(* 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[ 20494, 565] NotebookOptionsPosition[ 18807, 513] NotebookOutlinePosition[ 19210, 530] CellTagsIndexPosition[ 19167, 527] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Lecture 5", "Section", CellChangeTimes->{{3.426947592244849*^9, 3.4269475961628494`*^9}, { 3.4315423019150314`*^9, 3.4315423352420316`*^9}, {3.4327427093176003`*^9, 3.4327427098915997`*^9}}], Cell["\<\ In the Lecture we discuss the use of complex numbers and complex functions to \ solve linear second order (in)homogeneous ordinary differential equations by \ using the Ansatz of complex exponential time dependence. Here we will \ discuss the tools available in Mathematica to study such equations. In \ particular, recall that we used the Solve command for algebraic equations. \ Now we want to employ the DSolve command for differential equations.\ \>", "Text", CellChangeTimes->{{3.4269476003598495`*^9, 3.4269476357228494`*^9}, 3.4315423175140314`*^9, {3.431542352626032*^9, 3.431542424331032*^9}, { 3.4327427140746*^9, 3.4327429812265997`*^9}, 3.4327511315046*^9, 3.4327511653416*^9}], Cell["\<\ First consider the homogeneous (undriven) equation for a mass on a spring \ with viscous damping. Note that DSolve expects derivatives to be indicated \ by primes, 's, and that the desire function or depedent variable (second \ argument) and independent variable (third argument) be clearly indicated. We \ have\ \>", "Text", CellChangeTimes->{{3.431542448713032*^9, 3.4315424510580316`*^9}, { 3.4315424941060314`*^9, 3.4315425335300317`*^9}, {3.4327430108306*^9, 3.4327432013976*^9}, {3.4327432946476*^9, 3.4327433118436003`*^9}, { 3.4327511969636*^9, 3.4327512099116*^9}}], Cell[BoxData[ RowBox[{"DSolve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"m", " ", RowBox[{ RowBox[{"x", "''"}], "[", "t", "]"}]}], "+", RowBox[{"b", " ", RowBox[{ RowBox[{"x", "'"}], "[", "t", "]"}]}], "+", " ", RowBox[{"k", " ", RowBox[{"x", "[", "t", "]"}]}]}], "\[Equal]", "0"}], ",", RowBox[{"x", "[", "t", "]"}], ",", "t"}], "]"}]], "Input", CellChangeTimes->{{3.4315425582480316`*^9, 3.4315425645630317`*^9}, { 3.431543892739032*^9, 3.4315439015950317`*^9}, {3.4315440829790316`*^9, 3.431544083211032*^9}, {3.4315472267070312`*^9, 3.4315472270590315`*^9}, { 3.4327432057306004`*^9, 3.4327432879056*^9}, {3.4327447790456*^9, 3.4327447994986*^9}}], Cell["\<\ If we don't tell Mathematica that x ia function t, it tells us \ \>", "Text", CellChangeTimes->{{3.4327448072546*^9, 3.4327448265846*^9}}], Cell[BoxData[ RowBox[{"DSolve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"m", " ", RowBox[{ RowBox[{"x", "''"}], "[", "t", "]"}]}], "+", RowBox[{"b", " ", RowBox[{ RowBox[{"x", "'"}], "[", "t", "]"}]}], "+", " ", RowBox[{"k", " ", RowBox[{"x", "[", "t", "]"}]}]}], "\[Equal]", "0"}], ",", "x", ",", "t"}], "]"}]], "Input", CellChangeTimes->{{3.4315425582480316`*^9, 3.4315425645630317`*^9}, { 3.431543892739032*^9, 3.4315439015950317`*^9}, {3.4315440829790316`*^9, 3.431544083211032*^9}, {3.4315472267070312`*^9, 3.4315472270590315`*^9}, { 3.4327432057306004`*^9, 3.4327432879056*^9}, {3.4327447790456*^9, 3.4327447994986*^9}, {3.4327448384046*^9, 3.4327448392066*^9}}], Cell[TextData[{ "We get just what we expect from our standard (real) exponential Ansatz, \ x[t] ~ ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["e", "\[Alpha]t"], "\[Rule]", " ", "\[Alpha]"}], " ", "=", " ", RowBox[{ RowBox[{"-", FractionBox["b", RowBox[{"2", "m"}]]}], "\[PlusMinus]", SqrtBox[ RowBox[{ FractionBox[ SuperscriptBox["b", "2"], RowBox[{"4", SuperscriptBox["m", "2"]}]], "-", FractionBox["k", "m"]}]]}]}], TraditionalForm]]], ". Note also that Mathematica includes the 2 undetermined constants \ (labeled in the form C[1], C[2]) which will be determined by initial \ conditions. This confirms the analysis in Eq. (5.29) and following in the \ Lecture. Mathematica will also solve for these constants directly if the \ inital conditions are provided in extra equations, as in " }], "Text", CellChangeTimes->{{3.4327433402876*^9, 3.4327436108986*^9}, { 3.4327436546205997`*^9, 3.4327437152626*^9}, {3.4327437735786*^9, 3.4327438496256*^9}, 3.4327512370926*^9}], Cell[BoxData[ RowBox[{"sol", "=", RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"m", " ", RowBox[{ RowBox[{"x", "''"}], "[", "t", "]"}]}], "+", RowBox[{"b", " ", RowBox[{ RowBox[{"x", "'"}], "[", "t", "]"}]}], "+", " ", RowBox[{"k", " ", RowBox[{"x", "[", "t", "]"}]}]}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{"x", "[", "0", "]"}], "==", "1"}], ",", RowBox[{ RowBox[{ RowBox[{"x", "'"}], "[", "0", "]"}], "\[Equal]", "0"}]}], "}"}], ",", RowBox[{"x", "[", "t", "]"}], ",", "t"}], "]"}]}]], "Input", CellChangeTimes->{{3.4315425582480316`*^9, 3.4315425645630317`*^9}, { 3.431543892739032*^9, 3.4315439015950317`*^9}, {3.4315440829790316`*^9, 3.431544083211032*^9}, {3.4315472267070312`*^9, 3.4315472270590315`*^9}, { 3.4327432057306004`*^9, 3.4327432879056*^9}, {3.4327438702706003`*^9, 3.4327438961706*^9}, {3.4327448646636*^9, 3.4327448738106003`*^9}}], Cell["To pull on the solution for manipulation we define", "Text", CellChangeTimes->{{3.4327454753826*^9, 3.4327454891776*^9}}], Cell[BoxData[ RowBox[{"sol1", "=", RowBox[{ RowBox[{"x", "[", "t", "]"}], "/.", RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.4327439104096003`*^9, 3.4327439427826*^9}, { 3.4327441516986*^9, 3.4327441660116*^9}, {3.4327448803326*^9, 3.4327448832146*^9}, {3.4327449601535997`*^9, 3.4327450306306*^9}, { 3.4327450822326*^9, 3.4327451330236*^9}, {3.4327451807906*^9, 3.4327451829196*^9}, {3.4327475991716003`*^9, 3.4327476004616003`*^9}, { 3.4327476843297997`*^9, 3.4327476855137997`*^9}}], Cell["And test the initial conditions", "Text", CellChangeTimes->{{3.4327454961696*^9, 3.4327455067846003`*^9}}], Cell[BoxData[ RowBox[{"sol1", "/.", RowBox[{"t", "\[Rule]", "0"}]}]], "Input", CellChangeTimes->{ 3.4327445311136*^9, {3.4327448922296*^9, 3.4327448935966*^9}, { 3.4327451925146*^9, 3.4327451960875998`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"sol1", "'"}], "/.", RowBox[{"t", "\[Rule]", "0"}]}]], "Input", CellChangeTimes->{{3.4327449095076*^9, 3.4327449158115997`*^9}, { 3.4327452073506002`*^9, 3.4327452104736*^9}}], Cell["Let's try it with constants defined via", "Text", CellChangeTimes->{{3.4327452715506*^9, 3.4327452836546*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"k", "=", "1"}], ";", RowBox[{"b", "=", "1"}], ";", RowBox[{"m", "=", "1"}], ";"}]], "Input", CellChangeTimes->{{3.4327452875025997`*^9, 3.4327453072066*^9}}], Cell[BoxData[ RowBox[{"sol2", "=", RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"m", " ", RowBox[{ RowBox[{"x", "''"}], "[", "t", "]"}]}], "+", RowBox[{"b", " ", RowBox[{ RowBox[{"x", "'"}], "[", "t", "]"}]}], "+", " ", RowBox[{"k", " ", RowBox[{"x", "[", "t", "]"}]}]}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{"x", "[", "0", "]"}], "==", "1"}], ",", RowBox[{ RowBox[{ RowBox[{"x", "'"}], "[", "0", "]"}], "\[Equal]", "0"}]}], "}"}], ",", RowBox[{"x", "[", "t", "]"}], ",", "t"}], "]"}]}]], "Input", CellChangeTimes->{{3.4315425582480316`*^9, 3.4315425645630317`*^9}, { 3.431543892739032*^9, 3.4315439015950317`*^9}, {3.4315440829790316`*^9, 3.431544083211032*^9}, {3.4315472267070312`*^9, 3.4315472270590315`*^9}, { 3.4327432057306004`*^9, 3.4327432879056*^9}, {3.4327438702706003`*^9, 3.4327438961706*^9}, {3.4327448646636*^9, 3.4327448738106003`*^9}, 3.4327453271176*^9}], Cell["\<\ The expected damped oscillation that looks like (note the initial conditions \ are explicit in the plot).