(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 75251, 2036] NotebookOptionsPosition[ 29808, 1044] NotebookOutlinePosition[ 73493, 1980] CellTagsIndexPosition[ 73389, 1974] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ StyleBox["Lab 5B - Line Integrals of Vector Fields", FontSize->24, FontWeight->"Bold", FontVariations->{"Underline"->True}], "\nMath 2374 - University of Minnesota\nhttp://www.math.umn.edu/math2374\n\ Questions to: swenson@math.umn.edu" }], "Text", TextAlignment->Center, FontSize->12, FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Introduction", FontSize->14]], "Section", Background->None], Cell[TextData[{ "In last week's lab, we looked at arc length. You should remember from \ class that the arc length of a curve is calculated with a line integral: the \ arc length of the curve C is ", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "C", " "], RowBox[{"1", RowBox[{"\[DifferentialD]", "s"}]}]}]], "Text"], ", which is the same as ", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "a", "b"], RowBox[{"\[DoubleVerticalBar]", RowBox[{ RowBox[{ StyleBox["f", FontWeight->"Bold"], "'"}], RowBox[{"(", "t", ")"}]}], "\[DoubleVerticalBar]", RowBox[{"\[DifferentialD]", "t"}]}]}]], "Text"], " when C is parametrized by ", StyleBox["f", FontWeight->"Bold"], "(t), a\[LessEqual]t\[LessEqual]b. This time, we're looking at something \ which is related, but distinct: line integrals of vector fields. You should \ clearly separate these two concepts in your mind, and ask yourself, each time \ you calculate a line integral, \"What kind of function am I integrating?\" \ Your textbook often helps by writing line integrals of scalar functions with \ the differential \"\[DifferentialD]s,\" and line integrals of vector fields \ with the differential \"\[FilledSmallCircle] \[DifferentialD]", StyleBox["s ", FontWeight->"Bold"], ".\" Helpful, isn't it? (Sigh... just because the bold ", StyleBox["s", FontWeight->"Bold"], " can be hard to see, we might occasionally write ", Cell[BoxData[ FormBox[ RowBox[{"\[DifferentialD]", OverscriptBox["s", "\[RightVector]"]}], TraditionalForm]]], "to indicate when the 's' represents a vector. That's still not that \ different from the scalar case, though.)" }], "Text", FontSize->14], Cell[TextData[{ "On the other hand, you should not let yourself be confused by the symbols \ ", Cell[BoxData[ FormBox[ SubscriptBox["\[Integral]", RowBox[{" ", StyleBox["C", FontSlant->"Italic"]}]], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["\[ContourIntegral]", RowBox[{" ", "C"}]], TraditionalForm]]], "- they mean the same thing! The ", Cell[BoxData[ FormBox[ SubscriptBox["\[Integral]", RowBox[{" ", "C"}]], TraditionalForm]]], " symbol can always be used in place of ", Cell[BoxData[ FormBox[ SubscriptBox["\[ContourIntegral]", RowBox[{" ", "C"}]], TraditionalForm]]], ". We write ", Cell[BoxData[ FormBox[ SubscriptBox["\[ContourIntegral]", RowBox[{" ", "C"}]], TraditionalForm]]], " when we want an extra reminder that C is a closed path (it ends at the \ same point where it began). Of course, we don't write ", Cell[BoxData[ FormBox[ SubscriptBox["\[ContourIntegral]", RowBox[{" ", "C"}]], TraditionalForm]]], " if C is not closed!" }], "Text", FontSize->14], Cell[TextData[{ "Finally, be forewarned that, in some books, \"line integral\" and \"path \ integral\" are synonymous, whereas other books call them \"line integrals of \ vector fields\" and \"line integrals of scalar functions,\" respectively. \ Ask your TA if you need help sorting this out. In this lab we're working \ with the integrals of vector fields; recall from class and your textbook that \ these are typically used to measure Work. For a vector field ", StyleBox["F", FontWeight->"Bold"], " and a curve parametrized by ", StyleBox["f", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["(t)", FontSlant->"Italic"], ", ", Cell[BoxData[ FormBox[ RowBox[{ "a", " ", "\[LessEqual]", " ", "t", " ", "\[LessEqual]", " ", "b"}], TraditionalForm]]], ", they are defined as:" }], "Text", FontSize->14], Cell[BoxData[ RowBox[{" ", RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "C", " "], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["s", FontWeight->"Bold"]}]}]}], StyleBox[" ", FontWeight->"Bold"], StyleBox["=", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], RowBox[{ SubsuperscriptBox["\[Integral]", "a", "b"], RowBox[{ StyleBox["F", FontWeight->"Bold"], RowBox[{ RowBox[{"(", RowBox[{ StyleBox["f", FontWeight->"Bold"], RowBox[{"(", "t", ")"}]}], ")"}], "\[CenterDot]", RowBox[{ StyleBox["f", FontWeight->"Bold"], "'"}]}], RowBox[{"(", "t", ")"}], RowBox[{"\[DifferentialD]", "t"}]}]}]}]}]], "Text", TextAlignment->Center, FontSize->14, FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Example - Integration Using Mathematica", FontSize->14]], "Section", Background->None], Cell[TextData[{ "Here's a sample problem to illustrate some of the ", StyleBox["Mathematica", FontSlant->"Italic"], " commands we'll need this week. You should evaluate each \"evaluatable\" \ cell as you come to it." }], "Text"], Cell[TextData[{ "Calculate the integral ", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "C", " "], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["s", FontWeight->"Bold"]}]}]}]], "Text"], ", where ", StyleBox["F", FontWeight->"Bold"], "(x,y,z)=(", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["z", "3"], ",", SuperscriptBox["y", "2"], ",", "x"}], TraditionalForm]]], ") and C is parametrized by ", StyleBox["f", FontWeight->"Bold"], "(t)=(t,sin(t),1), 0\[LessEqual]t\[LessEqual]\[Pi]." }], "Text"], Cell["First, let's define and graph our functions.", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"F", "[", RowBox[{"x_", ",", "y_", ",", "z_"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"z", "^", "3"}], ",", RowBox[{"y", "^", "2"}], ",", "x"}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"f", "[", "t_", "]"}], "=", RowBox[{"{", RowBox[{"t", ",", RowBox[{"Sin", "[", "t", "]"}], ",", "1"}], "}"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"vf1", "=", RowBox[{"VectorFieldPlot3D", "[", RowBox[{ RowBox[{"F", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "Pi"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", "0", ",", "2"}], "}"}], ",", RowBox[{"VectorHeads", "\[Rule]", "True"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"c1", "=", RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "Pi"}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"Show", "[", RowBox[{"vf1", ",", "c1", ",", RowBox[{"Axes", "\[Rule]", "True"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}]}], "Input", CellChangeTimes->{{3.41235141526834*^9, 3.412351433092118*^9}}], Cell[TextData[{ "Which direction does C go? Look at the definition of ", StyleBox["f,", FontWeight->"Bold"], " and ask your TA if you're not sure. Then evaluate the cell below to \ create an animation and check your answer. (This is the same sort of \ animation that you created at the end of Lab 2A; ask your TA if you don't \ remember how to create and view them.) Should we expect the value of ", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "C", " "], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["s", FontWeight->"Bold"]}]}]}]]], " to be positive or negative?" }], "Text"], Cell[BoxData[ RowBox[{"PathAnimate3D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "\[Pi]"}], "}"}], ",", "40"}], "]"}]], "Input",\ GeneratedCell->False, CellAutoOverwrite->False], Cell[TextData[{ "Now that we know what we're working with, let's calculate. By definition, \ ", Cell[BoxData[ RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "C", " "], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["x", FontWeight->"Bold"]}]}]}], StyleBox["=", FontWeight->"Bold"], RowBox[{ SubsuperscriptBox["\[Integral]", "a", "b"], RowBox[{ StyleBox["F", FontWeight->"Bold"], RowBox[{ RowBox[{"(", RowBox[{ StyleBox["f", FontWeight->"Bold"], RowBox[{"(", "t", ")"}]}], ")"}], "\[CenterDot]", RowBox[{ StyleBox["f", FontWeight->"Bold"], "'"}]}], RowBox[{"(", "t", ")"}], RowBox[{"\[DifferentialD]", "t"}]}]}]}]], "Text"], ", so we must first plug ", StyleBox["f", FontWeight->"Bold"], " into ", StyleBox["F,", FontWeight->"Bold"], " and find the derivative of ", StyleBox["f", FontWeight->"Bold"], ". Here are the appropriate commands." }], "Text"], Cell[BoxData[{ RowBox[{"Apply", "[", RowBox[{"F", ",", RowBox[{"f", "[", "t", "]"}]}], "]"}], "\[IndentingNewLine]", RowBox[{"D", "[", RowBox[{ RowBox[{"f", "[", "t", "]"}], ",", "t"}], "]"}]}], "Input"], Cell[TextData[{ "Remember that we use the command ", StyleBox["Apply[F,f[t]]", FontFamily->"Courier", FontWeight->"Bold"], ", where you'd think that we could just use ", StyleBox["F[f[t]]", FontFamily->"Courier", FontWeight->"Bold"], ". We mentioned before that this is because ", StyleBox["Mathematica", FontSlant->"Italic"], " reads ", StyleBox["F[f[t]]", FontFamily->"Courier", FontWeight->"Bold"], " as ", StyleBox["F[{t,Sin[t],1}]", FontFamily->"Courier", FontWeight->"Bold"], ", but what we really want is ", StyleBox["F[t,Sin[t],1]", FontFamily->"Courier", FontWeight->"Bold"], ". Look closely until you see the difference!" }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell["\<\ Next, we need to take a dot product. To do this, we just put a dot (a \ period) between the two vectors:\ \>", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"{", RowBox[{"1", ",", RowBox[{ RowBox[{"Sin", "[", "t", "]"}], "^", "2"}], ",", "t"}], "}"}], ".", RowBox[{"{", RowBox[{"1", ",", RowBox[{"Cos", "[", "t", "]"}], ",", "0"}], "}"}]}]], "Input"], Cell["Finally, we integrate.", "Text"], Cell[BoxData[ RowBox[{"Integrate", "[", RowBox[{ RowBox[{"1", "+", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], " ", SuperscriptBox[ RowBox[{"Sin", "[", "t", "]"}], "2"]}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "Pi"}], "}"}]}], "]"}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["The Gravitational Force, Near Earth", FontSize->14]], "Section", Background->None, CellTags->"flatearth"], Cell[TextData[{ "In this example, we consider the gravitational force field for small \ objects near Earth. We approximate Earth by the ", StyleBox["xy-", FontSlant->"Italic"], "plane, and assume that the acceleration due to gravity is the constant ", StyleBox["g", FontSlant->"Italic"], "=9.8 ", Cell[BoxData[ FormBox[ FractionBox[ StyleBox["m", FontSlant->"Plain"], SuperscriptBox[ StyleBox["s", FontSlant->"Plain"], "2"]], TraditionalForm]]], ". Newton's first law of motion, ", StyleBox["F=ma", FontSlant->"Italic"], ", gives an expression for the gravitational force on an object: ", StyleBox["mg", FontSlant->"Italic"], ", where ", StyleBox["m", FontSlant->"Italic"], " is the mass of the object. This force pulls straight down, so the \ gravitational force field is ", StyleBox["F", FontWeight->"Bold"], "(x,y,z)=(0, 0, -", StyleBox["mg", FontSlant->"Italic"], "), which should not be a surprise. Here's a graph, for a specific value of \ ", StyleBox["m.", FontSlant->"Italic"] }], "Text", CellTags->"flatearth"], Cell[BoxData[{ RowBox[{ RowBox[{"m", "=", "1"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"g", "=", "9.8"}], ";"}], "\[IndentingNewLine]", RowBox[{" ", StyleBox[ RowBox[{ RowBox[{ RowBox[{"F", "[", RowBox[{"x_", ",", "y_", ",", "z_"}], "]"}], "=", RowBox[{"{", RowBox[{"0", ",", "0", ",", RowBox[{ RowBox[{"-", "m"}], "*", "g"}]}], "}"}]}], ";"}], FontFamily->"Courier", FontWeight->"Bold"], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"vf2", "=", RowBox[{"VectorFieldPlot3D", "[", RowBox[{ RowBox[{"F", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "50"}], ",", "50"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "50"}], ",", "50"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", RowBox[{"-", "50"}], ",", "50"}], "}"}], ",", " ", RowBox[{"VectorHeads", "\[Rule]", "True"}], ",", RowBox[{"ScaleFactor", "\[Rule]", "Max"}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.412352527857798*^9, 3.41235257991551*^9}}, CellTags->"flatearth"], Cell[TextData[{ StyleBox["\"Points for Style\"", FontWeight->"Bold"], ": Since we're picturing Earth as flat, it would make more sense for the \ gravitational force to stop at Earth's surface. The ", StyleBox["If", FontFamily->"Courier", FontWeight->"Bold"], " command will help (you can look it up in the Help Browser). Try it by \ replacing the above definition of ", StyleBox["F", FontWeight->"Bold"], " with ", StyleBox["F[x_,y_,z_]:={0,0,-m*g}*If[(z\[LessEqual]0),0,1]", FontFamily->"Courier", FontWeight->"Bold"], "." }], "Text", CellFrame->True, Background->GrayLevel[0.849989], CellTags->"flatearth"], Cell[TextData[{ "Let ", StyleBox["f", FontWeight->"Bold"], "(t), a\[LessEqual]t\[LessEqual]b, parametrize a curve C. Then ", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "C", " "], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["x", FontWeight->"Bold"]}]}]}]], "Text"], " represents the work done by gravity on a particle moving from the \ beginning of C to the end. In other words, this is the gravitational \ potential energy that the particle loses (or \"uses up\") by following the \ curve C. If you've studied physics, you should expect that this equals ", StyleBox["mg\[CapitalDelta]h = mg", FontSlant->"Italic"], "(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["f", FontSlant->"Plain"], "3"], TraditionalForm]]], "(a)-", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["f", FontSlant->"Plain"], "3"], TraditionalForm]]], "(b)), where ", StyleBox["f", FontWeight->"Bold"], "=(", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["f", FontSlant->"Plain"], "1"], TraditionalForm]]], ",", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["f", FontSlant->"Plain"], "2"], TraditionalForm]]], ",", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["f", FontSlant->"Plain"], "3"], TraditionalForm]]], "). Let's prove this." }], "Text", CellTags->"flatearth"], Cell[TextData[{ "By definition, ", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "C", " "], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["s", FontWeight->"Bold"]}]}]}]]], "=", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "a", "b"], RowBox[{ StyleBox["F", FontWeight->"Bold"], RowBox[{ RowBox[{"(", RowBox[{ StyleBox["f", FontWeight->"Bold"], RowBox[{"(", "t", ")"}]}], ")"}], "\[CenterDot]", RowBox[{ StyleBox["f", FontWeight->"Bold"], "'"}]}], RowBox[{"(", "t", ")"}], RowBox[{"\[DifferentialD]", "t"}]}]}]], "Text"], ", and since ", StyleBox["F", FontWeight->"Bold"], " is constant, ", StyleBox["F", FontWeight->"Bold"], "(", StyleBox["f", FontWeight->"Bold"], "(t)) is (0,0,-", StyleBox["mg", FontSlant->"Italic"], ")=-", StyleBox["mg", FontSlant->"Italic"], StyleBox["k,", FontWeight->"Bold"], " where ", StyleBox["k", FontWeight->"Bold"], "=(0,0,1)", ". Then we have ", Cell[BoxData[ RowBox[{ RowBox[{ SubsuperscriptBox["\[Integral]", "a", "b"], RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"-", StyleBox[ RowBox[{ StyleBox["mg", FontSlant->"Italic"], StyleBox["k", FontWeight->"Bold"]}]]}], ")"}], "\[CenterDot]", RowBox[{ StyleBox["f", FontWeight->"Bold"], "'"}]}], RowBox[{"(", "t", ")"}], RowBox[{"\[DifferentialD]", "t"}]}]}], "=", RowBox[{ RowBox[{ RowBox[{"-", StyleBox["mg", FontSlant->"Italic"]}], RowBox[{ SubsuperscriptBox["\[Integral]", "a", "b"], RowBox[{ RowBox[{ SubscriptBox["f", "3"], "'"}], RowBox[{"(", "t", ")"}], RowBox[{"\[DifferentialD]", "t"}]}]}]}], "=", RowBox[{ RowBox[{"-", StyleBox["mg", FontSlant->"Italic"]}], RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["f", "3"], RowBox[{"(", "b", ")"}]}], "-", RowBox[{ SubscriptBox["f", "3"], RowBox[{"(", "a", ")"}]}]}], ")"}]}]}]}]], "Text"], ", using the Fundamental Theorem of Calculus." }], "Text", CellTags->"flatearth"], Cell[TextData[{ StyleBox["Exercise 1", FontWeight->"Bold"], ": Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to check this calculation, for the path parametrized by ", StyleBox["f", FontWeight->"Bold"], "(t)=(", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"50", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], RowBox[{"cos", "(", "t", ")"}]}], ",", RowBox[{"50", SuperscriptBox["\[ExponentialE]", RowBox[{"-", "t"}]], RowBox[{"sin", "(", "t", ")"}]}], ",", " ", RowBox[{ SuperscriptBox["t", "2"], "+", RowBox[{"5", "t"}]}]}], TraditionalForm]]], "), 0\[LessEqual]t\[LessEqual]5. [Specifically: Use ", StyleBox["ParametricPlot3D", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["Show", FontFamily->"Courier", FontWeight->"Bold"], " to plot the path and the vector field together, then evaluate the \ integral. Finally, check that the answer is what you expected! Use the \ values ", StyleBox["m", FontSlant->"Italic"], "=1, ", StyleBox["g", FontSlant->"Italic"], "=9.8, as given above.]" }], "Text", CellFrame->True, Background->RGBColor[1, 0.501961, 0.501961], CellTags->"flatearth"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["The Gravitational Force, Farther From Earth", FontSize->14]], "Section", Background->None, CellTags->"spaceflight"], Cell[TextData[{ "Newton's law of gravitation shows how the force of Earth's gravity on an \ object diminishes as the object moves farther from Earth. If the center of \ the earth is at the origin (0,0,0), then the exact formula is ", StyleBox["F", FontWeight->"Bold"], "(", StyleBox["v", FontWeight->"Bold"], ")=", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{"(", FractionBox["GmM", RowBox[{"\[DoubleVerticalBar]", "v", SuperscriptBox["\[DoubleVerticalBar]", "2"]}]], ")"}], RowBox[{"(", RowBox[{"-", FractionBox["v", RowBox[{"\[DoubleVerticalBar]", "v", "\[DoubleVerticalBar]"}]]}], ")"}]}], FontSize->16], TraditionalForm]]], ".