 2 displayed. ", StyleBox["Mathematica", FontSlant->"Italic"], " often does not start the Kernel until you request a calculation to be \ done. (This is a memory and time saving feature, in case you are only typing \ in text and commands which you are not interested in evaluating immediately, \ but when you evaluate your first cell in a ", StyleBox["Mathematica", FontSlant->"Italic"], " session it then takes extra time to get the Kernel started.)" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "To create a new cell, simply move your cursor to a place just between two \ cells, before the first cell in the notebook, or after the last cell in the \ notebook. The cursor will change from vertical to horizontal. If you click \ the mouse, you will get a horizontal line across the window, marking a new \ cell into which you can type or paste. Once it is created, you can choose \ (or change) the type of a cell using the ", StyleBox["Format..Style", FontColor->RGBColor[0, 0, 1]], " menu item. (By default, ", StyleBox["Mathematica", FontSlant->"Italic"], " creates an input cell, unless you tell it otherwise. If the toolbar is \ open at the top of the notebook, you can also choose the cell type by using \ the drop-down box at the left of the toolbar.) ", StyleBox["Create a text cell just following this cell, and type something \ profound in it.", FontColor->RGBColor[1, 0, 1]] }], "Text"], Cell[TextData[{ "By the way, if you forgot to choose the cell type for a cell when you \ created it, you can change the cell type by clicking on the cell bracket at \ the right of the cell, and either using the ", StyleBox["Format..Style", FontColor->RGBColor[0, 0, 1]], " menu item or the drop-down box on the toolbar (if it is present at the \ top of the notebook)." }], "Text"], Cell[TextData[{ "Note that there is a color scheme in this notebook. Things in ", StyleBox["magenta", FontColor->RGBColor[1, 0, 1]], " are steps you need to perform or values you may wish to change. Words in \ ", StyleBox["blue", FontColor->RGBColor[0, 0, 1]], " are menu items or keys on the keyboard. Words in ", StyleBox["red", FontColor->RGBColor[1, 0, 0]], " are ", StyleBox["Mathematica", FontSlant->"Italic"], " commands, syntax, or names." }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " allows you to \"group\" cells together. This gives a \"table of contents\ \" look to the screen. Groups of cells have their cell brackets enclosed by \ additional brackets. To open up a group of cells, you can double-click the \ bracket which includes a \"filled-in\" triangle at the bottom of the bracket. \ You can re-close the group by double-clicking that same bracket. For \ example, the cell you are currently reading is the first cell of a group of \ two cells. To open the group, ", StyleBox["double-click the bracket at the right of this cell that has the \ \"filled-in\" triangle (the rightmost bracket). ", FontColor->RGBColor[1, 0, 1]], "Then close the group by ", StyleBox["double-clicking the bracket enclosing both cells.", FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[StyleBox["Boo!!!", Evaluatable->False, AspectRatioFixed->True, FontSize->72, FontColor->RGBColor[0, 1, 0]]], "SmallText", Evaluatable->False, AspectRatioFixed->True, FontSize->12] }, Closed]], Cell[TextData[{ "In later versions of ", StyleBox["Mathematica", FontSlant->"Italic"], " there is an alternate way to open and close groups of cells. If you go \ to the ", StyleBox["Edit...Preferences", FontColor->RGBColor[0, 0, 1]], " menu item (Windows) or the ", StyleBox["Mathematica.", FontColor->RGBColor[0, 0, 1], FontVariations->{"CompatibilityType"->0}], StyleBox["..Preferences", FontColor->RGBColor[0, 0, 1]], " menu item (Mac OS X), you can adjust many of ", StyleBox["Mathematica", FontSlant->"Italic"], "'s features. The one you may wish to change is in the ", StyleBox["Cell Options...Display Options", FontColor->RGBColor[0, 0, 1]], " section. Setting ", StyleBox["ShowGroupOpenCloseIcon", FontColor->RGBColor[0, 0, 1]], " to ", StyleBox["True", FontColor->RGBColor[0, 0, 1]], " (enter a check in the box) causes a triangle to appear to the left of any \ group of cells. Clicking on the triangle cycles between opening the group \ (triangle pointing down) and closing the group (triangle pointing right). If \ you prefer this to navigating the cell brackets, then you should set it now." }], "Text"], Cell[TextData[{ "The remainder of this notebook consists of groups of cells you will need \ to open in order to read. Notice that there is also a hierarchy of text \ cell types (Title, Subtitle, Section, Subsection, etc.) that are used in \ organizing notebooks. The cell types are used by ", StyleBox["Mathematica", FontSlant->"Italic"], " if the ", StyleBox["Automatic Grouping", FontColor->RGBColor[0, 0, 1]], " option is chosen in the ", StyleBox["Cell..Cell Grouping", FontColor->RGBColor[0, 0, 1]], " menu item. If the toolbar is open at the top of the notebook, you can \ easily see the type of cell by clicking in the cell, and looking in the \ drop-down box at the left of the toolbar." }], "Text"], Cell[TextData[{ "A continuation of the above introduction is available in the", Cell[BoxData[ FormBox[ ButtonBox[\(\(\ \)\(Introduction_to _Mathematica . nb\)\), ButtonData:>{"Introduction_to_Mathematica.nb", None}, ButtonStyle->"Hyperlink"], TraditionalForm]]], " tutorial in your course folder. If you have the time and interest, take \ a look. If you copied that notebook to the same folder as this notebook, you \ can open this tutorial by clicking on the hyperlink above in this paragraph. \ (This will also apply to the hyperlink in the next section.)" }], "Text"], Cell[CellGroupData[{ Cell["Some one-variable Calculus stuff", "Section"], Cell[TextData[{ "For a more complete introduction