(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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[ 37507, 1176]*) (*NotebookOutlinePosition[ 38456, 1212]*) (* CellTagsIndexPosition[ 38363, 1205]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Work in Conservative and Non-Conservative Force Fields", "Title"], Cell[TextData[StyleBox["Chapter 13, Section 3", FontFamily->"Arial", FontSize->16, FontWeight->"Bold"]], "Text"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell["\<\ OBJECTIVE: Visualize and evaluate work integrals along different paths, and \ observe the effect of following different paths through conservative and \ non-conservative force fields.\ \>", "Text"], Cell["\<\ You will explore integration over vector fields and experiment with both \ conservative and non-conservative force functions along different paths. \ These explorations should help you understand line integrals, as well as \ better appreciate situations when the work done is independent of the path \ taken.\ \>", "Text"], Cell[CellGroupData[{ Cell["Technology Guidelines", "Subsection", CellDingbat->"\[LightBulb]"], Cell[TextData[{ StyleBox["NOTE: If you have just finished a module, restart ", CellFrame->True, Background->None], StyleBox["Mathematica", CellFrame->True, FontSlant->"Italic", Background->None], StyleBox[ " before executing a new module.\nTO OPEN CELLS, put your cursor on the \ right cell bracket and double click.", CellFrame->True, Background->None], "\nTO STOP AN EXECUTION\n\tSelect the ", StyleBox["Kernel", FontSlant->"Italic"], " pull-down menu and click on ", StyleBox["Abort Evaluation.\n", FontSlant->"Italic"], "ORDER OF EXECUTION\n\tExecute cells in the order given. Do not skip any \ Input cells within a given notebook.\nSAVING NOTEBOOKS\n\tYou can save \ anytime to any directory you choose, and it is wise to save often.\n\t\ However, before you do your final save, delete all your output by selecting \ the \n\t ", StyleBox["Delete All Output", FontSlant->"Italic"], " selection under the ", StyleBox["Kernel", FontSlant->"Italic"], " pull-down menu.\nEXPERIENCING MAJOR PROBLEMS\n\tSave if appropriate, and \ then shut down ", StyleBox["Mathematica", FontSlant->"Italic"], " and start it up again." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Part I: Examples in Two Dimensions", "Section"], Cell[CellGroupData[{ Cell["Conservative Force", "Subsection"], Cell[CellGroupData[{ Cell["Verifying That the Force Is Conservative", "Subsubsection"], Cell[TextData[{ "Consider the following force defined by ", Cell[BoxData[ StyleBox[\({\(-x\)\ Cos[2 y], \(x\^2\) Sin[2 y]}, \), "DisplayFormula", FontFamily->"Times New Roman", FontWeight->"Bold"]]], " and verify that it is conservative." }], "Text"], Cell[BoxData[{ \(Clear[x, y, z, force]\ \), "\n", \(Off[General::spell]\ \), "\n", \(Off[General::spell1]\ \), "\n", \(\(force := {\(-x\)\ Cos[2 y], \(x\^2\) Sin[2 y]};\)\), "\n", \(my = D[force[\([1]\)], y] // Simplify\), "\n", \(nx = D[force[\([2]\)], x] // Simplify\), "\n", \(If[my == nx, Print["\"], Print["\"]]\)}], "Input"], Cell[BoxData[ ButtonBox[ ButtonBox[ RowBox[{ StyleBox["\[MathematicaIcon]", FontWeight->"Bold", FontColor->RGBColor[0.792981, 0.777356, 0.144533], FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["About", FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.500008]], StyleBox[" ", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.500008, 0, 0.500008]]}], ButtonStyle->"Paste"], ButtonData:>"h1", ButtonStyle->"Hyperlink"]], "Input", Evaluatable->False, CellTags->"hb1"] }, Closed]], Cell[CellGroupData[{ Cell["Visualizing the Force Field and Different Paths", "Subsubsection"], Cell["\<\ To visualize the force field, you need to first load a graphing package.