(*********************************************************************** 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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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[ 75032, 2246]*) (*NotebookOutlinePosition[ 75870, 2273]*) (* CellTagsIndexPosition[ 75826, 2269]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["How to plot direction fields", "Title"], Cell["\<\ For plotting direction fields, you must first load a package:\ \>", "Text"], Cell[BoxData[ \(Needs["\"]\)], "Input"], Cell["\<\ Let us say your differential equation is y'=3y^2-t^3. Your \ direction field consists of arrow with direction (1,y') and unit length. The \ length of the vector (1,y') is sqrt(1+y'^2), so a unit vector is given by \ (1/sqrt(1+y'^2),y'/sqrt(1+y'^2)). The following commands define a vector \ function that yields this vector.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(yprime[t_, y_] = 3 y^2 - t^3\)], "Input"], Cell[BoxData[ \(\(-t\^3\) + 3\ y\^2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f[t_, y_] = {1/Sqrt[1 + yprime[t, y]^2], yprime[t, y]/Sqrt[1 + yprime[t, y]^2]}\)], "Input"], Cell[BoxData[ \({1\/\@\(1 + \((\(-t\^3\) + 3\ y\^2)\)\^2\), \(\(-t\^3\) + 3\ y\^2\)\/\@\(1 + \((\(-t\^3\) + 3\ y\^2)\)\^2\)}\)], "Output"] }, Open ]], Cell["\<\ Now the command PlotVectorField will plot the direction field. You \ have to input the range of t and y which you want. 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3P03o`h01?l=00Co3P04oa800olc00001P03o`h00ol>00?o3P03o`D000?o00001P03o`h00ol>00?o 3`02o`l00ol>00?o3`02oa@00_lc00001P02o`l00_l?00;o3`02o`H000?o00001`000ol0000>0003 o`0000h000?o00003P02o`l00_l?00;o4002oaD000?o0000<@0000H00_l?00;o3`02o`l00_l60003 o`0000L000?o00003P000ol0000>0003o`0000h00_l?00;o3`02oa4000?o0000A`0000L000?o0000 3P000ol0000>0003o`0000h000?o000010000ol000070003o`0000h000?o00003P000ol0000?0003 o`0000h000?o00003P000ol0001I0000@@000ol0002i0000@@000ol0002i0000@@000ol0002i0000 @@000ol0002i0000@@000ol0002i0000@@000ol0002i0000\ \>"], ImageRangeCache->{{{0, 252.25}, {287, 0}} -> {-1.1005, -2.40077, 0.0167294, 0.0167294}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ Warning: If you want to define a function of t and y, you must make \ sure t and y do not have values assigned to them. Let us see what happens if \ I try to define a new function g(t,y)=t-y, but t and y have values assigned \ to them.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(t = 1\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(y = 3\)], "Input"], Cell[BoxData[ \(3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[t_, y_] = t - y\)], "Input"], Cell[BoxData[ \(\(-2\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[2, 2]\)], "Input"], Cell[BoxData[ \(\(-2\)\)], "Output"] }, Open ]], Cell["\<\ You see that the values t=1, y=3 get used in the definition of the \ function g, so that g gets defined to be -2 instead of t-y. You can avoid \ this by clearing variables before you define a function.\ \>", "Text"], Cell[BoxData[ \(Clear[t, y]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(g[t_, y_] = t - y\)], "Input"], Cell[BoxData[ \(t - y\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(g[2, 2]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell["\<\ Now we get the correct value g(2,2)=0. If you are in doubt whether \ variables have values assigned to them, it is always a good idea to clear \ them before defining new functions.\ \>", "Text"] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 640}, {0, 460}}, WindowSize->{520, 365}, WindowMargins->{{4, Automatic}, {Automatic, 1}}, MacintoshSystemPageSetup->"\<\ 0@00D01804P000000]X2:?oQon82n@96D005:0?l0040D01804P000000]X2:000 0000I0000000000000000BL?00400@00000000000000000B0002I0000aP00000 8W800@0100<00P0000000000oU`10000\>" ] (*********************************************************************** 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. 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