(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 35836, 1040] NotebookOptionsPosition[ 33996, 978] NotebookOutlinePosition[ 34356, 994] CellTagsIndexPosition[ 34313, 991] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ StyleBox["MAT 331:", FontSize->24], " ", StyleBox["Homework 3", FontSize->24] }], "Section", CellChangeTimes->{{3.618884780257655*^9, 3.618884795191327*^9}, { 3.619013741815179*^9, 3.619013742915434*^9}, {3.6190639773624067`*^9, 3.6190639931423893`*^9}, 3.6190640316767807`*^9, {3.620115017220894*^9, 3.620115018260913*^9}, {3.622348203306175*^9, 3.622348203752513*^9}}], Cell[CellGroupData[{ Cell[TextData[StyleBox["Problem 3.1", "Subsection"]], "Section", CellChangeTimes->{{3.6188834470493517`*^9, 3.618883458254191*^9}, { 3.6188841737965927`*^9, 3.618884177712782*^9}, {3.6188846295181627`*^9, 3.6188846297983932`*^9}, {3.619056965231044*^9, 3.619056965742797*^9}, { 3.620184812387269*^9, 3.620184812898695*^9}, {3.620185327776074*^9, 3.620185327888328*^9}, {3.622348234232251*^9, 3.622348237600342*^9}, { 3.623068761984556*^9, 3.6230687632311697`*^9}}], Cell["\<\ The replacement operator /. (slash-dot) applies a transformation rule to an \ expression. If you give a list of rules, each rule will be tried once on each \ part of the expression. If you give a list of lists of rules, you get a list \ of results; each sublist is treated like an independent set of rules. \ \>", "Text", CellChangeTimes->{{3.6190083437795057`*^9, 3.619008619778887*^9}, { 3.6190086997717037`*^9, 3.619008742318366*^9}, {3.61900877337542*^9, 3.619008846690468*^9}, {3.619063502179244*^9, 3.619063504155314*^9}, { 3.619064140835752*^9, 3.619064155080373*^9}, {3.6194770073674107`*^9, 3.6194770491064157`*^9}, {3.620075409161674*^9, 3.620075433432282*^9}, { 3.620075524069899*^9, 3.620075731127187*^9}, {3.620075775367186*^9, 3.6200758388256407`*^9}, {3.620075869795953*^9, 3.620076143652358*^9}, { 3.620076194766493*^9, 3.620076199750688*^9}, {3.620184195501205*^9, 3.6201842312101793`*^9}, {3.620184266043998*^9, 3.620184324595572*^9}, { 3.620184405840395*^9, 3.620184408351913*^9}, 3.620184440649316*^9, { 3.620184824715065*^9, 3.620184871272209*^9}, {3.620197754934945*^9, 3.620197830056409*^9}, {3.622359325175309*^9, 3.622359432517733*^9}, { 3.62235955393786*^9, 3.622359554581221*^9}, {3.622371481972286*^9, 3.622371509950182*^9}, {3.622371543295084*^9, 3.622371551638085*^9}, { 3.6224072157134237`*^9, 3.622407247968191*^9}, {3.622998078870495*^9, 3.622998121459565*^9}, 3.6230007636654882`*^9}, TextJustification->1.], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"x", "^", "3"}], "+", RowBox[{"y", "^", "2"}], "+", "z"}], " ", "/.", " ", RowBox[{"{", RowBox[{"x_", "->", "2"}], "}"}]}]], "Input", CellChangeTimes->{{3.622359444260406*^9, 3.622359444262418*^9}, { 3.6223715559302273`*^9, 3.622371587102578*^9}, {3.623069888186159*^9, 3.623069892722011*^9}, {3.6236760535614223`*^9, 3.6236760568227787`*^9}, { 3.623676186394165*^9, 3.623676186857806*^9}, 3.623676228617916*^9, { 3.623676306171558*^9, 3.623676306370545*^9}}, Background->GrayLevel[0.85]], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"x", "^", "3"}], "+", RowBox[{"y", "^", "2"}], "+", "z"}], " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{"x", "->", "2"}], ",", " ", RowBox[{"y", "\[Rule]", "3"}], ",", " ", RowBox[{"z", "\[Rule]", "1"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.6223594631682463`*^9, 3.622359471456312*^9}, { 3.6223715915109463`*^9, 3.6223715918946342`*^9}}, Background->GrayLevel[0.85]], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"x", "^", "3"}], "+", RowBox[{"y", "^", "2"}], "+", "z"}], " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"x", "\[Rule]", "2"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "2"}], ",", " ", RowBox[{"y", "\[Rule]", "3"}], ",", RowBox[{"z", "\[Rule]", "1"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "0"}], ",", RowBox[{"z", "\[Rule]", "0"}]}], "}"}]}], "}"}]}]], "Input", CellChangeTimes->{{3.622359521173123*^9, 