\ \>", "Text"], Cell[BoxData[ \(\(\(<< Graphics`PlotField`\)\(\ \)\)\)], "Input"], Cell[TextData[{ "The force field is plotted together with three paths between (0, 0) and \ (1, 1): ", StyleBox["y = ", FontSlant->"Italic"], Cell[BoxData[ RowBox[{ StyleBox["x", FontSlant->"Italic"], ",", " ", StyleBox[\(\@x\), FontFamily->"Times New Roman", FontSlant->"Italic"], ",", RowBox[{ StyleBox["and", FontFamily->"Times New Roman"], StyleBox[ RowBox[{ StyleBox[" ", FontFamily->"Times New Roman"], " "}]], StyleBox[\(x\^3\), FontFamily->"Times New Roman", FontSlant->"Italic"]}]}]]], "." }], "Text"], Cell[BoxData[{\(Clear[x, y]\), "\n", RowBox[{ RowBox[{"pv", "=", RowBox[{"PlotVectorField", "[", RowBox[{"force", ",", \({x, 0, 1}\), ",", \({y, 0, 1}\), ",", StyleBox[\(DisplayFunction -> Identity\), "MR"]}], "]"}]}], ";", "\n", RowBox[{"pc", "=", RowBox[{"Plot", "[", RowBox[{\({x, \@x, x\^3}\), ",", \({x, 0, 1}\), ",", StyleBox[\(DisplayFunction -> Identity\), "MR"]}], "]"}]}], ";", "\n", \(Show[pv, pc, DisplayFunction\ :> \ $DisplayFunction]\), ";"}]}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Writing Parametrizations and Computing Work Integrals", "Subsubsection"], Cell[TextData[{ "Find the work done in traveling along the straight line ", Cell[BoxData[ \(TraditionalForm\`y\ = \ x\)]], ". Choose an appropriate parametrization. We write the position and \ velocity vector to assist in computing the work integral. Note that by \ setting ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " equal to particular functions of ", StyleBox["t,", FontSlant->"Italic"], " the force function will reflect that parametrization when it appears in \ the line integral." }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = t;\)\), "\n", \(\(y = t;\)\), "\n", \(\(r1 = {x, y};\)\), "\n", \(Print["\", v1 = D[r1, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w1 = Integrate[force . v1, {t, 0, 1}] // N\ ]\)}], "Input"], Cell[TextData[{ "Now find the work done in traveling along the lower curve ", Cell[BoxData[ FormBox[ RowBox[{"y", " ", "=", " ", FormBox[\(x\^3\), "TraditionalForm"]}], TraditionalForm]]], ". Choose an appropriate parametrization." }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = t;\)\), "\n", \(\(y = t\^3;\)\), "\n", \(\(r2 = {x, y};\)\), "\n", \(Print["\", v2 = D[r2, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w2 = Integrate[force . v2, {t, 0, 1}] // N\ ]\)}], "Input"], Cell[TextData[{ "Now find the work done in traveling along the upper curve ", Cell[BoxData[ FormBox[ RowBox[{"y", " ", "=", " ", FormBox[\(\@x\), "TraditionalForm"]}], TraditionalForm]]], ". As before, choose an appropriate parametrization. " }], "Text"], Cell[BoxData[{ \(\nClear[x, y, t]\ \), "\n", \(\(x = t;\)\), "\n", \(\(y = \@t;\)\), "\n", \(\(r3 = {x, y};\)\), "\n", \(Print["\", v3 = D[r3, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w3 = Integrate[force . v3, {t, 0, 1}] // N\ ]\)}], "Input"], Cell["Was the work done along each path the same?", "Text"], Cell[BoxData[ \(w1 == w2 == w3\)], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Non-Conservative Force", "Subsection"], Cell[CellGroupData[{ Cell["Defining and Visualizing the Force Field", "Subsubsection"], Cell["\<\ Consider the a spinning force and verify that it is not conservative.\ \>", "Text"], Cell[BoxData[{ \(Clear[x, y, z, force]\ \), "\n", \(Off[General::spell]\ \), "\n", \(Off[General::spell1]\ \), "\n", \(\(force = {\(-y\)/\@\(x\^2 + y\^2\), x/\@\(x\^2 + y\^2\)};\)\), "\n", \(my = D[force[\([1]\)], y] // Simplify\), "\n", \(nx = D[force[\([2]\)], x] // Simplify\), "\n", \(my == nx // Simplify\)}], "Input"], Cell[TextData[{ "You can see that for arbitrary values of ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], ", this last equation will not be true. Therefore, by the definition \ provided in this Section of your text, the force is not conservative." }], "Text"], Cell["\<\ There is no need to read in the following package if you have already done \ so.