(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 13.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 145429, 3850] NotebookOptionsPosition[ 122352, 3492] NotebookOutlinePosition[ 122834, 3510] CellTagsIndexPosition[ 122791, 3507] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ "F. Fass\[OGrave]. University of Padova. \n", StyleBox["Introduction to the use of Mathematica in Mathematics and Science", FontSlant->"Italic"], ". SoftSkills course for the PhD program in Mathematics\nLecture 3 (October \ 20, 2023)" }], "Text", CellFrame->{{0, 0}, {0.5, 0}}, CellChangeTimes->{{3.6030403380477047`*^9, 3.6030403496377206`*^9}, { 3.6042342921245947`*^9, 3.604234293844597*^9}, {3.6659842250605717`*^9, 3.665984234810585*^9}, {3.665988776604652*^9, 3.6659887774046535`*^9}, { 3.666106805730517*^9, 3.666106807983646*^9}, {3.686715090311494*^9, 3.68671511399888*^9}, {3.6973875672817383`*^9, 3.6973875784126663`*^9}, { 3.697434883527854*^9, 3.697434886700633*^9}, {3.7288472476388865`*^9, 3.728847250442567*^9}, 3.7290563713306975`*^9, {3.7603265886670856`*^9, 3.760326590879668*^9}, {3.792953448635523*^9, 3.7929534512106314`*^9}, { 3.793001443067047*^9, 3.793001457902358*^9}, {3.793125487673247*^9, 3.7931254933125515`*^9}, {3.823794609368146*^9, 3.8237946203338003`*^9}, { 3.905514199116931*^9, 3.90551422872608*^9}, {3.9055144580037365`*^9, 3.9055145289462028`*^9}, 3.905524053605975*^9, {3.9056541789884577`*^9, 3.90565419524717*^9}, {3.906167256174015*^9, 3.9061672588858814`*^9}, { 3.9067247773353424`*^9, 3.906724780763774*^9}}, FontSize->12, Background->None,ExpressionUUID->"fb6f17c5-199e-449a-82b1-a1440b2aa22a"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["A first look at ", FontColor->GrayLevel[0]], StyleBox["Mathematica", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[": Graphics (1)", FontColor->GrayLevel[0]] }], "Title", CellChangeTimes->{{3.7603304738687916`*^9, 3.7603304848477964`*^9}, { 3.823865007385685*^9, 3.823865007955243*^9}, {3.90551453503039*^9, 3.9055145398655195`*^9}, {3.905514613214245*^9, 3.905514628990396*^9}, 3.9055696380856085`*^9, {3.9061215880998898`*^9, 3.9061215901132812`*^9}, { 3.906167274132251*^9, 3.906167276303677*^9}, {3.9067247930267067`*^9, 3.906724801281706*^9}, {3.9067248768676667`*^9, 3.9067248911871243`*^9}, { 3.907064949973484*^9, 3.9070649515354977`*^9}}, Background->RGBColor[ 1., 0., 0.],ExpressionUUID->"28ee57e5-5f27-4eaa-84ad-d1d2593e6cad"], Cell[CellGroupData[{ Cell["0. 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Plots of functions", "Chapter", CellChangeTimes->{{3.4500076661265*^9, 3.4500076743945*^9}, { 3.542529108003581*^9, 3.5425291082375813`*^9}, 3.5439146999804525`*^9, { 3.906725453737033*^9, 3.9067254958639235`*^9}, {3.9067255849068375`*^9, 3.9067256148240414`*^9}}, Background->RGBColor[ 0., 1., 1.],ExpressionUUID->"c6cd339f-7364-4361-acd0-e1af746ca0e3"], Cell[TextData[{ "There are several commands to create plots of functions, each one in 2D and \ 3D versions.\nWhich one to use?