3.6223595330278997`*^9}, { 3.6223715989268208`*^9, 3.622371640316669*^9}, {3.622371679494577*^9, 3.622371701221772*^9}, {3.623089392852899*^9, 3.623089394716387*^9}, { 3.623676109647258*^9, 3.623676109878537*^9}}, Background->GrayLevel[0.85]], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"x", "^", "3"}], "+", RowBox[{"y", "^", "2"}], "+", "z"}], " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{"x_", "^", "n_"}], " ", "\[Rule]", " ", "a"}], "}"}]}]], "Input", CellDingbat->"\[LightBulb]", CellChangeTimes->{{3.623068593703718*^9, 3.623068617973507*^9}, { 3.6230710985260353`*^9, 3.6230711060370903`*^9}}, Background->GrayLevel[0.85]], Cell[CellGroupData[{ Cell["\<\ Write a replacement rule that when applied to the expression f[x] + g[x] \ outputs Sin[x] + Cos[3]. \ \>", "ItemNumbered", CellChangeTimes->{{3.623070065841669*^9, 3.623070128210013*^9}, { 3.623070180157112*^9, 3.623070232904689*^9}, {3.6230704373362923`*^9, 3.6230704452470007`*^9}, {3.623071042152134*^9, 3.623071069292366*^9}, { 3.623071170628931*^9, 3.623071176355859*^9}, {3.623072259750602*^9, 3.623072275365745*^9}, {3.623072321378746*^9, 3.623072326666456*^9}, { 3.6230773844685373`*^9, 3.623077403909594*^9}, {3.623676669801736*^9, 3.6236766698044*^9}}, TextJustification->1.], Cell["\<\ Write a transformation rule that replaces any expression of the form \ Function[variable] with Cos[3].\ \>", "ItemNumbered", CellChangeTimes->{{3.623070065841669*^9, 3.623070128210013*^9}, { 3.623070180157112*^9, 3.623070232904689*^9}, {3.6230704373362923`*^9, 3.6230704452470007`*^9}, {3.623071042152134*^9, 3.623071069292366*^9}, { 3.623071170628931*^9, 3.623071176355859*^9}, {3.623072259750602*^9, 3.623072275365745*^9}, {3.623072321378746*^9, 3.623072326666456*^9}, { 3.6230773844685373`*^9, 3.623077499395648*^9}, 3.6230776220323687`*^9, { 3.6230892349971037`*^9, 3.623089235000449*^9}, {3.623106591441285*^9, 3.6231066462479687`*^9}, {3.6236026478069353`*^9, 3.623602647810088*^9}}, TextJustification->1.] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Problem 3.2", "Subsection"]], "Section", CellChangeTimes->{{3.6188834205755587`*^9, 3.618883429719199*^9}, { 3.618884597065878*^9, 3.618884597215721*^9}, {3.619056954579022*^9, 3.619056955327201*^9}, {3.620185334072404*^9, 3.6201853357283173`*^9}, { 3.6223482422723494`*^9, 3.622348245192368*^9}, {3.623068790663968*^9, 3.6230687909111433`*^9}}], Cell["\<\ Mathematica can solve various kinds of equations, symbolically or \ numerically. The result will be displayed as a list of transformation rules. \ We can then use the replacement operator /. (slash-dot) to apply these rules \ to any given expression.\ \>", "Text", CellChangeTimes->{ 3.618884689235755*^9, {3.6189578925440702`*^9, 3.6189578957037983`*^9}, { 3.619973398056472*^9, 3.6199734360271807`*^9}, {3.619973569266963*^9, 3.6199735743609457`*^9}, {3.6199743475198603`*^9, 3.619974351799468*^9}, { 3.620197925712225*^9, 3.620198030044511*^9}, {3.622474053364324*^9, 3.622474113603716*^9}, {3.622821359911496*^9, 3.622821396149352*^9}, { 3.622821426621347*^9, 3.622821579570013*^9}, {3.6228216570682373`*^9, 3.622821661620681*^9}, {3.6228222082421713`*^9, 3.62282223644938*^9}, 3.622822434654705*^9, {3.623000635373149*^9, 3.623000635837368*^9}, 3.699312700817292*^9}, TextJustification->1.], Cell[TextData[{ "Consider now the cubic polynomial ", Cell[BoxData[ SuperscriptBox["x", "3"]], CellChangeTimes->{{3.6217832683954487`*^9, 3.621783282814507*^9}}, FontSize->14], "+a", Cell[BoxData[ SuperscriptBox["x", "2"]], CellChangeTimes->{{3.6217832683954487`*^9, 3.621783282814507*^9}}, FontSize->14], "+bx+c. Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to find the roots ", Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["x", "2"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["x", "3"], TraditionalForm]]], ". Then use transformation rules to compute the three symmetric expressions ", Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SubscriptBox["x", "2"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SubscriptBox["x", "3"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["x", "2"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SubscriptBox["x", "2"], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["x", "3"], TraditionalForm]]], "+", Cell[BoxData[ FormBox[ SubscriptBox["x", "3"], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["x", "2"], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["x", "3"], TraditionalForm]]], ". Use Simplify[...] to simplify the computations. What can you conclude \ about the relation between the three expressions and the coefficients of the \ polynomial?" }], "Text", CellChangeTimes->{{3.6228219658968782`*^9, 3.62282207868148*^9}, { 3.622822135504136*^9, 3.622822187776968*^9}, {3.622998146218347*^9, 3.622998148610546*^9}, {3.623000728392962*^9, 3.623000732728651*^9}, { 3.6230010962793837`*^9, 3.623001178357239*^9}, {3.623066556990986*^9, 3.6230668280236*^9}, {3.623106881945846*^9, 3.623107075242744*^9}}, TextJustification->1., FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Problem 3.3 ", "Subsection"]], "Section", CellChangeTimes->{{3.6188834640457153`*^9, 3.6188834693014603`*^9}, { 3.618884641389806*^9, 3.618884641797852*^9}, {3.619056973278646*^9, 3.619056973774465*^9}, {3.699312724272875*^9, 3.699312727143592*^9}}], Cell[TextData[{ "One of the main applications of matrix algebra is to solve systems of \ linear equations, usually in a large number of variables. A system of m such \ equations in the n variables ", Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["x", "2"], TraditionalForm]]], ", ..., ", Cell[BoxData[ FormBox[ SubscriptBox["x", "n"], TraditionalForm]]], ", can be written explicitly:\n", Cell[BoxData[ FormBox[ SubscriptBox["a", "11"], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SubscriptBox["a", "12"], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["x", "2"], "+"}], "..."}], "+", RowBox[{ SubscriptBox["a", RowBox[{"1", "n"}]], SubscriptBox["x", "n"]}]}], "=", SubscriptBox["b", "1"]}], TraditionalForm]]], "\n", Cell[BoxData[ FormBox[ SubscriptBox["a", "21"], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SubscriptBox["a", "22"], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["x", "2"], "+"}], "..."}], "+", RowBox[{ SubscriptBox["a", RowBox[{"2", "n"}]], SubscriptBox["x", "n"]}]}], "=", SubscriptBox["b", "2"]}], TraditionalForm]]], " \n.................................................\n", Cell[BoxData[ FormBox[ SubscriptBox["a", "m1"], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["x", "1"], TraditionalForm]]], " + ", Cell[BoxData[ FormBox[ SubscriptBox["a", "m2"], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["x", "2"], "+"}], "..."}], "+", RowBox[{ SubscriptBox["a", "mn"], SubscriptBox["x", "n"]}]}], "=", SubscriptBox["b", "m"]}], TraditionalForm]]], "\n\nThe same system can be written in the more convenient matrix notation \ as Ax=b, where A is the m \[Times] n coefficient matrix, x is the (column) \ vector of length n containing the variables, and b is the (column) vector of \ length m of the coefficients on the right hand sides of the equalities.\nBy \ general theory, a system of linear equations has no solution, exactly one \ solution or infinitely many solutions. \nIn ", StyleBox["Mathematica", FontSlant->"Italic"], ", we can use the command Solve[expression,variables], to solve the system \ Ax=b for the variable x. \n\nIn the code below we define a 2x2 matrix A and a \ vector b of length 2, both with random entries 0 or 1. Then we use the \ function Solve[A.x==b,x] to solve the system Ax=b. ", StyleBox["Mathematica", FontSlant->"Italic"], " knows that x should be a vector of 2 variables x[1] and x[2]. The function \ Solve ", StyleBox["r", FontSlant->"Italic"], "eturns an empty list if there is no solution, a list with one solution if \ the system has one solution, or a list which contains the dependency between \ the free variables and the dependent variables if the system has infinitely \ many solutions (like