\ \>", "Text"], Cell[BoxData[ \(\(\(<< Graphics`PlotField`\)\(\ \)\)\)], "Input"], Cell["\<\ You will get an error message here since the force field is not defined at \ (0, 0), but just proceed. For this function, the work integrals will be \ improper, but they can still be found. The error messages remind us how \ difficult integration can be. We will use the same paths as above.\ \>", "Text"], Cell[BoxData[{\(Clear[x, y]\), "\n", RowBox[{ RowBox[{"pv", "=", RowBox[{"PlotVectorField", "[", RowBox[{"force", ",", \({x, 0.01, 1}\), ",", \({y, 0, 1}\), ",", StyleBox[\(DisplayFunction -> Identity\), "MR"]}], "]"}]}], ";", "\n", RowBox[{"pc", "=", RowBox[{"Plot", "[", RowBox[{\({x, \@x, x\^3}\), ",", \({x, 0, 1}\), ",", StyleBox[\(DisplayFunction -> Identity\), "MR"]}], "]"}]}], ";", "\n", \(Show[pv, pc, DisplayFunction\ :> \ $DisplayFunction]\), ";"}]}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Writing Parametrizations and Computing Work Integrals", "Subsubsection"], Cell[TextData[{ "As you have learned from hand computation, line integrals can be difficult \ to evaluate. Even ", StyleBox["Mathematica", FontSlant->"Italic"], " has problems with some line integrals. When you experiment with your own \ force or path functions, sometimes you may have to use numerical integration \ (", StyleBox["NIntegrate", FontWeight->"Bold"], "), and sometimes you may have to use symbolic integration (", StyleBox["Integrate", FontWeight->"Bold"], "). You will also get error messages at times, warning you about problems \ associated with the convergence of the integration technique. In these cases, \ you will frequently, but not always, be given an answer that is reasonably \ accurate.\n\nIn this example, we have a problem at", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(t = 0\)\)\)]], ", because the force function has a 0 denominator for the parametrizations \ given." }], "Text"], Cell[TextData[{ "Find the work done in traveling along the straight line ", StyleBox["y = x", FontSlant->"Italic"], ". Choose an appropriate parametrization. We write the position and \ velocity vector to assist in computing the work integral. Note that by \ setting ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " equal to particular functions of ", StyleBox["t", FontSlant->"Italic"], ", the force function will reflect that parametrization when it appears in \ the line integral." }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = t;\)\), "\n", \(\(y = t;\)\), "\n", \(\(r1 = {x, y};\)\), "\n", \(Print["\", v1 = D[r1, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w1 = Integrate[force . v1, {t, 0, 1}] // N\ ]\)}], "Input"], Cell["\<\ Could you have predicted this answer by looking at the graph?\ \>", "Text"], Cell[TextData[{ "Now find the work done in traveling along the lower curve ", StyleBox["y = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`x\^3\)], FontSlant->"Italic"], ". Choose an appropriate parametrization." }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = t;\)\), "\n", \(\(y = t\^3;\)\), "\n", \(\(r2 = {x, y};\)\), "\n", \(Print["\", v2 = D[r2, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w2 = NIntegrate[force . v2, {t, 0, 1}]]\)}], "Input"], Cell[TextData[{ "The result is very different from our previous value. Now find the work \ done in traveling along the upper curve ", StyleBox["y = ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\@x\)], FontSlant->"Italic"], ". As before, choose an appropriate parametrization. " }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\ \), "\n", \(\(x = t;\)\), "\n", \(\(y = \@t;\)\), "\n", \(\(r3 = {x, y};\)\), "\n", \(Print["\", v3 = D[r3, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w3 = Integrate[force . v3, {t, 0, 1}] // N\ ]\)}], "Input"], Cell["\<\ What does it mean when our work done went from 0 on the first path to a \ positive number on the second path and now is negative? Could you have \ predicted that from the graph showing the force field? This example \ demonstrates how for non-conservative forces, the work done in getting from \ one point to another is not independent of the path taken.