on how to use ", StyleBox["Mathematica", FontSlant->"Italic"], " to perform calculations from one-variable Calculus, you might want to \ open the ", Cell[BoxData[ FormBox[ ButtonBox[\(Calculus_ \((one_variable)\) . nb\), ButtonData:>{"Calculus_(one_variable).nb", None}, ButtonStyle->"Hyperlink"], TraditionalForm]]], " tutorial in your course folder. Here are some of the basics." }], "Text"], Cell[TextData[{ "You can easily plot functions of one variable using the ", StyleBox["Plot", FontColor->RGBColor[1, 0, 0]], " command. ", StyleBox["Evaluate", FontColor->RGBColor[1, 0, 1]], " the next cell to see a graph of the sine function." }], "Text"], Cell[BoxData[ \(Plot[Sin[x], {x, \(-Pi\), Pi}]\)], "Input"], Cell[TextData[{ "You can keep ", StyleBox["Mathematica", FontSlant->"Italic"], " from displaying the output cell containing only the text \"-Graphics-\" \ by putting a semicolon (", StyleBox[";", FontColor->RGBColor[1, 0, 0]], ") at the end of the syntax, as in the cell below. ", StyleBox["Evaluate it ", FontColor->RGBColor[1, 0, 1]], "to see the difference", ". In general, putting a semicolon at the end of a line of syntax tells ", StyleBox["Mathematica", FontSlant->"Italic"], " not to display the output cell for that syntax (although in the case of \ plots the graph itself is still displayed, as ", StyleBox["Mathematica", FontSlant->"Italic"], " doesn't consider it output for some strange reason)." }], "Text"], Cell[BoxData[ \(\(Plot[Sin[x], {x, \(-Pi\), Pi}];\)\)], "Input"], Cell["To zoom in (or out) on a plot, simply change the interval.", "Text"], Cell[BoxData[ \(\(Plot[Sin[x], {x, \(-\(Pi\/2\)\), Pi\/2}];\)\)], "Input"], Cell[TextData[{ "If you click on a plot to select it, you will see a dashed line box or \ window appear around it. You can resize the plot by moving the mouse to one \ of the squares on the border, holding down the left mouse button, and then \ dragging the box. Also with the window around the plot you can find the \ coordinates of any point in the window. Simply hold down the Ctrl key (on a \ Windows machine) or the Apple key (on a Mac) and the cursor changes to a \ crosshair. The coordinates of the point in the cross hair appear at the \ lower left bottom edge of your ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook window (not the plot window). Unfortunately there is no way to \ trace a graph, so you have to put the cursor where you want it. " }], "Text"], Cell[TextData[{ "There are so many options to the ", StyleBox["Plot", FontColor->RGBColor[1, 0, 0]], " command that there isn't enough time to do justice to them here. To see \ some of the default values for these options, ", StyleBox["evaluate the next cell", FontColor->RGBColor[1, 0, 1]], "." }], "Text"], Cell[BoxData[ \(?? Plot\)], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can easily calculate the derivative of a function. For instance, to find \ the derivative of x sin(x), ", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], " the ", StyleBox["Mathematica", FontSlant->"Italic"], " syntax" }], "Text"], Cell[BoxData[ \(D[x\ Sin[x], x]\)], "Input"], Cell[TextData[{ "You could also do this by using the palette symbol ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\[Placeholder]\ \[Placeholder]\)], FontColor->RGBColor[1, 0, 0]], ", found on the ", StyleBox["BasicInput", FontColor->RGBColor[0, 0, 1]], " palette, or on the ", Cell[BoxData[ FormBox[ ButtonBox["Calculus", ButtonData:>{ FrontEnd`FileName[ {"Extra Palettes"}, "Calculus.nb", CharacterEncoding -> "WindowsANSI"], None}, ButtonStyle->"Hyperlink"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ ButtonBox["MultivariableCalculus", ButtonData:>{ FrontEnd`FileName[ {"Extra Palettes"}, "MultivariableCalculus.nb", CharacterEncoding -> "WindowsANSI"], None}, ButtonStyle->"Hyperlink"], TraditionalForm]]], " Palettes in your course folder. (If the BasicInput palette is not open, \ you can open it from the ", StyleBox["File...Palettes", FontColor->RGBColor[0, 0, 1]], " menu item. The other two palettes are Math/Stat Department created \ palettes and are probably only in that menu on Math/Stat Lab computers, but \ you can open them on other computers by double-clicking on their icons in \ your course folder, or possibly by clicking on their buttons in this \ paragraph.) You click on the desired palette symbol to paste it into your \ current cursor location in the notebook. ", StyleBox["Evaluate the next cell", FontColor->RGBColor[1, 0, 1]], " to see the derivative (again)." }], "Text"], Cell[BoxData[ \(\[PartialD]\_x\ \((x\ Sin[x])\)\)], "Input"], Cell[TextData[{ StyleBox["Exercise:", FontColor->RGBColor[1, 0, 1]], " Make a new cell after this one, and click on the palette symbol ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\_\[Placeholder]\ \[Placeholder]\)], FontColor->RGBColor[1, 0, 0]], " ", "to paste it into that cell. Then use it to find the derivative of the \ function tan(x). Did you get the correct answer (", Cell[BoxData[ FormBox[ RowBox[{\(sec\^2\), "(", StyleBox["x", FontSlant->"Plain"], ")"}], TraditionalForm]]], ")?" }], "Text"], Cell[TextData[{ "Similarly, ", StyleBox["Mathematica", FontSlant->"Italic"], " can calculate many integrals. Here are a few examples to ", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], ", both without using palette symbols and with using palette symbols." }], "Text"], Cell[BoxData[ \(Integrate[Sin[x], x]\)], "Input"], Cell[BoxData[ \(\[Integral]Sin[t] \[DifferentialD]t\)], "Input"], Cell[BoxData[ \(Integrate[Sin[z], {z, 0, Pi}]\)], "Input"], Cell[BoxData[ \(\[Integral]\_0\%\[Pi] Sin[u] \[DifferentialD]u\)], "Input"], Cell[BoxData[ \(\[Integral]x\ Sin[x] \[DifferentialD]x\)], "Input"], Cell[BoxData[ \(\[Integral]\_0\%1 x\ \(\[ExponentialE]\^x\) \[DifferentialD]x\)], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can even do some integrals you probably cannot, such as (", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], ")" }], "Text"], Cell[BoxData[ \(\[Integral]Sin[x\^2] \[DifferentialD]x\)], "Input"], Cell[TextData[{ "although the output might look like gibberish to you. (If you wish, you \ can look up information about the function name that appeared in the output \ by highlighting it with the mouse, and choosing ", StyleBox["Help...Find Selected Function", FontColor->RGBColor[0, 0, 1]], " from the menu, although this might still not be very helpful.)" }], "Text"], Cell[TextData[{ "Even ", StyleBox["Mathematica", FontSlant->"Italic"], " has its limitations, as is seen if you ", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], " the next cell." }], "Text"], Cell[BoxData[ \(\[Integral]\_0\%1\(\@ Sin[\(x + x\^3\)\/2]\) \[DifferentialD]x\)], "Input"], Cell[TextData[{ "What do you think ", StyleBox["Mathematica", FontSlant->"Italic"], " is saying by the output of that evaluation?" }], "Text"], Cell["\<\ When symbolic techniques fail, you can alway use numerical ones to get \ answers to definite integrals.\ \>", "Text"], Cell[BoxData[ \(N[\[Integral]\_0\%1\(\@ Sin[\(x + x\^3\)\/2]\) \[DifferentialD]x]\)], "Input"], Cell[BoxData[ \(NIntegrate[\@Sin[\(x + x\^3\)\/2], {x, 0, 1}]\)], "Input"], Cell[TextData[{ "Notice that the second of these seems to go a bit faster. That's because \ ", StyleBox["NIntegrate", FontColor->RGBColor[1, 0, 0]], " doesn't even try symbolic methods first." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["A few surfaces (3D plotting), including the quadric surfaces", "Section"], Cell[TextData[{ "The command to draw graphs of functions of two variables is ", StyleBox["Plot3D", FontColor->RGBColor[1, 0, 0]], ". Evaluating the next cell shows an example of its use." }], "Text"], Cell[BoxData[ \(\(Plot3D[ Sin[x] Cos[2\ y], {x, \(-2\), 2}, {y, \(-2\), 2}];\)\)], "Input"], Cell[TextData[{ "(As with the ", StyleBox["Plot", FontColor->RGBColor[1, 0, 0]], " command, the semicolon at the end of the line just keeps ", StyleBox["Mathematica", FontSlant->"Italic"], " from responding with an output cell containing only the text \ \"-SurfaceGraphics-.\") The graph is a static figure, but you can change the \ viewpoint by with the ", StyleBox["3D Viewpoint Selector", FontColor->RGBColor[0, 0, 1]], ". You use it by clicking in a Plot3D input cell (such as the one below) \ just before the final ", StyleBox["]", FontColor->RGBColor[1, 0, 0]], ", typing a comma (", StyleBox[",", FontColor->RGBColor[1, 0, 0]], "), and then choosing Input...3D Viewpoint Selector. In the resulting \ window, use your mouse to rotate the box any desired orientation, and click \ the ", StyleBox["Paste", FontColor->RGBColor[0, 0, 1]], " button. ", StyleBox["Try this in the cell below", FontColor->RGBColor[1, 0, 1]], ". You should get something that looks like ", StyleBox["Plot3D[Sin[x]Sin[2 y],{x,-2,2},{y,-2,2},ViewPoint\[Rule]{-2.756, \ -0.536, 1.889}]", FontColor->RGBColor[1, 0, 0]], " (but with different numbers after the word ", StyleBox["ViewPoint", FontColor->RGBColor[1, 0, 0]], "). Then ", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], " the cell to produce the graph from the new viewpoint." }], "Text"], Cell[BoxData[ \(\(Plot3D[ Sin[x] Cos[2\ y], {x, \(-2\), 2}, {y, \(-2\), 2}];\)\)], "Input"], Cell[TextData[{ "You can change the viewpoint once again by highlighting the part of the \ syntax the ", StyleBox["3D Viewpoint Selector", FontColor->RGBColor[0, 0, 1]], " pasted in, and then rotating the box again and pasting in a different \ location. You may wish to try this." }], "Text"], Cell[TextData[{ "There is another way to rotate the graph. A new (experimental) package \ has been included in recent versions of ", StyleBox["Mathematica", FontSlant->"Italic"], " (but without documentation in the ", StyleBox["Help", FontColor->RGBColor[0, 0, 1]], " menu). You can load this package by evaluating the cell below. Check \ the output of the $Packages command to make sure ", StyleBox["RealTime3D", FontColor->RGBColor[0, 0, 1]], " is now loaded." }], "Text"], Cell[BoxData[{ \(Needs["\"]\), "\[IndentingNewLine]", \($Packages\)}], "Input"], Cell[TextData[{ "Once this package has been loaded, any additional 3D graphs you create can \ be rotated in real time. (Note that graphs you created before loading the \ package still cannot be rotated this way.) For instance, ", StyleBox["evaluate", FontColor->RGBColor[1, 0, 1]], " the syntax below to be able to rotate the graph from the beginning of \ this section. Click and hold the (left) mouse button anywhere in the graph \ and move the mouse to rotate the graph. When you release the mouse button, \ the graph \"freezes\" in the current position. " }], "Text"], Cell[BoxData[ \(\(Plot3D[ Sin[x] Cos[2\ y], {x, \(-2\), 2}, {y, \(-2\), 2}];\)\)], "Input"], Cell[TextData[{ "If you wish, you can change the region on which the graph is plotted, for \ instance, to zoom in. All you need to do are to put in different values for \ the range on x and y. The values in ", StyleBox["magenta", FontColor->RGBColor[1, 0, 1]], " below can be changed to do this. ", StyleBox["Change them in the next cell in order to redraw the graph on the \ square -0.5 \[LessEqual] x \[LessEqual] 0.5 and -0.5 \[LessEqual] y \ \[LessEqual] 0.5, and then evaluate the cell to see the \"zoomed-in\" graph", FontColor->RGBColor[1, 0, 1]], "." }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(Sin[x] Cos[2\ y]\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}], ";"}]], "Input"], Cell[TextData[{ "Another interesting type of plot, although not 3D, is a ", StyleBox["ContourPlot", FontColor->RGBColor[1, 0, 0]], ". This plot shows graphs of the