\ \>", "Text", CellChangeTimes->{{3.4327454490446*^9, 3.4327454597706003`*^9}, { 3.4327455344256*^9, 3.4327455452995996`*^9}, {3.4327512651116*^9, 3.4327512784846*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"x", "[", "t", "]"}], "/.", RowBox[{"sol2", "[", RowBox[{"[", "1", "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "10"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", ".2"}], ",", "1"}], "}"}]}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.4327453624486*^9, 3.4327454292186003`*^9}, { 3.4327455592216*^9, 3.4327456018486*^9}}], Cell["\<\ Now consider the inhomogeneous (harmonically driven) form of the equation \ (see Eqns. (5.23) to (5.25) (note that Mathematica still rememebers our \ values for m, b and k)\ \>", "Text", CellChangeTimes->{{3.4327456568386*^9, 3.4327456716126003`*^9}, { 3.4327458102276*^9, 3.4327458265126*^9}, {3.4327471856296*^9, 3.4327472068766003`*^9}, {3.4327512909765997`*^9, 3.4327513216426*^9}}], Cell[BoxData[ RowBox[{"sol3", "=", RowBox[{"DSolve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"m", " ", RowBox[{ RowBox[{"x", "''"}], "[", "t", "]"}]}], "+", RowBox[{"b", " ", RowBox[{ RowBox[{"x", "'"}], "[", "t", "]"}]}], "+", " ", RowBox[{"k", " ", RowBox[{"x", "[", "t", "]"}]}]}], "\[Equal]", RowBox[{ SubscriptBox[ RowBox[{"F", " "}], "1"], RowBox[{"Cos", "[", RowBox[{ RowBox[{ SubscriptBox["\[Omega]", "0"], " ", "t"}], " ", "+", " ", SubscriptBox["\[CurlyPhi]", "1"]}], "]"}]}]}], ",", RowBox[{"x", "[", "t", "]"}], ",", "t"}], "]"}]}]], "Input", CellChangeTimes->{{3.4315425582480316`*^9, 3.4315425645630317`*^9}, { 3.431543892739032*^9, 3.4315439015950317`*^9}, {3.4315440829790316`*^9, 3.431544083211032*^9}, {3.4315472267070312`*^9, 3.4315472270590315`*^9}, { 3.4327432057306004`*^9, 3.4327432879056*^9}, {3.4327447790456*^9, 3.4327447994986*^9}, {3.4327457223556004`*^9, 3.4327457595956*^9}, { 3.4327458430366*^9, 3.4327458934695997`*^9}, {3.4327459714596*^9, 3.4327459729726*^9}}], Cell["Yikes - lots of apparent complexity so simplify", "Text", CellChangeTimes->{{3.4327459884026003`*^9, 3.4327459914835997`*^9}, { 3.4327461126146*^9, 3.4327461347066*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"x", "[", "t", "]"}], "/.", RowBox[{"sol3", "[", RowBox[{"[", "1", "]"}], "]"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.4327459995106*^9, 3.4327460093316*^9}, { 3.4327461380256*^9, 3.4327461717016*^9}, {3.4327462094806004`*^9, 3.4327462103096*^9}, 3.4327513719336*^9}], Cell["More muscle", "Text", CellChangeTimes->{{3.4327462199126*^9, 3.4327462226776*^9}}], Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"x", "[", "t", "]"}], "/.", RowBox[{"sol3", "[", RowBox[{"[", "1", "]"}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.4327459995106*^9, 3.4327460093316*^9}, { 3.4327461380256*^9, 3.4327461717016*^9}}], Cell["\<\ Now we recognize the first (damped) term as the complementary solution, i.e., \ the solution to the homogeneous equation (which knows about the damping and \ the natural frequency), while the second term is the particular solution for \ the specific driving function (and driving frequency). To see this is the \ same as Eq. (5.25) we would have to expand the single cosine function (with \ the 2 phases) in terms of the phase due to the spring system.