\n\nHere's what the letters mean:\n", StyleBox["F", FontWeight->"Bold"], " is the gravitational force; a vector\n", StyleBox["v", FontWeight->"Bold"], " is the position of the object -- the vector (x,y,z)\nG is the universal \ gravitational constant, 6.67\[CenterDot]", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"-", "11"}]], TraditionalForm]]], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ StyleBox["N", FontSlant->"Plain"], StyleBox["\[CenterDot]", FontSlant->"Plain"], SuperscriptBox[ StyleBox["m", FontSlant->"Plain"], "2"]}], SuperscriptBox["kg", "2"]], TraditionalForm]]], "\nm is the mass of the object\nM is the mass of Earth, 5.98\[CenterDot]", Cell[BoxData[ FormBox[ SuperscriptBox["10", "24"], TraditionalForm]]], "kg\n\nIn this formula, the first factor, ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["(", FontSize->16], StyleBox[ FractionBox["GmM", RowBox[{"\[DoubleVerticalBar]", "v", SuperscriptBox["\[DoubleVerticalBar]", "2"]}]], FontSize->16], StyleBox[")", FontSize->18]}], TraditionalForm]]], ", is a scalar, which measures how strong the gravitational force will be. \ The second factor, ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"-", FractionBox["v", RowBox[{"\[DoubleVerticalBar]", "v", "\[DoubleVerticalBar]"}]]}], ")"}], TraditionalForm]]], ", is the unit vector pointing from the object toward Earth; naturally, this \ is the direction of the gravitational force!" }], "Text", CellTags->"spaceflight"], Cell[TextData[{ "Suppose that the position of our object at time t is given by ", StyleBox["f", FontWeight->"Bold"], "(t), a\[LessEqual]t\[LessEqual]b. Then the work done by Earth's gravity \ between time a and time b is given by the integral ", Cell[BoxData[ RowBox[{ SubscriptBox["\[Integral]", "C"], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["x", FontWeight->"Bold"]}]}]}]], "Text"], ", where C is the curve parametrized by ", StyleBox["f", FontWeight->"Bold"], "." }], "Text", CellTags->"spaceflight"], Cell[TextData[{ StyleBox["Exercise 2", FontWeight->"Bold"], ": Let C be parametrized by ", StyleBox["f", FontWeight->"Bold"], "(t)=(0,0,", Cell[BoxData[ FormBox[ FractionBox["R", RowBox[{"1", "-", "t"}]], TraditionalForm]]], "), 0"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["s", FontWeight->"Bold"]}]}]}]]], " , the work done by Earth's gravity, as a TA with mass 90 kg travels along \ C. [Hint: Your answer should be negative: this represents work against \ gravity.]\nc) Calculate the work done by gravity as a Boeing 747-400 with \ mass 396890 kg climbs to a cruising altitude of 10000 m. [Hint: you may use \ any path with initial height R and final height R+10000.]\nd) Compare your \ answers to parts b) and c). What could you do to the TA with the energy from \ 57,285 gallons of jet fuel?\n[For airliner-related questions, see ", ButtonBox["http://www.boeing.com/commercial/747family/technical.html", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://www.boeing.com/commercial/747family/technical.html"], None}], ".]" }], "Text", CellFrame->True, Background->RGBColor[1, 0.501961, 0.501961], CellTags->"spaceflight"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["A Radical New Vector Field", FontSize->14]], "Section", Background->None], Cell[TextData[{ "In the ", ButtonBox["first gravitational-force example", BaseStyle->"Hyperlink", ButtonData:>"flatearth"], ", we found that the value of the line integral of the gravitational force \ field depended only on the heights of the endpoints. The same is true in the \ ", ButtonBox["second gravitational-force example", BaseStyle->"Hyperlink", ButtonData:>"spaceflight"], ", as Barr proves on pp. 289-290. When a vector field's line integrals \ depend only on the endpoints (and not on the route connecting them), the \ vector field is called ", StyleBox["conservative", FontSlant->"Italic"], ". From the first two examples, a person might get the idea that this \ always happens, but it's actually pretty rare. In the following exercise, \ for example, ", StyleBox["F", FontWeight->"Bold"], " is not conservative (hence the title of this section)." }], "Text"], Cell[TextData[{ StyleBox["Exercise 3", FontWeight->"Bold"], ": Let ", StyleBox["F", FontWeight->"Bold"], "(x,y,z)=(y,-x,z). For each curve C: first, animate the curve using the ", StyleBox["PathAnimate3D", FontFamily->"Courier", FontWeight->"Bold"], " and/or ", StyleBox["PathTangentAnimate3D", FontFamily->"Courier", FontWeight->"Bold"], " commands. Then find the endpoints of the curve (specify which is the \ beginning and which is the end), and evaluate ", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", "C", " "], RowBox[{ StyleBox["F", FontWeight->"Bold"], "\[CenterDot]", RowBox[{"\[DifferentialD]", StyleBox["s", FontWeight->"Bold"]}]}]}]]], ".