\ \>", "Text"] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["You Try It: Part I", "Section"], Cell["\<\ Try a different path in going from (0, 0) to (1, 1). Suppose you go from (0, \ 0) to (1, 0) and then to (1, 1), all along parallel and perpendicular lines. \ The following commands specify appropriate parameterizations. You can execute \ them for any two-dimensional force. Begin by entering any force you wish by \ replacing the terms in red.\ \>", "Text"], Cell[BoxData[{\(Clear[x, y, t]\), "\n", RowBox[{ RowBox[{"newforce", ":=", RowBox[{"{", RowBox[{ StyleBox[\(Cos[5 x]\), FontColor->RGBColor[1, 0, 0]], ",", StyleBox[\(\(-3\) x\ y\), FontColor->RGBColor[1, 0, 0]]}], "}"}]}], ";"}]}], "Input"], Cell["First, go from (0, 0) to (1, 0).", "Text"], Cell[BoxData[{ \(\(x = t;\)\), "\n", \(\(y = 0;\)\), "\n", \(\(r1 = {x, y};\)\), "\n", \(\(dr1 = D[r1, t];\)\), "\n", \(Print["\", w4a = Integrate[newforce . dr1, {t, 0, 1}] // N\ ]\)}], "Input"], Cell["Next, go from (1, 0) to (1, 1).", "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(x = 1\), "\n", \(\(y = t;\)\), "\n", \(\(r2 = {x, y};\)\), "\n", \(\(dr2 = D[r2, t];\)\), "\n", \(Print["\", w4b = Integrate[newforce . dr2, {t, 0, 1}] // N\ ]\)}], "Input"], Cell["\<\ Add your results, and contrast them to what you would get with the paths used \ earlier. Do this for both conservative and non-conservative forces.\ \>", "Text"], Cell[BoxData[ \(w4 = w4a + w4b\)], "Input"], Cell[TextData[{ "Compute the work done along the line ", Cell[BoxData[ \(TraditionalForm\`y\ = \ x\)]], "." }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = t;\)\), "\n", \(\(y = t;\)\), "\n", \(\(r1 = {x, y};\)\), "\n", \(Print["\", v1 = D[r1, t]]\), "\n", \(Print["\", newforce]\), "\n", \(Print["\", w1 = Integrate[newforce . v1, {t, 0, 1}] // N\ ]\)}], "Input"], Cell[TextData[{ "Compute the work done along the curve ", Cell[BoxData[ \(TraditionalForm\`y\ = \ x\^3\)]], "." }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = t;\)\), "\n", \(\(y = t\^3;\)\), "\n", \(\(r2 = {x, y};\)\), "\n", \(Print["\", v2 = D[r2, t]]\), "\n", \(Print["\", newforce]\), "\n", \(Print["\", w2 = NIntegrate[newforce . v2, {t, 0, 1}]]\)}], "Input"], Cell[TextData[{ "Compute the work done along the curve ", Cell[BoxData[ \(TraditionalForm\`y\ = \@\(\ x\)\)]], "." }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\ \), "\n", \(\(x = t;\)\), "\n", \(\(y = \@t;\)\), "\n", \(\(r3 = {x, y};\)\), "\n", \(Print["\", v3 = D[r3, t]]\), "\n", \(Print["\", newforce]\), "\n", \(Print["\", w3 = Integrate[newforce . v3, {t, 0, 1}] // N\ ]\)}], "Input"], Cell["\<\ Is the work done in going from (0, 0) to (1, 1) the same for all the \ paths?\ \>", "Text"], Cell[BoxData[{ \({w1, w2, w3, w4} // TableForm\), "\n", \(w1 == w2 == w3 == w4\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Part II: Example in Three Dimensions", "Section"], Cell[TextData[StyleBox["Section 13.3, Exercise 30 ", FontWeight->"Bold"]], "Text"], Cell[TextData[{ "Given the force with components: ", StyleBox["{", FontSlant->"Italic"], Cell[BoxData[ FormBox[ StyleBox[\(e\^\(y\ z\), x\ z\ e\^\(y\ z\) + \ z\ Cos \((y)\), x\ y\ e\^\(y\ z\) + \ Sin \((y)\)\), FontSlant->"Italic", FontColor->GrayLevel[0]], TextForm]]], StyleBox["}", FontSlant->"Italic"], ", find the work done in going from ", Cell[BoxData[ \(TraditionalForm\`\((1, 0, 1)\)\)]], " to ", Cell[BoxData[ \(TraditionalForm\`\(\((1, \[Pi]/2, 0)\)\(\ \)\)\)]], " by traveling along three different paths. The parametrizations for these \ paths are detailed below and the work is computed for each. The results from \ the three paths are then compared and visualized." }], "Text"], Cell[BoxData[{ RowBox[{\(Clear[x, y, z, force]\), " "}], "\n", RowBox[{ RowBox[{"force", ":=", StyleBox[\({E\^\(y\ z\), x\ z\ E\^\(y\ z\) + \ z\ Cos[y], x\ y\ E\^\(y\ z\) + \ Sin[y]\ }\), FontColor->GrayLevel[0]]}], StyleBox[";", FontColor->GrayLevel[0]]}]}], "Input"], Cell[CellGroupData[{ Cell["Verifying That the Force Is Conservative", "Subsubsection"], Cell["\<\ By computing and comparing the appropriate first partial derivatives, verify \ that the force defined is conservative. \ \>", "Text"], Cell[BoxData[{ \(my = D[force[\([1]\)], y] // Simplify\), "\n", \(nx = D[force[\([2]\)], x] // Simplify\), "\n", \(mz = D[force[\([1]\)], z] // Simplify\), "\n", \(px = D[force[\([3]\)], x] // Simplify\), "\n", \(nz = D[force[\([2]\)], z] // Simplify\), "\n", \(py = D[force[\([3]\)], y] // Simplify\), "\n", \(my == nx\), "\n", \(mz == px\), "\n", \(nz == py\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Writing the Parameterization and Computing the Work Integral for 13.3, 30 a\ \>", "Subsubsection"], Cell[TextData[{ "Find the work done in traveling along the straight line from (1, 0, 1) to \ (1, \[Pi]/2, 0), as specified in 13.30 a. Choose an appropriate \ parametrization. Write the position and velocity vector to assist in \ computing the work integral. When you set ", StyleBox["x", FontSlant->"Italic"], ", ", StyleBox["y", FontSlant->"Italic"], ", and", StyleBox[" z", FontSlant->"Italic"], " equal to particular functions of ", StyleBox["t", FontSlant->"Italic"], ", the force function will reflect that parametrization when it appears in \ the line integral." }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = 1;\)\), "\n", \(\(y = \[Pi]\ t/2;\)\), "\n", \(\(z = 1 - t;\)\), "\n", \(\(r1 = {x, y, z};\)\), "\n", \(Print["\", v1 = D[r1, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w1 = Integrate[force . v1, {t, 0, 1}] // N\ ]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Writing Parametrizations and Computing Work Integrals for 13.3, 30 b\ \>", "Subsubsection"], Cell["\<\ Find the work done in the two straight line paths specified in part b. Choose \ appropriate parametrizations. First, go from (1,0,1) to the origin.\ \>", "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = 1 - t;\)\), "\n", \(\(y = 0;\)\), "\n", \(z = 1 - t\), "\n", \(\(r21 = {x, y, z};\)\), "\n", \(Print["\", v2 = D[r21, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w21 = Integrate[force . v2, {t, 0, 1}] // N\ ]\)}], "Input"], Cell[TextData["Go from the origin to (1, \[Pi]/2, 0)."], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = t;\)\), "\n", \(\(y = \[Pi]\ t/2;\)\), "\n", \(z = 0\), "\n", \(\(r22 = {x, y, z};\)\), "\n", \(Print["\", v2 = D[r22, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w22 = Integrate[force . v2, {t, 0, 1}] // N\ ]\)}], "Input"], Cell["So the total work done for part b is as follows", "Text"], Cell[BoxData[ \(w2 = w21 + w22\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Writing Parameterizations and Computing Work Integrals for 13.3, 30 c\ \>", "Subsubsection"], Cell["\<\ Find the work done for the two straight line paths plus the parabolic path \ specified in part c. Choose appropriate parametrizations. First, go from (1, \ 0, 1) to (1, 0, 0).