\nYou should think of the ", StyleBox["matematical", FontSlant->"Italic"], " object you want to create" }], "Text", CellChangeTimes->{{3.4500054340101*^9, 3.4500054411705*^9}, { 3.4500065257605*^9, 3.4500065465085*^9}, {3.4503480691775*^9, 3.4503480707999*^9}, {3.5439122445809402`*^9, 3.54391227300419*^9}, { 3.5439123233142786`*^9, 3.543912392157199*^9}, {3.5439124338560724`*^9, 3.543912614129989*^9}, 3.5439127007257414`*^9, {3.543912758679843*^9, 3.5439128399715853`*^9}, {3.9067255085564413`*^9, 3.906725516564505*^9}, { 3.90672555410596*^9, 3.9067255790075808`*^9}, {3.906725618197062*^9, 3.9067256802019587`*^9}}, FontSize->18,ExpressionUUID->"5cb54967-ea1e-4324-ad80-a62a07e0d0b4"], Cell[CellGroupData[{ Cell[TextData[{ "1.a. 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", Cell[BoxData[ StyleBox[ FractionBox[ SuperscriptBox["x", "2"], RowBox[{"1", "+", SuperscriptBox["x", "2"]}]], FontSize->24]], CellChangeTimes->{{3.543913055190564*^9, 3.543913103051448*^9}, 3.5910850197794023`*^9},ExpressionUUID-> "9552b327-37f4-4e65-b202-998f13ba4f23"], " for -2 < x < 2\n 2. ", Cell[BoxData[ RowBox[{"Sin", "[", RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], "]"}]], CellChangeTimes->{{3.5439130026018715`*^9, 3.5439130388729353`*^9}, { 3.591085027751016*^9, 3.591085143222419*^9}}, FontSize->14,ExpressionUUID->"c9481ebb-b020-4101-aff4-0fc5153aca77"], " in the square (-\[Pi],\[Pi]) \[Cross] (-\[Pi], \[Pi]) \n 3. 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Cell[TextData[StyleBox["2.b Directives", FontWeight->"Plain"]], "Section", CellChangeTimes->{ 3.906725926030693*^9, {3.9067263806294017`*^9, 3.9067264236239843`*^9}, { 3.906726477216266*^9, 3.9067266593956227`*^9}, {3.906726786905038*^9, 3.906726805742013*^9}, 3.9067285901766567`*^9, 3.906728657746028*^9, { 3.9067333018194427`*^9, 3.9067333060267735`*^9}, {3.906734729446068*^9, 3.906734739577947*^9}}, Background->RGBColor[ 0., 1., 1.],ExpressionUUID->"29ea3583-17a9-468e-8c3e-2aa53d91f14b"], Cell["\<\ Graphics directives specify properties of the invidual graphics primitives (thickness of lines, colour of points, lines and surfaces, size of points, \ etc)\ \>", "Text", CellChangeTimes->{{3.906734745169407*^9, 3.9067348370514054`*^9}, { 3.906773579606683*^9, 3.906773624111174*^9}, {3.906774202468364*^9, 3.90677421841848*^9}}, FontSize->18, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"0b8f1eff-e20f-4313-a37a-6353c3c721ba"], Cell[CellGroupData[{ Cell["", "Subsubsection", CellChangeTimes->{3.54391571752444*^9}, FontSize->14,ExpressionUUID->"061cc676-5417-4a21-b255-36ad110fc648"], Cell[TextData[{ StyleBox["How to specify the graphics directives?", FontWeight->"Bold"], "\n\[FilledSmallCircle] If you build the graphics \ \[OpenCurlyDoubleQuote]assembling\[CloseCurlyDoubleQuote] the graphics \ primitives:\n \t", StyleBox["{ directive1 , primitive1, directive2, primitive 2, .... }\n", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], "\[FilledSmallCircle] Within commands such as Plot, Plot3D, ... using \ certain ", StyleBox["options", FontWeight->"Bold"], ": e.g.