x[1]=-x[2]). " }], "Text", CellChangeTimes->{{3.618885731824938*^9, 3.618886180431584*^9}, { 3.618886210982359*^9, 3.618886310898436*^9}, {3.618886481756562*^9, 3.618886541920966*^9}, {3.618886594520981*^9, 3.618886648820651*^9}, { 3.618886731729341*^9, 3.6188867343531637`*^9}, {3.618887584248918*^9, 3.61888759011757*^9}, {3.618889481601304*^9, 3.618889542086615*^9}, { 3.618889573694559*^9, 3.618889651537706*^9}, {3.61888972562352*^9, 3.6188900139793043`*^9}, {3.618948260021578*^9, 3.618948281339781*^9}, { 3.695106726060544*^9, 3.6951067304847307`*^9}}], Cell[BoxData[{ RowBox[{"Clear", "[", RowBox[{"A", ",", "b", ",", " ", "X", ",", " ", "x"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"A", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"RandomInteger", "[", "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"j", ",", "2"}], "}"}]}], "]"}]}], ";", " ", RowBox[{"Print", "[", RowBox[{"\"\\"", RowBox[{"MatrixForm", "[", "A", "]"}]}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"b", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"RandomInteger", "[", "]"}], ",", RowBox[{"{", RowBox[{"j", ",", "2"}], "}"}]}], "]"}]}], ";", " ", RowBox[{"Print", "[", RowBox[{"\"\\"", RowBox[{"MatrixForm", "[", "b", "]"}]}], "]"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"X", "=", RowBox[{"Table", "[", RowBox[{ RowBox[{"x", "[", "j", "]"}], ",", RowBox[{"{", RowBox[{"j", ",", "2"}], "}"}]}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"A", ".", "X"}], "\[Equal]", "b"}], ",", "X"}], "]"}]}], "Input", CellChangeTimes->{{3.6188868771498938`*^9, 3.618886970343876*^9}, { 3.618887048542817*^9, 3.6188871790156603`*^9}, {3.618887226840837*^9, 3.618887227732189*^9}, {3.618888049184164*^9, 3.6188880528077507`*^9}, { 3.618888127594583*^9, 3.618888136335723*^9}, {3.618888269829574*^9, 3.618888285052102*^9}, {3.618888823763673*^9, 3.618888890649508*^9}, { 3.618888981217669*^9, 3.6188890451473303`*^9}, {3.6188891383208923`*^9, 3.618889153607192*^9}, {3.618889276988312*^9, 3.6188892775318613`*^9}, { 3.618889366622489*^9, 3.6188893740062*^9}, {3.619051878122259*^9, 3.619051881631794*^9}, {3.69931276774936*^9, 3.699312782514035*^9}, { 3.6993128248608513`*^9, 3.699312825267432*^9}, {3.699312887663506*^9, 3.6993128879259157`*^9}, {3.699312975721856*^9, 3.699312983256947*^9}}, Background->GrayLevel[0.85]], Cell["\<\ Run the code several time to generate a couple of random 2x2 matrices and see \ how the solution set is displayed. Then modify the code above to answer the \ following questions:\ \>", "Text", CellChangeTimes->{{3.6190621296199007`*^9, 3.619062264309627*^9}, { 3.6190629344141207`*^9, 3.619062935418233*^9}}], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Let A be a random 3x3 matrix with integer coefficients. Find the \ solutions set of the system ", FontSize->14], "Ax=", Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1"}, {"0"}, {"0"} }], "\[NoBreak]", ")"}], TraditionalForm]], FontSize->12], StyleBox[". ", FontSize->12], StyleBox["Then repeat the procedure to find the the solutions set of the \ system Ax=", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0"}, {"1"}, {"0"} }], "\[NoBreak]", ")"}], TraditionalForm]], FontSize->12], ", and respectively of ", StyleBox["Ax", FontSize->14], StyleBox["=", FontSize->12], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0"}, {"0"}, {"1"} }], "\[NoBreak]", ")"}], TraditionalForm]], FontSize->12], StyleBox[". ", FontSize->12] }], "ItemNumbered", CellChangeTimes->{{3.6190517744973392`*^9, 3.619051811418685*^9}, { 3.619060288730673*^9, 3.619060301174124*^9}, {3.619060331908966*^9, 3.619060345748253*^9}, {3.6190618192341957`*^9, 3.6190618192406607`*^9}, { 3.619061951482662*^9, 3.619062012192177*^9}, {3.619062043135117*^9, 3.619062064597917*^9}, {3.6190622683418617`*^9, 3.6190623721534986`*^9}, 3.619062917107295*^9, {3.695164948349226*^9, 3.695165024188538*^9}, { 3.69516505741322*^9, 3.695165102838667*^9}}], Cell[TextData[{ StyleBox["Find the inverse of a random 3x3 matrix A with integer \ coefficients, if it exists, without using the ", FontSize->14], StyleBox["Mathematica", FontSize->14, FontSlant->"Italic"], StyleBox[" function Inverse[...]