\ \>", "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = 1;\)\), "\n", \(\(y = 0;\)\), "\n", \(\(z = 1 - t;\)\), "\n", \(\(r31 = {x, y, z};\)\), "\n", \(Print["\", v3 = D[r31, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w31 = Integrate[force . v3, {t, 0, 1}] // N\ ]\)}], "Input"], Cell[TextData[{ "Now follow the ", StyleBox["x", FontSlant->"Italic"], " axis in going from (1,0, 0) to the origin." }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = 1 - t;\)\), "\n", \(\(y = 0;\)\), "\n", \(\(z = 0;\)\), "\n", \(\(r32 = {x, y, z};\)\), "\n", \(Print["\", v3 = D[r32, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w32 = Integrate[force . v3, {t, 0, 1}] // N\ ]\)}], "Input"], Cell[TextData[ "Now follow a parabola from the origin to (1, \[Pi]/2, 0)."], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = t;\)\), "\n", \(\(y = \[Pi]\ t\^2/2;\)\), "\n", \(z = 0\), "\n", \(\(r33 = {x, y, z};\)\), "\n", \(Print["\", v3 = D[r33, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w33 = Integrate[force . v3, {t, 0, 1}] // N\ ]\)}], "Input"], Cell["So the total work done for part c is as follows.", "Text"], Cell[BoxData[ \(w3 = w31 + w32 + w33\)], "Input"], Cell["Was the work done along each path the same?", "Text"], Cell[BoxData[ \(w1 == w2 == w3\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Visualizing the Force Field and Different Paths", "Subsubsection"], Cell["\<\ To visualize the force field and the 3-D curves, first load two packages.\ \>", "Text"], Cell[BoxData[{ \(<< Graphics`ParametricPlot3D`\), "\n", \(<< Graphics`PlotField3D`\)}], "Input"], Cell[TextData[{ "The purpose of the next set of commands is to add color to each different \ part", Cell[BoxData[ \(TraditionalForm\`\(\ -\ \)\)]], "red to the part a curve, green to part b curves, and blue to part c \ curves. DO NOT EXECUTE THE FOLLOWING COMMANDS MORE THAN ONCE. If you do, the \ graphs below will not work unless you go back and re-execute all the cells \ that first defined the lists", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`r1, \ r21\)], FontSlant->"Italic"], ", etc." }], "Text"], Cell[BoxData[{ \(\(AppendTo[r1, RGBColor[1, 0, 0]];\)\), "\n", \(\(AppendTo[r21, RGBColor[0, 1, 0]];\)\), "\n", \(\(AppendTo[r22, RGBColor[0, 1, 0]];\)\), "\n", \(\(AppendTo[r31, RGBColor[0, 0, 1]];\)\), "\n", \(\(AppendTo[r32, RGBColor[0, 0, 1]];\)\), "\n", \(\(AppendTo[r33, RGBColor[0, 0, 1]];\)\)}], "Input"], Cell["\<\ The following set of commands will plot the force field, together with the \ paths in red (part a), green (part b), and blue (part c).\ \>", "Text"], Cell[BoxData[{ \(\(pp1 = ParametricPlot3D[Evaluate[r1], {t, 0, 1}, DisplayFunction -> Identity];\)\), "\n", \(\(pp21 = ParametricPlot3D[Evaluate[r21], {t, 0, 1}, DisplayFunction -> Identity];\)\), "\n", \(\(pp22 = ParametricPlot3D[Evaluate[r22], {t, 0, 1}, DisplayFunction -> Identity];\)\), "\n", \(\(pp31 = ParametricPlot3D[Evaluate[r31], {t, 0, 1}, DisplayFunction -> Identity];\)\), "\n", \(\(pp32 = ParametricPlot3D[Evaluate[r32], {t, 0, 1}, DisplayFunction -> Identity];\)\), "\n", \(\(pp33 = ParametricPlot3D[Evaluate[r33], {t, 0, 1}, DisplayFunction -> Identity];\)\), "\n", \(Clear[x, y, z]\), "\n", \(\(forceplot = PlotVectorField3D[force, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}, VectorHeads -> True, ScaleFactor -> .5, PlotPoints -> 4, DisplayFunction -> Identity];\)\), "\n", \(\(Show[pp1, pp21, pp22, pp31, pp32, pp33, forceplot, AxesLabel -> {x, y, z}, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["You Try It: Part II", "Section"], Cell["\<\ Redefine your force in the previous example, and re-execute all the cells. \ Only the terms in red in the force function need to be changed. This time try \ a non-conservative force. You could begin by checking out the function given; \ we have followed the same paths as in Part II. Be careful to use correct \ terminolgy.