\n\t", StyleBox["PlotStyle \[Rule] lista di direttive", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]] }], "Text", CellChangeTimes->{{3.906734850715457*^9, 3.9067348654210825`*^9}, { 3.9067742371482754`*^9, 3.906774439446162*^9}}, FontSize->18, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"d0cb15a1-c954-42d2-a9ff-364bce606f50"], Cell[TextData[{ StyleBox["Note:", FontWeight->"Bold"], "\nEach 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Tables of graphics ", StyleBox["(GraphicsRow, GraphicsGrid,....)", FontSize->18] }], "Chapter", CellChangeTimes->{{3.4500076661265*^9, 3.4500076743945*^9}, { 3.542529108003581*^9, 3.5425291082375813`*^9}, 3.5439146999804525`*^9, { 3.906725453737033*^9, 3.9067254958639235`*^9}, {3.9067255849068375`*^9, 3.9067256148240414`*^9}, {3.906730190579305*^9, 3.9067302280226007`*^9}, { 3.9067750428584385`*^9, 3.9067750653065767`*^9}, {3.9067795844670258`*^9, 3.9067795940748625`*^9}}, Background->RGBColor[ 0., 1., 1.],ExpressionUUID->"09dcd8e7-a249-435b-9cbb-99c6287ccd05"], Cell["\<\ To display different graphics in a 1-d or 2-d assembly, can simply use Table[ \ ]\ \>", "Text", CellChangeTimes->{{3.906775086114628*^9, 3.906775150099149*^9}}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"910900f4-9394-44de-b494-069ff0288ffd"], Cell[BoxData[ RowBox[{"gr", "=", RowBox[{"Table", "[", " ", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"x", "^", "n"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"n", ",", "0", ",", "5"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.687538916056591*^9, 3.687538984465994*^9}, { 3.6876065004046583`*^9, 3.6876065008744807`*^9}, {3.9067750814431095`*^9, 3.90677508464622*^9}, {3.9067751582532454`*^9, 3.9067751604409432`*^9}, 3.906791402082037*^9},ExpressionUUID->"2c46eaed-1b3f-4fdb-a43a-\ 2526411d8489"], Cell[CellGroupData[{ Cell["", "Subsubsection",ExpressionUUID->"a25e6f35-e087-464c-952d-0973f5186127"], Cell[TextData[{ "but formatting is poor.\nBetter using ", StyleBox[" GraphicsRow , GraphicsColumn, GraphicsGrid ", FontWeight->"Bold"] }], "Text", CellChangeTimes->{{3.906775086114628*^9, 3.906775150099149*^9}, { 3.906775181685278*^9, 3.906775218661651*^9}, {3.9067772096732516`*^9, 3.906777210016382*^9}}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"4e60fa51-556c-4726-9c0f-42ba12384203"], Cell[BoxData[ RowBox[{"Take", "[", RowBox[{"gr", ",", RowBox[{"-", "3"}]}], "]"}]], "Input", CellChangeTimes->{{3.906791749447487*^9, 3.9067917715096116`*^9}}, CellLabel->"In[46]:=",ExpressionUUID->"47b5f881-b2aa-4833-a8a2-d4480f28cdf8"], Cell[BoxData[ RowBox[{"GraphicsGrid", "[", " ", RowBox[{ RowBox[{"{", " ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"gr", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"gr", "[", RowBox[{"[", "2", "]"}], "]"}], ",", " ", RowBox[{"gr", "[", RowBox[{"[", "3", "]"}], "]"}]}], " ", "}"}], ",", " ", RowBox[{"Take", "[", RowBox[{"gr", ",", RowBox[{"-", "3"}]}], "]"}]}], "}"}], " ", ",", RowBox[{"Spacings", "\[Rule]", RowBox[{"{", RowBox[{"0", ",", "20"}], "}"}]}], ",", RowBox[{"ImageSize", "\[Rule]", "Full"}]}], "]"}]], "Input", CellChangeTimes->{{3.906791655799244*^9, 3.9067917442414823`*^9}, { 3.906791785329714*^9, 3.9067918880662146`*^9}}, CellLabel->"In[51]:=",ExpressionUUID->"bf390437-6375-4250-a590-7598ba1d0d9a"] }, Closed]], Cell[CellGroupData[{ Cell[" ", "Subsubsection", CellChangeTimes->{{3.9067915689142256`*^9, 3.90679157073759*^9}},ExpressionUUID->"405475b6-6d6a-46eb-b72c-\ 759a83cb0994"], Cell[BoxData[ RowBox[{"GraphicsRow", "[", " ", RowBox[{"{", RowBox[{ RowBox[{"gr", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"gr", "[", RowBox[{"[", "2", "]"}], "]"}]}], "}"}], " ", "]"}]], "Input", CellChangeTimes->{{3.4500093985275*^9, 3.4500094122711*^9}, 3.5910869499707923`*^9, {3.687538990036227*^9, 3.687538996572562*^9}, { 3.6876066473697853`*^9, 3.687606650635893*^9}}, CellLabel->"In[54]:=",ExpressionUUID->"8c827b23-1dac-44e3-8139-9c55ae09ab25"], Cell[BoxData[ RowBox[{ RowBox[{"Take", "[", RowBox[{"gr", ",", "2"}], "]"}], "//", "GraphicsRow"}]], "Input", CellChangeTimes->{{3.687606715793743*^9, 3.6876067210993595`*^9}, { 3.6876067522065277`*^9, 3.687606758484099*^9}}, CellLabel->"In[55]:=",ExpressionUUID->"766650d2-092f-4220-a578-acb279ea4e02"], Cell[BoxData[ RowBox[{"GraphicsGrid", "[", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"gr", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"gr", "[", RowBox[{"[", "2", "]"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", " ", RowBox[{"{", RowBox[{ RowBox[{"gr", "[", RowBox[{"[", "2", "]"}], "]"}], ",", " ", RowBox[{"gr", "[", RowBox[{"[", "3", "]"}], "]"}], ",", RowBox[{"gr", "[", RowBox[{"[", "4", "]"}], "]"}]}], "}"}]}], "}"}], "]"}]], "Input", CellChangeTimes->{{3.6875390004217744`*^9, 3.687539030424743*^9}, 3.687606777554228*^9}, CellLabel->"In[56]:=",ExpressionUUID->"6b1095b7-64d9-4ff0-a883-f94962f7cb20"] }, Closed]], Cell[CellGroupData[{ Cell["", "Subsubsection", CellChangeTimes->{{3.599556280585082*^9, 3.5995562888350935`*^9}},ExpressionUUID->"c5e5ce6b-b907-417a-b69e-\ 4948722d5931"], Cell[TextData[{ StyleBox["Exercise:\n", FontWeight->"Bold"], "1. ", "Write a function\n ", StyleBox["gtorus(R,r)\n", FontWeight->"Bold"], "that plots a torus (donught) of\n equatorial radius R \n \ meridian radius r", StyleBox["\n", FontWeight->"Bold"], "2. Assemble a number of them, with the same R and various r between 0 and R \ (or even >R,why not?) in a grid." }], "Text", ShowGroupOpener->False, CellChangeTimes->{{3.6874460956256742`*^9, 3.687446162756789*^9}, { 3.687446427062728*^9, 3.6874464894411564`*^9}, {3.6875388795092983`*^9, 3.6875388935618706`*^9}, 3.68754061326365*^9, {3.9067752676807146`*^9, 3.9067752825529156`*^9}, {3.906777218561158*^9, 3.906777314132493*^9}, { 3.906777877831765*^9, 3.9067778784104843`*^9}, {3.906791973392905*^9, 3.9067919956147776`*^9}}, Background->RGBColor[ 0, 1, 0],ExpressionUUID->"192a6a40-26ab-43dd-8517-fca7c9104b20"], Cell["E.g., use R=3 , r = 0,...,5 and assemble them in two rows", "Text", CellChangeTimes->{{3.9067773290826826`*^9, 3.906777352733761*^9}}, Background->GrayLevel[ 0.85],ExpressionUUID->"235e7623-5074-4168-84af-5e6a0fad8c43"], Cell["\<\ Might be useful to define two functions: torus(R,r,u,t) : the parametrization { (R+r cos(u) cos(t) , \ (R+r cos(u) sin(t) , r sin(t) } gtorus(R,r) : plots the torus, calling torus[R,r]\ \>", "Text", CellChangeTimes->{{3.9067773290826826`*^9, 3.906777352733761*^9}, { 3.9067777413991213`*^9, 3.906777871818282*^9}, {3.90706456364318*^9, 