curves you get by slicing the surface at \ various heights. (You may have seen maps of mountains and hills done this \ way.) Here is the contour plot for the above function." }], "Text"], Cell[BoxData[ \(\(ContourPlot[ Sin[x] Cos[2\ y], {x, \(-2\), 2}, {y, \(-2\), 2}];\)\)], "Input"], Cell[TextData[{ "The shading reflects the height of the contour. Darker is lower, while \ lighter is higher. If you just want to see the contours without the shading, \ use the ", StyleBox["ContourShading\[Rule]False", FontColor->RGBColor[1, 0, 0]], " option." }], "Text"], Cell[BoxData[ \(\(ContourPlot[Sin[x] Cos[2\ y], {x, \(-2\), 2}, {y, \(-2\), 2}, ContourShading \[Rule] False];\)\)], "Input"], Cell[TextData[{ "The subsections below contain some examples of 3D and contour plots. Keep \ in mind that you can change the plot region, if you wish, by adjusting the \ numbers colored ", StyleBox["magenta", FontColor->RGBColor[1, 0, 1]], "." }], "Text"], Cell[CellGroupData[{ Cell["Paraboloids", "Subsection"], Cell[CellGroupData[{ Cell["Elliptical (actually circular)", "Subsubsection"], Cell["Opening up:", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(x\^2 + \ y\^2\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(x\^2 + \ y\^2\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}], ";"}]], "Input"], Cell["Opening down:", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(\(-x\^2\) - \ y\^2\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(\(-x\^2\) - \ y\^2\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}], ";"}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Hyperbolic", "Subsubsection"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(x\^2 - \ y\^2\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(x\^2 - \ y\^2\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}], ";"}]], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Hyperboloids", "Subsection"], Cell[CellGroupData[{ Cell["Elliptical (actually circular) of 2 sheets", "Subsubsection"], Cell[BoxData[{\(Clear[plot1, plot2, plot1c, plot2c]\), "\n", RowBox[{ RowBox[{"plot1", "=", RowBox[{"Plot3D", "[", RowBox[{\(\@\(\(\ \)\(x\^2 + \ y\^2 + 1\)\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}]}], ";"}]}], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"plot2", "=", RowBox[{"Plot3D", "[", RowBox[{\(-\@\(\(\ \)\(x\^2 + \ y\^2 + 1\)\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}]}], ";"}]], "Input"], Cell[BoxData[ \(\(Show[plot1, plot2];\)\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"plot1c", "=", RowBox[{"ContourPlot", "[", RowBox[{\(\@\(\(\ \)\(x\^2 + \ y\^2 + 1\)\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}]}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"plot2c", "=", RowBox[{"ContourPlot", "[", RowBox[{\(-\@\(\(\ \)\(x\^2 + \ y\^2 + 1\)\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}]}], ";"}]], "Input"], Cell[BoxData[ \(\(Show[plot1c, plot2c];\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Elliptical of 1 sheet", "Subsubsection"], Cell[TextData[{ "To get a good graph of a hyperboloid of 1 sheet we need to use ", StyleBox["ParametricPlot", FontColor->RGBColor[1, 0, 0]], " rather than ", StyleBox["Plot3D", FontColor->RGBColor[1, 0, 0]], ". This will be saved for a future notebook, once parametrizations of \ surfaces has been discussed in class. (Note: even the other hyperboloids and \ the paraboloids in this notebook will look better using ", StyleBox["ParametricPlot", FontColor->RGBColor[1, 0, 0]], ".)" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Elliptical (actually circular) cone", "Subsubsection"], Cell[BoxData[{\(Clear[plot1, plot2, plot1c, plot2c]\), "\n", RowBox[{ RowBox[{"plot1", "=", RowBox[{"Plot3D", "[", RowBox[{\(3 \@\(\(\ \)\(x\^2 + \ y\^2\)\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}]}], ";"}]}], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"plot2", "=", RowBox[{"Plot3D", "[", RowBox[{\(\(-3\) \@\(x\^2 + \ y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}]}], ";"}]], "Input"], Cell[BoxData[ \(\(Show[plot1, plot2];\)\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"plot1c", "=", RowBox[{"ContourPlot", "[", RowBox[{\(3 \@\(\(\ \)\(x\^2 + \ y\^2\)\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}]}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"plot2c", "=", RowBox[{"ContourPlot", "[", RowBox[{\(\(-3\) \@\(x\^2 + \ y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}]}], ";"}]], "Input"], Cell[BoxData[ \(\(Show[plot1c, plot2c];\)\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Cylinders", "Subsection"], Cell[CellGroupData[{ Cell["Plane", "Subsubsection"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(2 x + 3 y\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(2 x + 3 y\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}], ";"}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Elliptical (actually circular)", "Subsubsection"], Cell[BoxData[{\(Clear[plot1, plot2, plot1c, plot2c]\), "\n", RowBox[{ RowBox[{"plot1", "=", RowBox[{"Plot3D", "[", RowBox[{\(\@\(\(\ \)\(1 - y\^2\)\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-1\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}]}], ";"}]}], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"plot2", "=", RowBox[{"Plot3D", "[", RowBox[{\(-\@\(\(\ \)\(1 - \ y\^2\)\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-1\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}]}], ";"}]], "Input"], Cell[BoxData[ \(\(Show[plot1, plot2];\)\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"plot1c", "=", RowBox[{"ContourPlot", "[", RowBox[{\(\@\(\(\ \)\(1 - y\^2\)\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-1\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}]}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"plot2c", "=", RowBox[{"ContourPlot", "[", RowBox[{\(-\@\(\(\ \)\(1 - \ y\^2\)\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-1\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}]}], ";"}]], "Input"], Cell[BoxData[ \(\(Show[plot1c, plot2c];\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Parabolic", "Subsubsection"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(x\^2\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(x\^2\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}], ";"}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exponential", "Subsubsection"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(\[ExponentialE]\^y\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(\[ExponentialE]\^y\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}], ";"}]], "Input"] }, Closed]], Cell["\<\ What is the common feature of all the contour plots drawn above? Is this how \ you could define a cylinder?\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Miscellaneous", "Subsection"], Cell[CellGroupData[{ Cell["Trigonometric", "Subsubsection"], Cell["One plot:", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(Sin[x\ y]\), " ", ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(Sin[x\ y]\), " ", ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}], ";"}]], "Input"], Cell["Another plot:", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(Cos[\@\(x\^2 + y\^2\)]\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-10\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["10", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-10\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["10", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(PlotPoints \[Rule] 40\)}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(Cos[\@\(x\^2 + y\^2\)]\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-10\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["10", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-10\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["10", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(PlotPoints \[Rule] 40\), ",", \(ContourShading \[Rule] False\)}], "]"}], ";"}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Exponential", "Subsubsection"], Cell["One example:", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(x\ \[ExponentialE]\^\(\(-x\^2\) - y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(x\ \[ExponentialE]\^\(\(-x\^2\) - y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}], ";"}]], "Input"], Cell["Another example:", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(\((x\^2 + 3 y\^2)\)\ \[ExponentialE]\^\(\(-2\) x\^2 - 2 y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(PlotPoints \[Rule] 30\)}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(\((x\^2 + 3 y\^2)\)\ \[ExponentialE]\^\(\(-2\) x\^2 - 2 y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(PlotPoints \[Rule] 30\), ",", \(ContourShading \[Rule] False\)}], "]"}], ";"}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["A Monkey Saddle", "Subsubsection"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(x \((x\^2 - 3\ y\^2)\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-3\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["3", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}]}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(x \((x\^2 - 3\ y\^2)\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-3\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["3", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\)}], "]"}], ";"}]], "Input"], Cell["Why do you think it is called a monkey saddle?", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["A Hat?", "Subsubsection"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(Sin[2 x\^2 + 3 y\^2]\/\(x\^2 + y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(PlotPoints \[Rule] 30\)}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(Sin[2 x\^2 + 3 y\^2]\/\(x\^2 + y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(PlotPoints \[Rule] 30\), ",", \(ContourShading \[Rule] False\)}], "]"}], ";"}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Two Peaks, no Valley", "Subsubsection"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(\(-\((x\^2 - 1)\)\^2\) - \((x\^2 - \[ExponentialE]\^y)\)\^2\ \), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-1.5\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["1.5", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(- .5\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox[".5", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(PlotPoints \[Rule] 30\)}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(\(-\((x\^2 - 1)\)\^2\) - \((x\^2 - \[ExponentialE]\^y)\)\^2\ \), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-1.5\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["1.5", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(- .5\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox[".5", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(PlotPoints \[Rule] 30\), ",", \(ContourShading \[Rule] False\)}], "]"}], ";"}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Only one Critical Point (local min), but no Absolute Min", \ "Subsubsection"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(\(x\^2\) \((y + 1)\)\^3 + y\^2\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-1\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-3\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(PlotPoints \[Rule] 30\)}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(\(x\^2\) \((y + 1)\)\^3 + y\^2\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-1\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-3\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["1", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(PlotPoints \[Rule] 30\), ",", \(ContourShading \[Rule] False\)}], "]"}], ";"}]], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Limit Examples", "Subsection"], Cell["\<\ For each of the functions below, use the graphs to try and determine whether \ or not the limit as (x,y)\[Rule](0,0) exists for the function. \ \"Zooming-in\" a bit might be very helpful here.