\ \>", "Text", CellChangeTimes->{{3.4327463833606*^9, 3.4327465234526*^9}, { 3.4327471339365997`*^9, 3.4327471506675997`*^9}, {3.4327475393426*^9, 3.4327475751036*^9}, {3.4327513472106*^9, 3.4327513772086*^9}}], Cell["\<\ Now with inital conditions and a specific diving force we have\ \>", "Text", CellChangeTimes->{{3.4327470446356*^9, 3.4327470693056*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["F", "1"], "=", "2"}], ";", " ", RowBox[{ SubscriptBox["\[Omega]", "0"], "=", "1"}], ";", RowBox[{ SubscriptBox["\[CurlyPhi]", "1"], "=", RowBox[{"\[Pi]", "/", "3"}]}], ";"}]], "Input", CellChangeTimes->{{3.4327470776156*^9, 3.4327471600216*^9}}], Cell[BoxData[ RowBox[{"sol4", "=", RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"m", " ", RowBox[{ RowBox[{"x", "''"}], "[", "t", "]"}]}], "+", RowBox[{"b", " ", RowBox[{ RowBox[{"x", "'"}], "[", "t", "]"}]}], "+", " ", RowBox[{"k", " ", RowBox[{"x", "[", "t", "]"}]}]}], "\[Equal]", RowBox[{ SubscriptBox[ RowBox[{"F", " "}], "1"], RowBox[{"Cos", "[", RowBox[{ RowBox[{ SubscriptBox["\[Omega]", "0"], " ", "t"}], " ", "+", " ", SubscriptBox["\[CurlyPhi]", "1"]}], "]"}]}]}], ",", RowBox[{ RowBox[{"x", "[", "0", "]"}], "==", "1"}], ",", RowBox[{ RowBox[{ RowBox[{"x", "'"}], "[", "0", "]"}], "\[Equal]", "0"}]}], "}"}], ",", RowBox[{"x", "[", "t", "]"}], ",", "t"}], "]"}]}]], "Input", CellChangeTimes->{{3.4315425582480316`*^9, 3.4315425645630317`*^9}, { 3.431543892739032*^9, 3.4315439015950317`*^9}, {3.4315440829790316`*^9, 3.431544083211032*^9}, {3.4315472267070312`*^9, 3.4315472270590315`*^9}, { 3.4327432057306004`*^9, 3.4327432879056*^9}, {3.4327447790456*^9, 3.4327447994986*^9}, {3.4327457223556004`*^9, 3.4327457595956*^9}, { 3.4327458430366*^9, 3.4327458934695997`*^9}, {3.4327459714596*^9, 3.4327459729726*^9}, {3.4327472194016*^9, 3.4327472530425997`*^9}}], Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{ RowBox[{"x", "[", "t", "]"}], "/.", RowBox[{"sol4", "[", RowBox[{"[", "1", "]"}], "]"}]}], "]"}]], "Input", CellChangeTimes->{{3.4327459995106*^9, 3.4327460093316*^9}, { 3.4327461380256*^9, 3.4327461717016*^9}, {3.4327474988166*^9, 3.4327475108576*^9}, {3.4327477354628*^9, 3.4327477365748*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"x", "[", "t", "]"}], "/.", RowBox[{"sol4", "[", RowBox[{"[", "1", "]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "15"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "3"}], "}"}]}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.4327453624486*^9, 3.4327454292186003`*^9}, { 3.4327455592216*^9, 3.4327456018486*^9}, {3.4327472814716*^9, 3.4327473417855997`*^9}}], Cell["\<\ Note there is a brief period (as in the previous plot) where the \ complementary solution damps out and after which the initial conditions are \ \"forgotten\". Then for large times we settle into the behavior given by the \ particular solution corresponding to the specific driving function. Note \ that the coefficients in the complementary solution (C[1]) & C[2]) are chosen \ so that the full solution (complementary plus particular) obeys the given \ inital conditions, as illustrated in the plot. This is not the case for the \ individual bits alone.