3.9070645650962114`*^9}}, Background->GrayLevel[ 0.85],ExpressionUUID->"4e9e5600-41ff-4a77-ab28-3d1f843d5dd8"] }, Closed]], Cell[CellGroupData[{ Cell["", "Subsubsection",ExpressionUUID->"bbbe2892-ae07-4689-b985-1db0c0c6efd5"], Cell[BoxData[ RowBox[{ RowBox[{"torus", "[", RowBox[{"R_", ",", "r_", ",", "u_", ",", "t_"}], "]"}], ":=", " ", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"R", "+", RowBox[{"r", " ", RowBox[{"Cos", "[", "u", "]"}]}]}], ")"}], RowBox[{"Cos", "[", "t", "]"}]}], ",", " ", RowBox[{ RowBox[{"(", RowBox[{"R", "+", RowBox[{"r", " ", RowBox[{"Cos", "[", "u", "]"}]}]}], ")"}], RowBox[{"Sin", "[", "t", "]"}]}], ",", " ", RowBox[{"r", " ", RowBox[{"Sin", "[", "u", "]"}]}]}], "}"}]}]], "Input", CellChangeTimes->{{3.906792698386016*^9, 3.9067927012529125`*^9}, { 3.90706458776326*^9, 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RowBox[{"gtorus", "[", RowBox[{"3", ",", "3"}], "]"}], ",", " ", RowBox[{"gtorus", "[", RowBox[{"3", ",", "4"}], "]"}], ",", " ", RowBox[{"gtorus", "[", RowBox[{"3", ",", "5"}], "]"}]}], "}"}]}], "}"}], " ", "]"}]], "Input", CellChangeTimes->{{3.6364596408733788`*^9, 3.636459684889896*^9}, 3.6364597240421357`*^9, {3.6875394123347716`*^9, 3.687539431717141*^9}, { 3.6875406328961306`*^9, 3.6875406397041154`*^9}, {3.906777928499096*^9, 3.9067779457290688`*^9}}, CellLabel->"In[7]:=",ExpressionUUID->"637e3df8-3ef3-49dd-916e-c8e7ecca3d9f"] }, Closed]], Cell[CellGroupData[{ Cell[" ", "Subsubsection", CellChangeTimes->{{3.5995562153349905`*^9, 3.5995562188449955`*^9}},ExpressionUUID->"3ff126b9-a197-4958-985c-\ de8629b1d64b"], Cell[TextData[{ "Too small?\nOption ", StyleBox[" ImageSize ", FontWeight->"Bold"], "\[Rule] Large, Full, 800, 1000,....\n\n(Need not recreating the graphics)" }], "Text", CellChangeTimes->{{3.9067779516964183`*^9, 3.90677799698341*^9}, { 3.906778031695406*^9, 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3.6875406397041154`*^9}, {3.906777928499096*^9, 3.9067779457290688`*^9}, {3.9067780506909494`*^9, 3.9067780574542503`*^9}}, CellLabel->"In[8]:=",ExpressionUUID->"591d5325-e7c9-40bc-83ae-f9c14a4d9744"], Cell["This can also be done without recreating the graphics", "Text", CellChangeTimes->{{3.9067779516964183`*^9, 3.90677799698341*^9}, { 3.906778031695406*^9, 3.9067780439893904`*^9}, {3.906778084475048*^9, 3.9067780872400317`*^9}, {3.9070646928872104`*^9, 3.907064707427862*^9}}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"d0b8f12a-7e23-4450-b9fc-225b46a09051"], Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"%%", ",", " ", RowBox[{"ImageSize", "->", "Full"}]}], "]"}]], "Input", CellChangeTimes->{{3.906778009105567*^9, 3.906778020337969*^9}, 3.9070647166679697`*^9},ExpressionUUID->"67f80682-9f60-40db-a8fb-\ 4231c4dfa8c4"], Cell[BoxData[" "], "Input", CellChangeTimes->{ 3.9067779874851856`*^9},ExpressionUUID->"ddfbcb23-8976-4e05-8517-\ 5fe286258307"] }, Closed]], Cell[CellGroupData[{ Cell["", "Subsubsection",ExpressionUUID->"cff9aaa9-f494-48d2-ada6-407fa5ba8f0e"], Cell["\<\ Don\[CloseCurlyQuote]t like the mesh? The axes? The frame (in 3D, \ \[OpenCurlyDoubleQuote]box\[CloseCurlyDoubleQuote])? Remove