\ \>", "Text"], Cell["Example 1.", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(\(x\^2 - y\^2\)\/\(x\^2 + y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-3\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["3", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(PlotPoints \[Rule] 50\)}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(\(x\^2 - y\^2\)\/\(x\^2 + y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-3\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["3", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\), ",", \(PlotPoints \[Rule] 50\)}], "]"}], ";"}]], "Input"], Cell["Example 2.", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(\(x\ y \((x\^2 - y\^2)\)\)\/\(x\^2 + y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-3\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["3", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(PlotPoints \[Rule] 50\)}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(\(x\ y \((x\^2 - y\^2)\)\)\/\(x\^2 + y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-3\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["3", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\), ",", \(PlotPoints \[Rule] 50\)}], "]"}], ";"}]], "Input"], Cell["Example 3.", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(\(x\^2\ y\)\/\(x\^4 + y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-3\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["3", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(PlotPoints \[Rule] 50\)}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(\(x\^2\ y\)\/\(x\^4 + y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-3\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["3", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\), ",", \(PlotPoints \[Rule] 50\)}], "]"}], ";"}]], "Input"], Cell["Example 4. Another hat?", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"Plot3D", "[", RowBox[{\(Sin[x\^2 + y\^2]\/\(x\^2 + y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-3\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["3", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(PlotPoints \[Rule] 50\)}], "]"}], ";"}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{\(Sin[x\^2 + y\^2]\/\(x\^2 + y\^2\)\), ",", RowBox[{"{", RowBox[{"x", ",", StyleBox[\(-3\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["3", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", StyleBox[\(-2\), FontColor->RGBColor[1, 0, 1]], ",", StyleBox["2", FontColor->RGBColor[1, 0, 1]]}], "}"}], ",", \(ContourShading \[Rule] False\), ",", \(PlotPoints \[Rule] 50\)}], "]"}], ";"}]], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Parametric Plots", "Section"], Cell[TextData[{ "You can use ", StyleBox["ParametricPlot", FontColor->RGBColor[1, 0, 0]], " and ", StyleBox["ParametricPlot3D", FontColor->RGBColor[1, 0, 0]], " to plot curves. Here is an ellipse done with ", StyleBox["ParametricPlot", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[ \(\(ParametricPlot[{3\ Sin[t], 4\ Cos[t]}, {t, 0, 2\ \[Pi]}];\)\)], "Input"], Cell[TextData[{ "Notice that to get a good representation of a circle the ", StyleBox["AspectRatio", FontColor->RGBColor[1, 0, 0]], " option might be necessary. Without this option the plot may look like an \ ellipse." }], "Text"], Cell[BoxData[ \(\(ParametricPlot[{4\ Sin[t], 4\ Cos[t]}, {t, 0, 2\ \[Pi]}];\)\)], "Input"], Cell["With the option you get the true shape of the circle.", "Text"], Cell[BoxData[ \(\(ParametricPlot[{4\ Sin[t], 4\ Cos[t]}, {t, 0, 2\ \[Pi]}, AspectRatio \[Rule] Automatic];\)\)], "Input"], Cell[TextData[{ "This plots a spiral in three-dimensional space using ", StyleBox["ParametricPlot3D", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[ \(\(ParametricPlot3D[{7\ Cos[u], 7\ Sin[u], u}, {u, 0, 8 \[Pi]}];\)\)], "Input"], Cell[TextData[{ "You can also use ", StyleBox["ParametricPlot3D", FontColor->RGBColor[1, 0, 0]], " to plot surfaces. Here is a paraboloid." }], "Text"], Cell[BoxData[ \(\(ParametricPlot3D[{2\ r\ Cos[\[Theta]], 2\ r\ Sin[\[Theta]], r\^2}, {r, 0, 4}, {\[Theta], 0, 2\ \[Pi]}];\)\)], "Input"], Cell[TextData[{ "Notice the advantages to plotting the paraboloid using ", StyleBox["ParametricPlot3D", FontColor->RGBColor[1, 0, 0]], ", rather than using ", StyleBox["Plot3D", FontColor->RGBColor[1, 0, 0]], "." }], "Text"], Cell[BoxData[ \(\(Plot3D[ 1\/4\ x\^2 + 1\/4\ y\^2, {x, \(-8\), 8}, {y, \(-8\), 8}];\)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Vector Field Plots", "Section"], Cell[TextData[{ "In order to use the commands in this section, the package ", StyleBox["\"Graphics", FontColor->RGBColor[1, 0, 0]], StyleBox["`", FontFamily->"Courier", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["PlotField", FontColor->RGBColor[1, 0, 0]], StyleBox["`", FontFamily->"Courier", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["\"", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[0.499992, 0.0658274, 0.17203]], "(for two dimensional fields) and/or the package ", StyleBox["\"Graphics", FontColor->RGBColor[1, 0, 0]], StyleBox["`", FontFamily->"Courier", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["PlotField3D", FontColor->RGBColor[1, 0, 0]], StyleBox["`", FontFamily->"Courier", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["\"", FontColor->RGBColor[1, 0, 0]], " (for three dimensional fields) must be loaded. ", StyleBox["Evaluate the next cell to load the first of these packages.", FontColor->RGBColor[1, 0, 1]], StyleBox[" ", FontColor->RGBColor[1, 0.0721599, 0.542763]], "You will have to load the package during any session in which you plan to \ use these commands. ", StyleBox["$Packages", FontColor->RGBColor[1, 0, 0]], " displays the current list of loaded packages." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(Needs["\"]\), "\[IndentingNewLine]", \($Packages\)}], "Input"], Cell["\<\ Using this package you can display gradient fields of functions of two \ variables.\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"PlotGradientField", "[", RowBox[{\(3 x + 12 y - x\^3 - y\^3\), ",", RowBox[{"{", RowBox[{"x", StyleBox[",", FontColor->GrayLevel[0]], \(-2\), ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", StyleBox[",", FontColor->GrayLevel[0]], \(-3\), ",", "3"}], "}"}]}], "]"}], ";"}]], "Input"], Cell["\<\ There are, of course, tons of customization options. You can find \ information on this by evaluating the following cell.\ \>", "Text"], Cell[BoxData[ \(?? PlotGradientField\)], "Input"], Cell["You can also directly plot vector fields.", "Text"], Cell[BoxData[ \(\(PlotVectorField[{\(-y\), x}, {x, \(-2\), 2}, {y, \(-2\), 2}];\)\)], "Input"], Cell[TextData[{ "One way to get more information on this is to highlight the words \ PlotVectorField in the previous cell using your mouse, and then click the \ menu item ", StyleBox["Help...Find Selected Function", FontColor->RGBColor[0, 0, 1]], ". Do so, then click the link to ", StyleBox["Graphics`PlotField`", FontColor->RGBColor[0, 0, 1]], " under Standard Packages, or the section number link under the ", StyleBox["Mathematica", FontSlant->"Italic"], " Book." }], "Text"], Cell["\<\ To work in three dimensions you need to load the second of the packages.\ \>", "Text"], Cell[BoxData[{ \(Needs["\"]\), "\[IndentingNewLine]", \($Packages\)}], "Input"], Cell["\<\ You can then display gradient fields of functions of three variables\ \>", "Text"], Cell[BoxData[ \(\(PlotGradientField3D[ x\ y\ z, {x, \(-1\), 1}, {y, \(-1\), 1}, {z, \(-1\), 1}, VectorHeads \[Rule] True];\)\)], "Input"], Cell["or other arbitrary vector fields with three components.", "Text"], Cell[BoxData[ \(\(PlotVectorField3D[{x, y, z}, {x, 0, 2}, {y, 0, 2}, {z, 0, 2}, PlotPoints \[Rule] 5, VectorHeads \[Rule] True];\)\)], "Input"], Cell["\<\ Notice the options in these cells. The default is to draw without arrows, as \ you can see by evaluating the following cell, which displays the default \ options.\ \>", "Text"], Cell[BoxData[ \(Options[PlotVectorField3D]\)], "Input"] }, Closed]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 971}}, AutoGeneratedPackage->None, WindowToolbars->"EditBar", CellGrouping->Manual, WindowSize->{698, 648}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, ShowGroupOpenCloseIcon->True ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 244, 9, 166, "Title", Evaluatable->False], Cell[CellGroupData[{ Cell[2045, 66, 68, 0, 38, "Subsection"], Cell[2116, 68, 141, 3, 50, "Input", InitializationCell->True] }, Closed]] }, Closed]], Cell[2284, 75, 250, 4, 42, "SmallText"], Cell[2537, 81, 715, 19, 90, "Text", Evaluatable->False], Cell[3255, 102, 154, 5, 33, "Text", Evaluatable->False], Cell[3412, 109, 64, 2, 30, "Input"], Cell[3479, 113, 239, 6, 52, "Text", Evaluatable->False], Cell[3721, 121, 1502, 41, 147, "Text", Evaluatable->False], Cell[5226, 164, 827, 18, 109, "Text", Evaluatable->False], Cell[6056, 184, 946, 18, 128, "Text"], Cell[7005, 204, 390, 8, 71, "Text"], Cell[7398, 214, 496, 15, 71, "Text"], Cell[CellGroupData[{ Cell[7919, 233, 923, 18, 128, "Text", Evaluatable->False], Cell[8845, 253, 207, 7, 97, "SmallText", Evaluatable->False] }, Closed]], Cell[9067, 263, 1188, 30, 125, "Text"], Cell[10258, 295, 739, 16, 109, "Text"], Cell[11000, 313, 614, 11, 90, "Text"], Cell[CellGroupData[{ Cell[11639, 328, 51, 0, 73, "Section"], Cell[11693, 330, 492, 12, 52, "Text"], Cell[12188, 344, 274, 8, 52, "Text"], Cell[12465, 354, 63, 1, 30, "Input"], Cell[12531, 357, 767, 20, 90, "Text"], Cell[13301, 379, 68, 1, 30, "Input"], Cell[13372, 382, 74, 0, 33, "Text"], Cell[13449, 384, 78, 1, 42, "Input"], Cell[13530, 387, 794, 13, 128, "Text"], Cell[14327, 402, 326, 9, 52, "Text"], Cell[14656, 413, 40, 1, 30, "Input"], Cell[14699, 416, 323, 11, 52, "Text"], Cell[15025, 429, 48, 1, 30, "Input"], Cell[15076, 432, 1613, 39, 128, "Text"], Cell[16692, 473, 64, 1, 30, "Input"], Cell[16759, 476, 574, 16, 52, "Text"], Cell[17336, 494, 292, 8, 52, "Text"], Cell[17631, 504, 53, 1, 30, "Input"], Cell[17687, 507, 68, 1, 41, "Input"], Cell[17758, 510, 62, 1, 30, "Input"], Cell[17823, 513, 79, 1, 41, "Input"], Cell[17905, 516, 71, 1, 41, "Input"], Cell[17979, 519, 103, 2, 42, "Input"], Cell[18085, 523, 207, 7, 33, "Text"], Cell[18295, 532, 71, 1, 41, "Input"], Cell[18369, 535, 381, 7, 71, "Text"], Cell[18753, 544, 214, 8, 33, "Text"], Cell[18970, 554, 110, 2, 50, "Input"], Cell[19083, 558, 152, 5, 33, "Text"], Cell[19238, 565, 127, 3, 33, "Text"], Cell[19368, 570, 115, 2, 50, "Input"], Cell[19486, 574, 78, 1, 50, "Input"], Cell[19567, 577, 215, 6, 33, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[19819, 588, 79, 0, 43, "Section"], Cell[19901, 590, 210, 5, 33, "Text"], Cell[20114, 597, 103, 2, 30, "Input"], Cell[20220, 601, 1419, 38, 147, "Text"], Cell[21642, 641, 103, 2, 30, "Input"], Cell[21748, 645, 303, 7, 52, "Text"], Cell[22054, 654, 503, 13, 71, "Text"], Cell[22560, 669, 102, 2, 50, "Input"], Cell[22665, 673, 584, 10, 90, "Text"], Cell[23252, 685, 103, 2, 30, "Input"], Cell[23358, 689, 591, 12, 71, "Text"], Cell[23952, 703, 585, 16, 30, "Input"], Cell[24540, 721, 363, 7, 71, "Text"], Cell[24906, 730, 108, 2, 30, "Input"], Cell[25017, 734, 284, 7, 52, "Text"], Cell[25304, 