. ", FontSize->14], StyleBox["Recall that the matrix A is called invertible is there exists a \ (unique) 3x3 matrix B such that AB=BA=", FontSize->14, FontSlant->"Italic"], Cell[BoxData[ FormBox[ SubscriptBox["I", "3"], TraditionalForm]], FontSize->14, FontSlant->"Italic"], StyleBox[", where ", FontSize->14, FontSlant->"Italic"], Cell[BoxData[ FormBox[ SubscriptBox["I", "3"], TraditionalForm]], FontSize->14, FontSlant->"Italic"], StyleBox[" is the identity matrix. If A is invertible, then the matrix B is \ called the inverse of the matrix A, and it is denoted by ", FontSize->14, FontSlant->"Italic"], Cell[BoxData[ FormBox[ SuperscriptBox["A", RowBox[{"-", "1"}]], TraditionalForm]], FontSize->14, FontSlant->"Italic"], StyleBox[". ", FontSize->14, FontSlant->"Italic"] }], "ItemNumbered", CellChangeTimes->{{3.6190517744973392`*^9, 3.619051811418685*^9}, { 3.619060288730673*^9, 3.619060301174124*^9}, {3.619060331908966*^9, 3.619060345748253*^9}, {3.6190618192341957`*^9, 3.619061899532473*^9}, { 3.619062415631969*^9, 3.619062643998418*^9}, {3.619062688136025*^9, 3.619062753065941*^9}, {3.619062790688437*^9, 3.619062797856069*^9}, { 3.6190628669756117`*^9, 3.619062897691861*^9}, {3.6190629423149853`*^9, 3.6190629771448*^9}, {3.6190630358881407`*^9, 3.619063070421206*^9}}, TextJustification->1.] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Problem 3.4", "Subsection"]], "Section", CellChangeTimes->{{3.6188834372387037`*^9, 3.618883440270814*^9}, { 3.618884618760844*^9, 3.618884619110783*^9}, {3.6189486900451593`*^9, 3.618948692316845*^9}, 3.6189488450714417`*^9, {3.619056958023197*^9, 3.6190569585670424`*^9}, {3.620185341416134*^9, 3.6201853431924257`*^9}, { 3.6223482481205187`*^9, 3.6223482504163923`*^9}, {3.623068833952813*^9, 3.62306883433496*^9}, {3.65458215341435*^9, 3.654582159425049*^9}, { 3.699313098197316*^9, 3.69931309875635*^9}}], Cell["Consider the differential equation y' (x) = 1 + y (x).", "Text", CellChangeTimes->{{3.6545821619309187`*^9, 3.6545821685015306`*^9}}], Cell[CellGroupData[{ Cell["\<\ Use VectorPlot[ ...] and StreamPlot[ ...] to plot the vector field of the \ differential equation. Try several display options (i.e. change the size of \ the arrows, make the picture larger, change the colors, etc.) to see which \ one gives the most accurate picture.\ \>", "ItemNumbered", CellChangeTimes->{{3.623069949806264*^9, 3.623069976667191*^9}, { 3.623107115770756*^9, 3.623107149833493*^9}, {3.654581850708012*^9, 3.654581861232935*^9}, {3.654582173575944*^9, 3.654582175463683*^9}}, TextJustification->1., FontSize->14], Cell[TextData[{ "Next use DSolve[...] to solve the differential equation \ y\[CloseCurlyQuote](x) = 1 + y(x), with initial condition y(0)=1. The result \ will be a list of transformation rules. Define a function ", StyleBox["YSol[x_]", FontColor->RGBColor[0.5, 0, 0.5]], " that returns the solution found by DSolve[...]. Evaluate YSol numerically \ at x=0 and x=0.1. Then plot the function YSol." }], "ItemNumbered", CellChangeTimes->{{3.623069949806264*^9, 3.623069977753894*^9}, { 3.623107138282069*^9, 3.623107168538146*^9}, {3.654581954656397*^9, 3.654581958440246*^9}, {3.654583745112525*^9, 3.6545838352498493`*^9}, { 3.654584023735867*^9, 3.6545840246078653`*^9}}, TextJustification->1., FontSize->14], Cell[TextData[{ "Check that YSol is indeed a solution of the differential equation y\ \[CloseCurlyQuote](x) = 1 + y(x) by using ", StyleBox["Mathematica", FontSlant->"Italic"], " to compute the derivative of the function YSol." }], "ItemNumbered", CellChangeTimes->{{3.623069949806264*^9, 3.623069977753894*^9}, { 3.623107138282069*^9, 3.623107168538146*^9}, {3.654581954656397*^9, 3.654581958440246*^9}, {3.654582336460436*^9, 3.654582491977178*^9}, { 3.654582553016284*^9, 3.654582576267128*^9}, {3.654583723422868*^9, 3.654583733770171*^9}, {3.6545837981113*^9, 3.6545837994960814`*^9}, { 3.654583865424904*^9, 3.654583912667473*^9}, {3.6545840343886137`*^9, 3.654584040575733*^9}}, TextJustification->1., FontSize->14], Cell[TextData[{ "Now go back to the picture that you have obtained in part ", StyleBox["1", FontWeight->"Bold"], ", using StreamPlot[..]