\n\na) C is the upper half of the unit circle in the ", StyleBox["xy", FontSlant->"Italic"], "-plane -- i.e. the half where ", StyleBox["y\[GreaterEqual]", FontSlant->"Italic"], "0. [", StyleBox["Hint", FontWeight->"Bold"], ": All points in the ", StyleBox["xy", FontSlant->"Italic"], "-plane have the same ", StyleBox["z", FontSlant->"Italic"], "-value -- what is it?]\nb) C is parametrized by ", StyleBox["f", FontWeight->"Bold"], "(t)=(cos t, -sin t, sin 4t), 0\[LessEqual]t\[LessEqual]\[Pi].\nc) C is \ contained in both the plane ", StyleBox["y", FontSlant->"Italic"], "=0 and the parabolic sheet ", StyleBox["z=", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "-", "1"}], TraditionalForm]]], "; it begins at ", StyleBox["x", FontSlant->"Italic"], "=1 and ends at ", StyleBox["x", FontSlant->"Italic"], "=-1.\nd) C is parametrized by ", StyleBox["f", FontWeight->"Bold"], "(t)=(-t, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"(", FractionBox[ RowBox[{"-", "1"}], RowBox[{"1", "-", SuperscriptBox["t", "2"]}]], ")"}]], "cos", " ", RowBox[{"(", RowBox[{"20", "t"}], ")"}]}], ","}], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"(", FractionBox[ RowBox[{"-", "1"}], RowBox[{"1", "-", SuperscriptBox["t", "2"]}]], ")"}]], "sin", " ", RowBox[{"(", RowBox[{"20", "t"}], ")"}]}], TraditionalForm]]], "), -1\[LessEqual]t\[LessEqual]1. [", StyleBox["Hint", FontWeight->"Bold"], ": This is a hard integral to compute, even for ", StyleBox["Mathematica", FontSlant->"Italic"], ". Instead of using the command ", StyleBox["Integrate", FontFamily->"Courier", FontWeight->"Bold"], " in this part, use ", StyleBox["NIntegrate", FontFamily->"Courier", FontWeight->"Bold"], ", which approximates the value of the integral. Notice that ", StyleBox["f", FontWeight->"Bold"], "(t) is not actually defined at the two endpoints, so you'll have to \ integrate from \"close to -1\" to \"close to 1.\" Try integrating from -.99 \ to .99, then -.999 to .999, and so on, to see if the values approach \ something. Ask your TA about using limits to evaluate improper integrals if \ this doesn't seem familiar.]" }], "Text", CellFrame->True, Background->RGBColor[1, 0.501961, 0.501961]], Cell[CellGroupData[{ Cell["Credits", "Subsection"], Cell["\<\ This lab was written from scratch in 2002 by James Swenson. In Spring 2004 \ Jonathan Rogness went through and updated a few minor things to reflect the \ use of our math2374.nb file. (Fall 2004: a few minor updates and changed \ notation to match the new textbook.) This lab is copyright 2002 by James Swenson (swenson@math.umn.edu) and is \ protected by the Creative Commons Attribution-NonCommercial-ShareAlike \ License. You can find more information on this license at \ http://creativecommons.org/licenses/by-nc-sa/1.0/. Although it's not specifically required by the license, I'd appreciate it if \ you let me know at rogness@math.umn.edu if you use parts of our labs, just so \ I can keep track of it. 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The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["Hyperlink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookLocate[#2]}]& ), ButtonNote -> ButtonData}], Cell[ StyleData["Hyperlink", "Presentation"], FontSize -> 16], Cell[ StyleData["Hyperlink", "Condensed"], FontSize -> 11], Cell[ StyleData["Hyperlink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ "The following styles are for linking automatically to the on-line \ help system.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["MainBookLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, ButtonFrame -> "None", ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["MainBook", #]}]& )}], Cell[ StyleData["MainBookLink", "Presentation"], FontSize -> 16], Cell[ StyleData["MainBookLink", "Condensed"], FontSize -> 11], Cell[ StyleData["MainBookLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["AddOnsLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontFamily -> "Courier", FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, ButtonFrame -> "None", ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["AddOns", #]}]& )}], Cell[ StyleData["AddOnsLink", "Presentation"], FontSize -> 16], Cell[ StyleData["AddOnsLink", "Condensed"], FontSize -> 11], Cell[ StyleData["AddOnsLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["RefGuideLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontFamily -> "Courier", FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, ButtonFrame -> "None", ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["RefGuide", #]}]& )}], Cell[ StyleData["RefGuideLink", "Presentation"], FontSize -> 16], Cell[ StyleData["RefGuideLink", "Condensed"], FontSize -> 11], Cell[ StyleData["RefGuideLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["GettingStartedLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, ButtonFrame -> "None", ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["GettingStarted", #]}]& )}], Cell[ StyleData["GettingStartedLink", "Presentation"], FontSize -> 16], Cell[ StyleData["GettingStartedLink", "Condensed"], FontSize -> 11], Cell[ StyleData["GettingStartedLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["OtherInformationLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, ButtonFrame -> "None", ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["OtherInformation", #]}]& )}], Cell[ StyleData["OtherInformationLink", "Presentation"], FontSize -> 16], Cell[ StyleData["OtherInformationLink", "Condensed"], FontSize -> 11], Cell[ StyleData["OtherInformationLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[ StyleData["Header"], CellMargins -> {{0, 0}, {4, 1}}, DefaultNewInlineCellStyle -> "None", LanguageCategory -> "NaturalLanguage", StyleMenuListing -> None, FontSize -> 10, FontSlant -> "Italic"], Cell[ StyleData["Footer"], CellMargins -> {{0, 0}, {0, 4}}, DefaultNewInlineCellStyle -> "None", LanguageCategory -> "NaturalLanguage", StyleMenuListing -> None, FontSize -> 9, FontSlant -> "Italic"], Cell[ StyleData["PageNumber"], CellMargins -> {{0, 0}, {4, 1}}, StyleMenuListing -> None, FontFamily -> "Times", FontSize -> 10]}, Closed]], Cell[ CellGroupData[{ Cell["Palette Styles", "Section"], Cell[ "The cells below define styles that define standard ButtonFunctions, \ for use in palette buttons.", "Text"], Cell[ StyleData["Paste"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, After]}]& )}], Cell[ StyleData["Evaluate"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["EvaluateCell"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["CopyEvaluate"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["CopyEvaluateCell"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[], All]}]& )}]}, Closed]], Cell[ CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell[ "The cells below define styles useful for making placeholder objects \ in palette templates.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["Placeholder"], Placeholder -> True, StyleMenuListing -> None, FontSlant -> "Italic", FontColor -> RGBColor[0.890623, 0.864698, 0.384756], TagBoxOptions -> { Editable -> False, Selectable -> False, StripWrapperBoxes -> False}], Cell[ StyleData["Placeholder", "Presentation"]], Cell[ StyleData["Placeholder", "Condensed"]], Cell[ StyleData["Placeholder", "Printout"]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["PrimaryPlaceholder"], StyleMenuListing -> None, DrawHighlighted -> True, FontSlant -> "Italic", Background -> RGBColor[0.912505, 0.891798, 0.507774], TagBoxOptions -> { Editable -> False, Selectable -> False, StripWrapperBoxes -> False}], Cell[ StyleData["PrimaryPlaceholder", "Presentation"]], Cell[ StyleData["PrimaryPlaceholder", "Condensed"]], Cell[ StyleData["PrimaryPlaceholder", "Printout"]]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["FormatType Styles", "Section"], Cell[ "The cells below define styles that are mixed in with the styles of \ most cells. If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options. This is particularly true of Input and Output.", "Text"], Cell[ StyleData["CellExpression"], PageWidth -> Infinity, CellMargins -> {{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel -> False, ShowSpecialCharacters -> False, AllowInlineCells -> False, Hyphenation -> False, AutoItalicWords -> {}, StyleMenuListing -> None, FontFamily -> "Courier", FontSize -> 12, Background -> GrayLevel[1]], Cell[ StyleData["InputForm"], InputAutoReplacements -> {}, AllowInlineCells -> False, Hyphenation -> False, StyleMenuListing -> None, FontFamily -> "Courier"], Cell[ StyleData["OutputForm"], PageWidth -> Infinity, TextAlignment -> Left, LineSpacing -> {0.6, 1}, StyleMenuListing -> None, FontFamily -> "Courier"], Cell[ StyleData["StandardForm"], InputAutoReplacements -> { "->" -> "\[Rule]", ":>" -> "\[RuleDelayed]", "<=" -> "\[LessEqual]", ">=" -> "\[GreaterEqual]", "!