743, 138, 2, 30, "Input"], Cell[25445, 747, 266, 7, 52, "Text"], Cell[CellGroupData[{ Cell[25736, 758, 33, 0, 47, "Subsection"], Cell[CellGroupData[{ Cell[25794, 762, 55, 0, 43, "Subsubsection"], Cell[25852, 764, 27, 0, 33, "Text"], Cell[25882, 766, 582, 16, 31, "Input"], Cell[26467, 784, 630, 16, 31, "Input"], Cell[27100, 802, 29, 0, 33, "Text"], Cell[27132, 804, 587, 16, 31, "Input"], Cell[27722, 822, 635, 16, 31, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[28394, 843, 35, 0, 29, "Subsubsection"], Cell[28432, 845, 582, 16, 31, "Input"], Cell[29017, 863, 630, 16, 31, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[29696, 885, 34, 0, 31, "Subsection"], Cell[CellGroupData[{ Cell[29755, 889, 67, 0, 43, "Subsubsection"], Cell[29825, 891, 710, 17, 54, "Input"], Cell[30538, 910, 662, 17, 34, "Input"], Cell[31203, 929, 56, 1, 30, "Input"], Cell[31262, 932, 712, 17, 34, "Input"], Cell[31977, 951, 713, 17, 34, "Input"], Cell[32693, 970, 58, 1, 30, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[32788, 976, 46, 0, 29, "Subsubsection"], Cell[32837, 978, 520, 13, 71, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[33394, 996, 60, 0, 29, "Subsubsection"], Cell[33457, 998, 708, 17, 54, "Input"], Cell[34168, 1017, 654, 17, 34, "Input"], Cell[34825, 1036, 56, 1, 30, "Input"], Cell[34884, 1039, 710, 17, 34, "Input"], Cell[35597, 1058, 705, 17, 34, "Input"], Cell[36305, 1077, 58, 1, 30, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[36412, 1084, 31, 0, 31, "Subsection"], Cell[CellGroupData[{ Cell[36468, 1088, 30, 0, 43, "Subsubsection"], Cell[36501, 1090, 580, 16, 30, "Input"], Cell[37084, 1108, 628, 16, 30, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[37749, 1129, 55, 0, 29, "Subsubsection"], Cell[37807, 1131, 701, 17, 54, "Input"], Cell[38511, 1150, 655, 17, 34, "Input"], Cell[39169, 1169, 56, 1, 30, "Input"], Cell[39228, 1172, 703, 17, 34, "Input"], Cell[39934, 1191, 706, 17, 34, "Input"], Cell[40643, 1210, 58, 1, 30, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[40738, 1216, 34, 0, 29, "Subsubsection"], Cell[40775, 1218, 573, 16, 31, "Input"], Cell[41351, 1236, 621, 16, 31, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[42009, 1257, 36, 0, 29, "Subsubsection"], Cell[42048, 1259, 587, 16, 30, "Input"], Cell[42638, 1277, 635, 16, 30, "Input"] }, Closed]], Cell[43288, 1296, 132, 3, 30, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[43457, 1304, 35, 0, 31, "Subsection"], Cell[CellGroupData[{ Cell[43517, 1308, 38, 0, 43, "Subsubsection"], Cell[43558, 1310, 25, 0, 33, "Text"], Cell[43586, 1312, 583, 16, 30, "Input"], Cell[44172, 1330, 631, 16, 30, "Input"], Cell[44806, 1348, 29, 0, 33, "Text"], Cell[44838, 1350, 631, 16, 34, "Input"], Cell[45472, 1368, 686, 17, 56, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[46195, 1390, 36, 0, 29, "Subsubsection"], Cell[46234, 1392, 28, 0, 33, "Text"], Cell[46265, 1394, 609, 16, 34, "Input"], Cell[46877, 1412, 657, 16, 34, "Input"], Cell[47537, 1430, 32, 0, 33, "Text"], Cell[47572, 1432, 697, 18, 34, "Input"], Cell[48272, 1452, 752, 19, 56, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[49061, 1476, 40, 0, 29, "Subsubsection"], Cell[49104, 1478, 591, 16, 31, "Input"], Cell[49698, 1496, 639, 16, 31, "Input"], Cell[50340, 1514, 62, 0, 33, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[50439, 1519, 31, 0, 29, "Subsubsection"], Cell[50473, 1521, 644, 16, 47, "Input"], Cell[51120, 1539, 699, 17, 70, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[51856, 1561, 45, 0, 29, "Subsubsection"], Cell[51904, 1563, 673, 17, 32, "Input"], Cell[52580, 1582, 728, 18, 53, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[53345, 1605, 83, 1, 29, "Subsubsection"], Cell[53431, 1608, 635, 16, 31, "Input"], Cell[54069, 1626, 690, 17, 52, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[54808, 1649, 36, 0, 31, "Subsection"], Cell[54847, 1651, 217, 4, 52, "Text"], Cell[55067, 1657, 26, 0, 33, "Text"], Cell[55096, 1659, 637, 16, 47, "Input"], Cell[55736, 1677, 692, 17, 47, "Input"], Cell[56431, 1696, 26, 0, 33, "Text"], Cell[56460, 1698, 648, 16, 47, "Input"], Cell[57111, 1716, 703, 17, 47, "Input"], Cell[57817, 1735, 26, 0, 33, "Text"], Cell[57846, 1737, 633, 16, 47, "Input"], Cell[58482, 1755, 688, 17, 47, "Input"], Cell[59173, 1774, 40, 0, 33, "Text"], Cell[59216, 1776, 638, 16, 47, "Input"], Cell[59857, 1794, 693, 17, 70, "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[60599, 1817, 35, 0, 43, "Section"], Cell[60637, 1819, 309, 11, 33, "Text"], Cell[60949, 1832, 105, 2, 30, "Input"], Cell[61057, 1836, 241, 6, 52, "Text"], Cell[61301, 1844, 105, 2, 30, "Input"], Cell[61409, 1848, 69, 0, 33, "Text"], Cell[61481, 1850, 134, 2, 30, "Input"], Cell[61618, 1854, 158, 5, 33, "Text"], Cell[61779, 1861, 110, 2, 30, "Input"], Cell[61892, 1865, 162, 5, 33, "Text"], Cell[62057, 1872, 151, 2, 31, "Input"], Cell[62211, 1876, 242, 8, 33, "Text"], Cell[62456, 1886, 121, 3, 42, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[62614, 1894, 37, 0, 43, "Section"], Cell[62654, 1896, 1461, 45, 92, "Text", Evaluatable->False], Cell[64118, 1943, 110, 2, 50, "Input"], Cell[64231, 1947, 107, 3, 33, "Text"], Cell[64341, 1952, 454, 12, 31, "Input"], Cell[64798, 1966, 146, 3, 33, "Text"], Cell[64947, 1971, 53, 1, 30, "Input"], Cell[65003, 1974, 57, 0, 33, "Text"], Cell[65063, 1976, 109, 2, 30, "Input"], Cell[65175, 1980, 506, 13, 71, "Text"], Cell[65684, 1995, 96, 2, 33, "Text"], Cell[65783, 1999, 112, 2, 50, "Input"], Cell[65898, 2003, 92, 2, 33, "Text"], Cell[65993, 2007, 158, 3, 30, "Input"], Cell[66154, 2012, 71, 0, 33, "Text"], Cell[66228, 2014, 156, 2, 50, "Input"], Cell[66387, 2018, 187, 4, 52, "Text"], Cell[66577, 2024, 59, 1, 30, "Input"] }, Closed]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)