. Color the solution of the differential equation y\ \[CloseCurlyQuote](x) = 1+y(x) corresponding to the initial condition y(0)=1 \ on top of the stream plot picture." }], "ItemNumbered", CellChangeTimes->{{3.623069949806264*^9, 3.623069977753894*^9}, { 3.623107138282069*^9, 3.623107168538146*^9}, {3.654581954656397*^9, 3.654581958440246*^9}, {3.654582336460436*^9, 3.654582491977178*^9}, { 3.654582553016284*^9, 3.654582576267128*^9}}, TextJustification->1., FontSize->14] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Problem 3.5", "Subsection"]], "Section", CellChangeTimes->{{3.6188834372387037`*^9, 3.618883440270814*^9}, { 3.618884618760844*^9, 3.618884619110783*^9}, {3.6189486900451593`*^9, 3.618948692316845*^9}, 3.6189488450714417`*^9, {3.619056958023197*^9, 3.6190569585670424`*^9}, {3.620185341416134*^9, 3.6201853431924257`*^9}, { 3.6223482481205187`*^9, 3.6223482504163923`*^9}, {3.623068833952813*^9, 3.62306883433496*^9}, {3.65458215341435*^9, 3.654582159425049*^9}, { 3.699313098197316*^9, 3.69931309875635*^9}, {3.699313133870502*^9, 3.699313134397613*^9}}], Cell[TextData[{ "Recall the definitions and notations from the lecture notes. Let c be a \ complex number and consider the quadratic polynomial ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["p", StyleBox["c", FontSize->12]], "(", "z", ")"}], "=", RowBox[{ SuperscriptBox["z", StyleBox["2", FontSize->10]], "+", "c"}]}], TraditionalForm]]], ". Denote by ", Cell[BoxData[ FormBox[ SubscriptBox["J", StyleBox["c", FontSize->12]], TraditionalForm]]], " the Julia set of the polynomial ", Cell[BoxData[ FormBox[ SubscriptBox["p", StyleBox["c", FontSize->12]], TraditionalForm]]], " . The Mandelbrot set is the set of parameter values ", StyleBox["c ", FontSlant->"Italic"], "for which the Julia set ", Cell[BoxData[ FormBox[ SubscriptBox["J", StyleBox["c", FontSize->12]], TraditionalForm]], FormatType->"TraditionalForm"], " of the polynomial ", Cell[BoxData[ FormBox[ SubscriptBox["p", StyleBox["c", FontSize->12]], TraditionalForm]]], " is connected. " }], "Text", CellChangeTimes->{{3.6545821619309187`*^9, 3.6545821685015306`*^9}, { 3.6993140472617702`*^9, 3.699314099829692*^9}, {3.699314137051939*^9, 3.699314227034088*^9}, {3.699314264523571*^9, 3.699314348358324*^9}, { 3.699314734947212*^9, 3.699314758924217*^9}}, TextJustification->1.], Cell[CellGroupData[{ Cell[TextData[{ "Use the function Solve[..] to find the two fixed points ", Cell[BoxData[ FormBox[ SubscriptBox["z", StyleBox["1", FontSize->10]], TraditionalForm]], FormatType->"TraditionalForm"], "and ", Cell[BoxData[ FormBox[ SubscriptBox["z", "2"], TraditionalForm]]], " of the polynomial ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", StyleBox["c", FontSize->12]], "(", "z", ")"}], TraditionalForm]]], ". " }], "ItemNumbered", CellChangeTimes->{{3.623069949806264*^9, 3.623069976667191*^9}, { 3.623107115770756*^9, 3.623107149833493*^9}, {3.654581850708012*^9, 3.654581861232935*^9}, {3.654582173575944*^9, 3.654582175463683*^9}, { 3.699314360262689*^9, 3.699314432848796*^9}, {3.6993145316202087`*^9, 3.6993147315950413`*^9}, {3.699314775492423*^9, 3.699314976135585*^9}, { 3.699315055326825*^9, 3.699315069326714*^9}, {3.69931522209276*^9, 3.699315264761785*^9}, {3.699315568753323*^9, 3.6993156380938272`*^9}}, TextJustification->1., FontSize->14], Cell[TextData[{ "Show that at most one fixed point of the polynomial ", Cell[BoxData[ FormBox[ SubscriptBox["p", StyleBox["c", FontSize->12]], TraditionalForm]]], " can be an attracting fixed point, that is, we cannot simultaneously have \ ", Cell[BoxData[ FormBox[ RowBox[{"|", RowBox[{ RowBox[{ SubscriptBox["p", StyleBox["c", FontSize->12]], "'"}], RowBox[{"(", SubscriptBox["z", StyleBox["1", FontSize->10]], ")"}]}], "|", RowBox[{"<", "1"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"|", RowBox[{ RowBox[{ SubscriptBox["p", StyleBox["c", FontSize->12]], "'"}], RowBox[{"(", SubscriptBox["z", "2"], ")"}]}], "|", RowBox[{"<", "1"}]}], TraditionalForm]]], ". Moreover, show that both fixed points ", Cell[BoxData[ FormBox[ SubscriptBox["z", StyleBox["1", FontSize->10]], TraditionalForm]], FormatType->"TraditionalForm"], "and ", Cell[BoxData[ FormBox[ SubscriptBox["z", "2"], TraditionalForm]]], " are attracting or indifferent (that is", Cell[BoxData[ FormBox[ RowBox[{"|", RowBox[{ RowBox[{ SubscriptBox["p", StyleBox["c", FontSize->12]], "'"}], RowBox[{"(", SubscriptBox["z", StyleBox["1", FontSize->10]], ")"}]}], "|", RowBox[{"\[LessSlantEqual]", "1"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"|", RowBox[{ RowBox[{ SubscriptBox["p", StyleBox["c", FontSize->12]], "'"}], RowBox[{"(", SubscriptBox["z", "2"], ")"}]}], "|", RowBox[{"\[LessSlantEqual]", "1"}]}], ")"}], TraditionalForm]]], " if and only if ", Cell[BoxData[ FormBox[ SubscriptBox["z", StyleBox["1", FontSize->10]], TraditionalForm]], FormatType->"TraditionalForm"], "=", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["z", "2"], "="}], TraditionalForm]]], Cell[BoxData[ FormBox[ StyleBox[ FractionBox["1", "2"], FontSize->16], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"c", "=", StyleBox[ FractionBox["1", "4"], FontSize->16]}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "ItemNumbered", CellChangeTimes->{{3.623069949806264*^9, 3.623069976667191*^9}, { 3.623107115770756*^9, 3.623107149833493*^9}, {3.654581850708012*^9, 3.654581861232935*^9}, {3.654582173575944*^9, 3.654582175463683*^9}, { 3.699314360262689*^9, 3.699314432848796*^9}, {3.6993145316202087`*^9, 3.6993147315950413`*^9}, {3.699314775492423*^9, 3.699314976135585*^9}, { 3.699315055326825*^9, 3.699315069326714*^9}, {3.69931522209276*^9, 3.699315264761785*^9}, {3.699315568753323*^9, 3.699315716133684*^9}, { 3.699315832853325*^9, 3.699315832853846*^9}}, TextJustification->1., FontSize->14], Cell[TextData[{ "Plot the Julia set of the polynomial ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["p", StyleBox["c", FontSize->12]], "(", "z", ")"}], "=", RowBox[{ SuperscriptBox["z", StyleBox["2", FontSize->10]], "+", "c"}]}], TraditionalForm]]], " for ", Cell[BoxData[ FormBox[ RowBox[{"c", "=", StyleBox[ FractionBox["1", "4"], FontSize->16]}], TraditionalForm]], FormatType->"TraditionalForm"], ". Highlight in red the unique fixed point ", Cell[BoxData[ FormBox[ SubscriptBox["z", StyleBox["1", FontSize->10]], TraditionalForm]]], " of this polynomial. Now use the functions defined in class to show (in \ different colors) a couple of typical orbits of the polynomial ", Cell[BoxData[ FormBox[ SubscriptBox["p", StyleBox["c", FontSize->12]], TraditionalForm]]], " on the same plot of its Julia set. What can you conclude? If you pick any \ point z sufficiently close to the fixed point ", Cell[BoxData[ FormBox[ SubscriptBox["z", StyleBox["1", FontSize->10]], TraditionalForm]]], ", is it true that its orbit z, ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", StyleBox["c", FontSize->12]], "(", "z", ")"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", StyleBox["c", FontSize->12]], "(", RowBox[{ SubscriptBox["p", StyleBox["c", FontSize->12]], "(", "z", ")"}], ")"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", StyleBox["c", FontSize->12]], "(", RowBox[{ SubscriptBox["p", StyleBox["c", FontSize->12]], "(", RowBox[{ SubscriptBox["p", StyleBox["c", FontSize->12]], "(", "z", ")"}], ")"}], ")"}], TraditionalForm]]], ", ... converges to the fixed point ", Cell[BoxData[ FormBox[ SubscriptBox["z", StyleBox["1", FontSize->10]], TraditionalForm]]], "?" }], "ItemNumbered", CellChangeTimes->{{3.623069949806264*^9, 3.623069976667191*^9}, { 3.623107115770756*^9, 3.623107149833493*^9}, {3.654581850708012*^9, 3.654581861232935*^9}, {3.654582173575944*^9, 3.654582175463683*^9}, { 3.699314360262689*^9, 3.699314432848796*^9}, {3.6993145316202087`*^9, 