\ \>", "Text", CellChangeTimes->{{3.4327473616506*^9, 3.4327474691516*^9}, { 3.4327477693898*^9, 3.4327477888758*^9}, {3.4327478309378*^9, 3.4327479603478003`*^9}, {3.4327514019016*^9, 3.4327514225425997`*^9}}], Cell["Complementary", "Text", CellChangeTimes->{{3.4327479834648*^9, 3.4327479861198*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ FractionBox["1", "3"], " ", SuperscriptBox["\[ExponentialE]", RowBox[{ RowBox[{"-", "t"}], "/", "2"}]], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "3"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", SqrtBox["3"]}], ")"}], " ", RowBox[{"Cos", "[", FractionBox[ RowBox[{ SqrtBox["3"], " ", "t"}], "2"], "]"}]}], "-", RowBox[{ RowBox[{"(", RowBox[{"3", "+", SqrtBox["3"]}], ")"}], " ", RowBox[{"Sin", "[", FractionBox[ RowBox[{ SqrtBox["3"], " ", "t"}], "2"], "]"}]}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "15"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "3"}], "}"}]}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.4327453624486*^9, 3.4327454292186003`*^9}, { 3.4327455592216*^9, 3.4327456018486*^9}, {3.4327472814716*^9, 3.4327473417855997`*^9}, {3.4327479735278*^9, 3.4327480032518*^9}, { 3.4327480638118*^9, 3.4327480651988*^9}}], Cell["Particular", "Text", CellChangeTimes->{{3.4327480118918*^9, 3.4327480146678*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{ SqrtBox["3"], " ", RowBox[{"Cos", "[", "t", "]"}]}], "+", RowBox[{"Sin", "[", "t", "]"}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "15"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "3"}], "}"}]}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{"\"\\"", ",", "\"\\""}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.4327453624486*^9, 3.4327454292186003`*^9}, { 3.4327455592216*^9, 3.4327456018486*^9}, {3.4327472814716*^9, 3.4327473417855997`*^9}, {3.4327480374688*^9, 3.4327480540758*^9}}], Cell["\<\ But the sum of the 2 (linear superposition) satisfies the inhomogeneous \ equation AND the initial condiitons.\ \>", "Text", CellChangeTimes->{{3.4327514320986*^9, 3.4327514850536003`*^9}}] }, Open ]] }, WindowSize->{929, 596}, WindowMargins->{{0, Automatic}, {-29, Automatic}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, FrontEndVersion->"6.0 for Microsoft Windows (32-bit) (June 19, 2007)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[590, 23, 202, 3, 71, "Section"], Cell[795, 28, 714, 11, 65, "Text"], Cell[1512, 41, 595, 10, 47, "Text"], Cell[2110, 53, 729, 18, 31, "Input"], Cell[2842, 73, 149, 3, 29, "Text"], Cell[2994, 78, 746, 18, 31, "Input"], Cell[3743, 98, 1097, 28, 89, "Text"], Cell[4843, 128, 1054, 27, 31, "Input"], Cell[5900, 157, 128, 1, 29, "Text"], Cell[6031, 160, 556, 11, 31, "Input"], Cell[6590, 173, 113, 1, 29, "Text"], Cell[6706, 176, 217, 5, 31, "Input"], Cell[6926, 183, 215, 5, 31, "Input"], Cell[7144, 190, 117, 1, 29, "Text"], Cell[7264, 193, 199, 5, 31, "Input"], Cell[7466, 200, 1083, 28, 31, "Input"], Cell[8552, 230, 289, 6, 29, "Text"], Cell[8844, 238, 605, 17, 31, "Input"], Cell[9452, 257, 402, 7, 29, "Text"], Cell[9857, 266, 1152, 29, 31, "Input"], Cell[11012, 297, 178, 2, 29, "Text"], Cell[11193, 301, 371, 9, 31, "Input"], Cell[11567, 312, 89, 1, 29, "Text"], Cell[11659, 315, 282, 7, 31, "Input"], Cell[11944, 324, 679, 10, 65, "Text"], Cell[12626, 336, 148, 3, 29, "Text"], Cell[12777, 341, 314, 9, 31, "Input"], Cell[13094, 352, 1452, 37, 31, "Input"], Cell[14549, 391, 369, 8, 31, "Input"], Cell[14921, 401, 653, 18, 31, "Input"], Cell[15577, 421, 784, 12, 83, "Text"], Cell[16364, 435, 91, 1, 29, "Text"], Cell[16458, 438, 1318, 41, 85, "Input"], Cell[17779, 481, 88, 1, 29, "Text"], Cell[17870, 484, 718, 20, 39, "Input"], Cell[18591, 506, 200, 4, 29, "Text"] }, Open ]] } ] *) (* End of internal cache information *)