=" -> "\[NotEqual]", "==" -> "\[Equal]", Inherited}, LineSpacing -> {1.25, 0}, StyleMenuListing -> None, FontFamily -> "Courier"], Cell[ StyleData["TraditionalForm"], InputAutoReplacements -> { "->" -> "\[Rule]", ":>" -> "\[RuleDelayed]", "<=" -> "\[LessEqual]", ">=" -> "\[GreaterEqual]", "!=" -> "\[NotEqual]", "==" -> "\[Equal]", Inherited}, LineSpacing -> {1.25, 0}, SingleLetterItalics -> True, TraditionalFunctionNotation -> True, DelimiterMatching -> None, StyleMenuListing -> None], Cell[ "The style defined below is mixed in to any cell that is in an \ inline cell within another.", "Text"], Cell[ StyleData["InlineCell"], TextAlignment -> Left, ScriptLevel -> 1, StyleMenuListing -> None], Cell[ StyleData["InlineCellEditing"], StyleMenuListing -> None, Background -> RGBColor[1, 0.749996, 0.8]]}, Closed]], Cell[ CellGroupData[{ Cell["Automatic Styles", "Section"], Cell[ "The cells below define styles that are used to affect the display \ of certain types of objects in typeset expressions. For example, \ \"UnmatchedBracket\" style defines how unmatched bracket, curly bracket, and \ parenthesis characters are displayed (typically by coloring them to make them \ stand out).", "Text"], Cell[ StyleData["UnmatchedBracket"], StyleMenuListing -> None, FontColor -> RGBColor[0.760006, 0.330007, 0.8]]}, Closed]]}, Open]]}, Visible -> False, FrontEndVersion -> "6.0 for Linux x86 (32-bit) (April 20, 2007)", StyleDefinitions -> "Default.nb"] ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{ "flatearth"->{ Cell[12012, 421, 132, 3, 30, "Section", CellTags->"flatearth"], Cell[12147, 426, 1088, 38, 87, "Text", CellTags->"flatearth"], Cell[13238, 466, 1161, 34, 129, "Input", CellTags->"flatearth"], Cell[14402, 502, 633, 20, 83, "Text", CellTags->"flatearth"], Cell[15038, 524, 1443, 55, 68, "Text", CellTags->"flatearth"], Cell[16484, 581, 2316, 96, 57, "Text", CellTags->"flatearth"], Cell[18803, 679, 1218, 44, 82, "Text", CellTags->"flatearth"]}, "spaceflight"->{ Cell[20058, 728, 142, 3, 30, "Section", CellTags->"spaceflight"], Cell[20203, 733, 2338, 77, 261, "Text", CellTags->"spaceflight"], Cell[22544, 812, 608, 20, 50, "Text", CellTags->"spaceflight"], Cell[23155, 834, 1518, 41, 180, "Text", CellTags->"spaceflight"]} } *) (*CellTagsIndex CellTagsIndex->{ {"flatearth", 72586, 1947}, {"spaceflight", 73083, 1962} } *) (*NotebookFileOutline Notebook[{ Cell[568, 21, 369, 11, 113, "Text"], Cell[CellGroupData[{ Cell[962, 36, 85, 2, 50, "Section"], Cell[1050, 40, 1738, 43, 169, "Text"], Cell[2791, 85, 1078, 37, 77, "Text"], Cell[3872, 124, 837, 23, 90, "Text"], Cell[4712, 149, 900, 35, 51, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[5649, 189, 112, 2, 30, "Section"], Cell[5764, 193, 234, 6, 47, "Text"], Cell[6001, 201, 622, 24, 32, "Text"], Cell[6626, 227, 60, 0, 29, "Text"], Cell[6689, 229, 1592, 49, 149, "Input"], Cell[8284, 280, 686, 18, 68, "Text"], Cell[8973, 300, 244, 8, 28, "Input"], Cell[9220, 310, 1078, 41, 52, "Text"], Cell[10301, 353, 219, 6, 49, "Input"], Cell[10523, 361, 732, 27, 83, "Text"], Cell[11258, 390, 129, 3, 29, "Text"], Cell[11390, 395, 253, 8, 28, "Input"], Cell[11646, 405, 38, 0, 29, "Text"], Cell[11687, 407, 288, 9, 28, "Input"] }, Closed]], Cell[CellGroupData[{ Cell[12012, 421, 132, 3, 30, "Section", CellTags->"flatearth"], Cell[12147, 426, 1088, 38, 87, "Text", CellTags->"flatearth"], Cell[13238, 466, 1161, 34, 129, "Input", CellTags->"flatearth"], Cell[14402, 502, 633, 20, 83, "Text", CellTags->"flatearth"], Cell[15038, 524, 1443, 55, 68, "Text", CellTags->"flatearth"], Cell[16484, 581, 2316, 96, 57, "Text", CellTags->"flatearth"], Cell[18803, 679, 1218, 44, 82, "Text", CellTags->"flatearth"] }, Closed]], Cell[CellGroupData[{ Cell[20058, 728, 142, 3, 30, "Section", CellTags->"spaceflight"], Cell[20203, 733, 2338, 77, 261, "Text", CellTags->"spaceflight"], Cell[22544, 812, 608, 20, 50, "Text", CellTags->"spaceflight"], Cell[23155, 834, 1518, 41, 180, "Text", CellTags->"spaceflight"] }, Closed]], Cell[CellGroupData[{ Cell[24710, 880, 99, 2, 30, "Section"], Cell[24812, 884, 897, 22, 101, "Text"], Cell[25712, 908, 3229, 112, 278, "Text"], Cell[CellGroupData[{ Cell[28966, 1024, 29, 0, 46, "Subsection"], Cell[28998, 1026, 782, 14, 191, "Text"] }, Closed]] }, Closed]] } ] *) (* End of internal cache information *)