3.6993147315950413`*^9}, {3.699314775492423*^9, 3.699314976135585*^9}, { 3.699315055326825*^9, 3.699315069326714*^9}, {3.69931522209276*^9, 3.699315264761785*^9}, {3.699315568753323*^9, 3.699315716133684*^9}, { 3.699315834879669*^9, 3.699316026014223*^9}, {3.6993164298859873`*^9, 3.699316497645771*^9}, {3.6993165396007137`*^9, 3.699316784962536*^9}}, TextJustification->1., FontSize->14], Cell[TextData[{ "Find all values of the parameter c for which the polynomial ", Cell[BoxData[ FormBox[ SubscriptBox["p", StyleBox["c", FontSize->12]], TraditionalForm]]], " has an attracting or indifferent fixed point. This region is called the \ main cardioid of the Mandelbrot set." }], "ItemNumbered", CellChangeTimes->{{3.623069949806264*^9, 3.623069976667191*^9}, { 3.623107115770756*^9, 3.623107149833493*^9}, {3.654581850708012*^9, 3.654581861232935*^9}, {3.654582173575944*^9, 3.654582175463683*^9}, { 3.699314360262689*^9, 3.699314432848796*^9}, {3.6993145316202087`*^9, 3.6993147315950413`*^9}, {3.699314775492423*^9, 3.699314787124504*^9}, { 3.699314991597868*^9, 3.6993150406852837`*^9}, {3.6993150726221323`*^9, 3.699315143760297*^9}, {3.699315296925646*^9, 3.699315351204965*^9}, { 3.6993153969050922`*^9, 3.699315432477947*^9}, {3.699315481252081*^9, 3.6993155273414183`*^9}}, TextJustification->1., FontSize->14], Cell["\<\ Plot the region you found in part 3 in the Re(c), Im(c) parameter plane. Next \ plot the same region on top of the Mandelbrot set, using the command \ Show[...].\ \>", "ItemNumbered", CellChangeTimes->{{3.623069949806264*^9, 3.623069976667191*^9}, { 3.623107115770756*^9, 3.623107149833493*^9}, {3.654581850708012*^9, 3.654581861232935*^9}, {3.654582173575944*^9, 3.654582175463683*^9}, { 3.699314360262689*^9, 3.699314432848796*^9}, {3.6993145316202087`*^9, 3.6993147315950413`*^9}, {3.699314775492423*^9, 3.699314787124504*^9}, { 3.699314991597868*^9, 3.6993150406852837`*^9}, {3.6993150726221323`*^9, 3.699315143760297*^9}, {3.699315296925646*^9, 3.699315351204965*^9}, { 3.6993153969050922`*^9, 3.6993154701715307`*^9}, {3.699315545742193*^9, 3.6993155557983837`*^9}}, TextJustification->1., FontSize->14] }, Open ]] }, Open ]] }, WindowSize->{823, 532}, WindowMargins->{{Automatic, 127}, {Automatic, 14}}, FrontEndVersion->"11.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (September \ 21, 2016)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[558, 20, 405, 10, 64, "Section"], Cell[CellGroupData[{ Cell[988, 34, 477, 6, 41, "Section"], Cell[1468, 42, 1505, 22, 68, "Text"], Cell[2976, 66, 594, 13, 48, "Input"], Cell[3573, 81, 481, 13, 48, "Input"], Cell[4057, 96, 864, 23, 48, "Input"], Cell[4924, 121, 444, 12, 49, "Input"], Cell[CellGroupData[{ Cell[5393, 137, 612, 11, 30, "ItemNumbered"], Cell[6008, 150, 748, 12, 30, "ItemNumbered"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[6805, 168, 380, 5, 55, "Section"], Cell[7188, 175, 938, 16, 68, "Text"], Cell[8129, 193, 2217, 79, 92, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[10383, 277, 279, 3, 55, "Section"], Cell[10665, 282, 3857, 113, 380, "Text"], Cell[14525, 397, 2024, 50, 133, "Input"], Cell[16552, 449, 321, 6, 49, "Text"], Cell[CellGroupData[{ Cell[16898, 459, 1432, 48, 111, "ItemNumbered"], Cell[18333, 509, 1631, 47, 82, "ItemNumbered"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[20013, 562, 555, 7, 55, "Section"], Cell[20571, 571, 140, 1, 30, "Text"], Cell[CellGroupData[{ Cell[20736, 576, 548, 10, 62, "ItemNumbered"], Cell[21287, 588, 723, 14, 62, "ItemNumbered"], Cell[22013, 604, 745, 15, 45, "ItemNumbered"], Cell[22761, 621, 626, 13, 45, "ItemNumbered"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[23436, 640, 605, 8, 55, "Section"], Cell[24044, 650, 1391, 48, 76, "Text"], Cell[CellGroupData[{ Cell[25460, 702, 1029, 29, 31, "ItemNumbered"], Cell[26492, 733, 2918, 108, 79, "ItemNumbered"], Cell[29413, 843, 2738, 92, 96, "ItemNumbered"], Cell[32154, 937, 969, 20, 48, "ItemNumbered"], Cell[33126, 959, 842, 15, 45, "ItemNumbered"] }, Open ]] }, Open ]] } ] *)