\ \>", "Text"], Cell[BoxData[{\(Clear[x, y, z, force]\), "\n", RowBox[{ RowBox[{ RowBox[{"force", ":=", RowBox[{"{", RowBox[{ StyleBox[\(\(x\^2\) Cos[y]\), FontColor->RGBColor[1, 0, 0]], ",", StyleBox[\(\(E\^\(y\ z\)\) \((1 - x)\)\), FontColor->RGBColor[1, 0, 0]], ",", StyleBox[\(x\ y\ \ Sin[z]\), FontColor->RGBColor[1, 0, 0]]}], " ", "}"}]}], ";"}], " "}]}], "Input"], Cell[CellGroupData[{ Cell["Is the Force Conservative or Not?", "Subsubsection"], Cell[BoxData[{ \(\(my = D[force[\([1]\)], y] // Simplify;\)\), "\n", \(\(nx = D[force[\([2]\)], x] // Simplify;\)\), "\n", \(\(mz = D[force[\([1]\)], z] // Simplify;\)\), "\n", \(\(px = D[force[\([3]\)], x] // Simplify;\)\), "\n", \(\(nz = D[force[\([2]\)], z] // Simplify;\)\), "\n", \(\(py = D[force[\([3]\)], y] // Simplify;\)\), "\n", \(my == nx\), "\n", \(mz == px\), "\n", \(nz == py\)}], "Input"], Cell["\<\ Are the corresponding partial derivatives equal to one another?\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Writing the Parametrization and Computing the Work Integral for 13.3, 30 a\ \>", "Subsubsection"], Cell[TextData[{ "Find the work done in traveling along the straight line from (1, 0, 1) to \ (1, \[Pi]/2, 0), as specified in part a). Choose an appropriate \ parametrization. Write the position and velocity vector to assist in \ computing the work integral. When you set ", StyleBox["x", FontSlant->"Italic"], ", ", StyleBox["y", FontSlant->"Italic"], ", and", StyleBox[" z", FontSlant->"Italic"], " equal to particular functions of ", StyleBox["t", FontSlant->"Italic"], ", the force function will reflect that parametrization when it appears in \ the line integral." }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = 1;\)\), "\n", \(\(y = \[Pi]\ t/2;\)\), "\n", \(\(z = 1 - t;\)\), "\n", \(\(r1 = {x, y, z};\)\), "\n", \(Print["\", v1 = D[r1, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w1 = Integrate[force . v1, {t, 0, 1}] // N\ ]\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Writing Parametrizations and Computing Work Integrals for 13.3, 30 b\ \>", "Subsubsection"], Cell["\<\ Now find the work done in the two straight line paths specified in part b. \ Choose appropriate parametrizations. First we will go from (1, 0, 1) to the \ origin.\ \>", "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = 1 - t;\)\), "\n", \(\(y = 0;\)\), "\n", \(z = 1 - t\), "\n", \(\(r21 = {x, y, z};\)\), "\n", \(Print["\", v2 = D[r21, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w21 = Integrate[force . v2, {t, 0, 1}] // N\ ]\)}], "Input"], Cell[TextData["Next, go from the origin to (1, \[Pi]/2, 0)."], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = t;\)\), "\n", \(\(y = \[Pi]\ t/2;\)\), "\n", \(z = 0\), "\n", \(\(r22 = {x, y, z};\)\), "\n", \(Print["\", v2 = D[r22, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w22 = Integrate[force . v2, {t, 0, 1}] // N\ ]\)}], "Input"], Cell["So the total work done for part b is as follows", "Text"], Cell[BoxData[ \(w2 = w21 + w22\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Writing Parametrizations and Computing Work Integrals for 13.3, 30 c\ \>", "Subsubsection"], Cell["\<\ Find the work done for the two straight line paths plus the parabolic path \ specified in part c. Choose appropriate parametrizations. First, go from (1, \ 0, 1) to (1, 0, 0).\ \>", "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = 1;\)\), "\n", \(\(y = 0;\)\), "\n", \(\(z = 1 - t;\)\), "\n", \(\(r31 = {x, y, z};\)\), "\n", \(Print["\", v3 = D[r31, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w31 = Integrate[force . v3, {t, 0, 1}] // N\ ]\)}], "Input"], Cell[TextData[{ "Now follow the ", StyleBox["x ", FontSlant->"Italic"], "axis in going from (1, 0, 0) to the origin." }], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = 1 - t;\)\), "\n", \(\(y = 0;\)\), "\n", \(\(z = 0;\)\), "\n", \(\(r32 = {x, y, z};\)\), "\n", \(Print["\", v3 = D[r32, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w32 = Integrate[force . v3, {t, 0, 1}] // N\ ]\)}], "Input"], Cell[TextData[ "Now follow a parabola from the origin to (1, \[Pi]/2, 0)."], "Text"], Cell[BoxData[{ \(Clear[x, y, t]\), "\n", \(\(x = t;\)\), "\n", \(\(y = \[Pi]\ t\^2/2;\)\), "\n", \(z = 0\), "\n", \(\(r33 = {x, y, z};\)\), "\n", \(Print["\", v3 = D[r33, t]]\), "\n", \(Print["\", force]\), "\n", \(Print["\", w33 = Integrate[force . v3, {t, 0, 1}] // N\ ]\)}], "Input"], Cell["So the total work done for partc is as follows", "Text"], Cell[BoxData[ \(w3 = w31 + w32 + w33\)], "Input"], Cell["Was the work done along each path the same?", "Text"], Cell[BoxData[ \(w1 == w2 == w3\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Visualizing the Force Field and Different Paths", "Subsubsection"], Cell["\<\ To visualize the force field and the 3-D curves, first load two packages.\ \>", "Text"], Cell[BoxData[{ \(<< Graphics`ParametricPlot3D`\), "\n", \(<< Graphics`PlotField3D`\)}], "Input"], Cell[TextData[{ "The purpose of the next set of commands is to add color to each different \ part", Cell[BoxData[ \(TraditionalForm\`\(\ -\ \)\)]], "red to the part a curve, green to part b curves, and blue to part c \ curves. DO NOT EXECUTE THE FOLLOWING COMMANDS MORE THAN ONCE. If you do, the \ graphs below will not work unless you go back and re-execute all the cells \ that first defined the lists", StyleBox[" r1, r21", FontSlant->"Italic"], ", etc." }], "Text"], Cell[BoxData[{ \(\(AppendTo[r1, RGBColor[1, 0, 0]];\)\), "\n", \(\(AppendTo[r21, RGBColor[0, 1, 0]];\)\), "\n", \(\(AppendTo[r22, RGBColor[0, 1, 0]];\)\), "\n", \(\(AppendTo[r31, RGBColor[0, 0, 1]];\)\), "\n", \(\(AppendTo[r32, RGBColor[0, 0, 1]];\)\), "\n", \(\(AppendTo[r33, RGBColor[0, 0, 1]];\)\)}], "Input"], Cell["\<\ The following set of commands will plot the force field, together with the \ paths in red (part a), green (part b), and blue (part c).\ \>", "Text"], Cell[BoxData[{ \(\(pp1 = ParametricPlot3D[Evaluate[r1], {t, 0, 1}, DisplayFunction -> Identity];\)\), "\n", \(\(pp21 = ParametricPlot3D[Evaluate[r21], {t, 0, 1}, DisplayFunction -> Identity];\)\), "\n", \(\(pp22 = ParametricPlot3D[Evaluate[r22], {t, 0, 1}, DisplayFunction -> Identity];\)\), "\n", \(\(pp31 = ParametricPlot3D[Evaluate[r31], {t, 0, 1}, DisplayFunction -> Identity];\)\), "\n", \(\(pp32 = ParametricPlot3D[Evaluate[r32], {t, 0, 1}, DisplayFunction -> Identity];\)\), "\n", \(\(pp33 = ParametricPlot3D[Evaluate[r33], {t, 0, 1}, DisplayFunction -> Identity];\)\), "\n", \(Clear[x, y, z]\), "\n", \(\(forceplot = PlotVectorField3D[force, {x, 0, 1}, {y, 0, 1}, {z, 0, 1}, VectorHeads -> True, ScaleFactor -> .5, PlotPoints -> 4, DisplayFunction -> Identity];\)\), "\n", \(\(Show[pp1, pp21, pp22, pp31, pp32, pp33, forceplot, AxesLabel -> {x, y, z}, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["\[MathematicaIcon]", FontWeight->"Bold", FontColor->RGBColor[0.792981, 0.777356, 0.144533], FontVariations->{"CompatibilityType"->0}], StyleBox[" ", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["About", FontWeight->"Bold", FontColor->RGBColor[0.500008, 0, 0.500008]], StyleBox[" ", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0.500008, 0, 0.500008]] }], "Section", CellDingbat->None], Cell[TextData[{ "The ", StyleBox["If", FontWeight->"Bold"], " statement used here is in standard programming format. If the statements \ in the first input is true, the second input is executed; if false, the third \ input is executed.\n", ButtonBox["Go back.", ButtonData:>"hb1", ButtonStyle->"Hyperlink"] }], "Text", PageWidth->WindowWidth, CellTags->"h1"] }, Closed]] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowSize->{767, 522}, WindowMargins->{{50, Automatic}, {Automatic, 18}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic} ] (*********************************************************************** 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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