(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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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Let's use it to plot a \ more interesting function: ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[") = ", Evaluatable->False, AspectRatioFixed->True], StyleBox["xy", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[". Here is the basic command:", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Plot3D[x y, {x,-3, 3}, {y,-3, 3}]"], "Input", AspectRatioFixed->True], Cell[TextData[ "The default view of this graph is a little misleading; the perspective \ doesn't show the shape of the surface well. To get another view, move the \ cursor so it falls into the space between this paragraph and your output \ above; the cursor should flop on its side, looking like a dog bone. Click \ the left mouse button and a horizontal line will appear. Hold down the Alt \ key, type \"L\", and a copy of the previous command will appear. Place the \ blinking insertion point just before the last bracket \"]\" in the command \ and type a comma. Now click on the \"Action\" Menu at the top of the screen, \ and select \"Prepare Input\". A submenu will appear; click on \"3D Viewpoint \ Selector\". In the resulting Dialog box is a picture of some coordinate axes \ that can be rotated to change the point of view we are given of the graph. \ Rotate the axes to get something like a \"usual\" view where we look at the \ graph from some point with all positive coordinates. Then click on the \ \"Paste\" button to insert the new viewpoint into the plot command. Close \ the dialog box and enter the modified command. Print out your plot."], "Text",\ Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " can generate level curves, as well. The following command does that", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "ContourPlot[x y,{x,-3,3},{y,-3,3},ContourShading->False]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox["Copy the command as you did above, delete the option ", Evaluatable->False, AspectRatioFixed->True], StyleBox["ContourShading->False", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[ " (and the preceding comma) and enter the new command. Explain the \ significance of light and dark contour shading below. (Compare this plot with \ the graph you found above.)", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell["", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Part 2: Level Curves and Limits"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "One of the best ways to find trouble spots for a scalar field is to look \ at their level curves. Recall that on a level curve of a scalar field ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", the function ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox["(", Evaluatable->False, AspectRatioFixed->True], StyleBox["x", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[", ", Evaluatable->False, AspectRatioFixed->True], StyleBox["y", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ") is constant. What would happen if two level curves for different values \ of ", Evaluatable->False, AspectRatioFixed->True], StyleBox["f", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ " were to intersect? On the next page, explain why a function cannot be \ continuous at a point in its domain where two level curves intersect. \ Consider all three parts of the definition of continuity given in class in \ your explanation. (After lab?)", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n\n\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Places where level curves intersect are called ", Evaluatable->False, AspectRatioFixed->True], StyleBox["singularities", Evaluatable->False, AspectRatioFixed->True, FontSlant->"Italic"], StyleBox[ ", and are very important in many applications of scalar fields. 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HINT: If you \ can't tell from your computation above, try re-entering the command above \ with the additional option: ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["Contours->40", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[".", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["\n\n"], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "To see what the singularity looks like, plot the graph of the function \ with the command below. The option forces ", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"], StyleBox["Mathematica", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago", FontSlant->"Italic"], StyleBox[ " to sample the function more often so the graph is more accurate.", Evaluatable->False, AspectRatioFixed->True, FontFamily->"Chicago"] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Plot3D[f[x,y],{x,-1,1},{y,-1,1},PlotPoints->25]"], "Input", AspectRatioFixed->True], Cell[TextData[ "Alter the view point and/or increase the number of plot points until you get \ a graph that you feel shows the singularity well. Print it out and describe \ in your own words what happens at the singularity."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Part 3: You are on your own!"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Find and investigate the singularity of the function below. 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