them with Options\ \>", "Text", CellChangeTimes->{{3.9067779516964183`*^9, 3.906777953024232*^9}, { 3.906778110579716*^9, 3.906778211974134*^9}, {3.906778261158924*^9, 3.906778289043022*^9}}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"b1ab99bf-1d76-4edc-a717-0bd1fe206476"], Cell[BoxData[ RowBox[{ RowBox[{"gtorus3", "[", RowBox[{"R_", ",", "r_"}], "]"}], ":=", RowBox[{"ParametricPlot3D", "[", " ", RowBox[{ RowBox[{"torus", "[", RowBox[{"R", ",", "r", ",", "u", ",", "t"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"u", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], StyleBox[",", FontColor->GrayLevel[0]], StyleBox["\[IndentingNewLine]", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[ RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Opacity", "[", ".5", "]"}], "}"}]}], FontColor->GrayLevel[0]], StyleBox[",", FontColor->GrayLevel[0]], StyleBox["\[IndentingNewLine]", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[ RowBox[{"Mesh", "\[Rule]", "False"}], FontColor->RGBColor[1, 0, 0]], ",", " ", StyleBox[ RowBox[{"Axes", "\[Rule]", "False"}], FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[",", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[ RowBox[{"Boxed", " ", "\[Rule]", "False"}], FontColor->RGBColor[1, 0, 0]]}], "]"}]}]], "Input", CellChangeTimes->{{3.687540686188027*^9, 3.687540706734062*^9}, { 3.906778169832167*^9, 3.9067781847612667`*^9}, {3.906778225033063*^9, 3.9067782255492764`*^9}, {3.9067783613229456`*^9, 3.906778415266557*^9}, { 3.907064747813552*^9, 3.907064750062561*^9}}, CellLabel->"In[12]:=",ExpressionUUID->"1459778e-d7dd-4ddf-9a98-7e4ce3d3b591"], Cell[BoxData[ RowBox[{"gtorus3", "[", RowBox[{"3", ",", "5"}], "]"}]], "Input", CellChangeTimes->{{3.906778427045777*^9, 3.9067784273418946`*^9}}, CellLabel->"In[13]:=",ExpressionUUID->"23e3584e-337d-45f4-a3f2-2237ac553ae9"], Cell[TextData[{ StyleBox["Exercise", FontWeight->"Bold"], ": Could we have changed all these three options with Show? Why?" }], "Text", CellChangeTimes->{{3.9067784601628265`*^9, 3.906778501433549*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"85c5d797-7d3d-4162-aa03-1c900a2b3adc"] }, Closed]], Cell[CellGroupData[{ Cell[" ", "Subsubsection", CellChangeTimes->{{3.63645966292764*^9, 3.6364596635456753`*^9}, 3.6875394465639057`*^9},ExpressionUUID->"e06d25a7-8ac3-43d6-9ad1-\ 16847565810f"], Cell[TextData[{ "The image is not smooth enough?\nOption ", StyleBox["PlotPoints \[Rule] n\n", FontWeight->"Bold"], "(try not too large values of n first)\n\nOr make new function\n \ gtorus(R,r,n)\nthat includes option PlotPoints \[Rule] n" }], "Text", CellChangeTimes->{{3.906778526536639*^9, 3.906778602176118*^9}, { 3.906778676774745*^9, 3.906778744431446*^9}}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"e6dc284b-73a2-42c9-ad67-ef57c18ab0bf"], Cell[BoxData[ RowBox[{ RowBox[{"gtorus", "[", RowBox[{"R_", ",", "r_", ",", "n_"}], "]"}], ":=", "\[IndentingNewLine]", RowBox[{"ParametricPlot3D", "[", " ", RowBox[{ RowBox[{"torus", "[", RowBox[{"R", ",", "r", ",", "u", ",", "t"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"u", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], StyleBox[",", FontColor->GrayLevel[0]], StyleBox["\[IndentingNewLine]", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[ RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Opacity", "[", ".5", "]"}], "}"}]}], FontColor->GrayLevel[0]], StyleBox[",", FontColor->GrayLevel[0]], StyleBox["\[IndentingNewLine]", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[ RowBox[{"Mesh", "\[Rule]", "False"}], FontColor->GrayLevel[0]], StyleBox[",", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[ RowBox[{"Axes", "\[Rule]", "False"}], FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[",", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[ RowBox[{"Boxed", " ", "\[Rule]", "False"}], FontColor->GrayLevel[0]], StyleBox[",", FontColor->GrayLevel[0]], StyleBox["\[IndentingNewLine]", FontColor->GrayLevel[0]], StyleBox[" ", FontColor->GrayLevel[0]], StyleBox[ RowBox[{"PlotPoints", "->", " ", "n"}], FontColor->GrayLevel[0]]}], "]"}]}]], "Input", CellChangeTimes->{{3.687540686188027*^9, 3.687540706734062*^9}, { 3.906778169832167*^9, 3.9067781847612667`*^9}, {3.906778225033063*^9, 3.9067782255492764`*^9}, {3.9067783613229456`*^9, 3.906778415266557*^9}, { 3.906778562372202*^9, 3.90677856285709*^9}, {3.90677861421945*^9, 3.906778631839346*^9}, {3.9067786644964905`*^9, 3.906778668433603*^9}, { 3.906778725310937*^9, 3.9067787476337924`*^9}, {3.9070647689754963`*^9, 3.907064774366479*^9}}, CellLabel->"In[14]:=",ExpressionUUID->"96e86beb-131d-42b9-a74f-f928c71d1e97"], Cell[BoxData[ RowBox[{"GraphicsRow", "[", " ", RowBox[{"{", RowBox[{ RowBox[{"gtorus", "[", RowBox[{"3", ",", "5", ",", "30"}], "]"}], " ", ",", " ", RowBox[{"gtorus", "[", RowBox[{"3", ",", "5", ",", "50"}], "]"}], " ", ",", " ", RowBox[{"gtorus", "[", RowBox[{"3", ",", "5", ",", "100"}], "]"}]}], " ", "}"}], " ", "]"}]], "Input", CellChangeTimes->{{3.687540686188027*^9, 3.687540706734062*^9}, { 3.906778169832167*^9, 3.9067781847612667`*^9}, {3.906778225033063*^9, 3.9067782255492764`*^9}, {3.9067783613229456`*^9, 3.906778415266557*^9}, { 3.9067786518061066`*^9, 3.9067786567481933`*^9}, {3.906778751694639*^9, 3.906778786748951*^9}}, CellLabel->"In[15]:=",ExpressionUUID->"c707dbb8-0abb-4ce6-b5ce-344407553157"], Cell[TextData[{ StyleBox["Exercise: ", FontWeight->"Bold"], "Automatize the ", "creation of the list\n { { gtorus[3,0], gtorus[3,1], gtorus[3,2] \ },\n { gtorus[3,3], gtorus[3,4], gtorus[3,5] } }" }], "Text", CellChangeTimes->{{3.6875394501868744`*^9, 3.6875394607099953`*^9}, { 3.687539491858039*^9, 3.68753953945287*^9}, {3.6875404208076277`*^9, 3.6875404243885956`*^9}, {3.9067788097903976`*^9, 3.906778882470952*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"a90882cf-841a-4af7-9e57-699a157c6d43"], Cell["\<\ As usual, can be done in different ways. E.g. - A list of two table (each with one iterator) - A single table (with two iterators)\ \>", "Text", CellChangeTimes->{{3.9067788961534677`*^9, 3.906778934772411*^9}}, Background->GrayLevel[ 0.85],ExpressionUUID->"5b1c2934-d468-43df-a3ca-d602aa5a3230"] }, Closed]], Cell[CellGroupData[{ Cell[" ", "Subsubsection", CellChangeTimes->{{3.6875395417821455`*^9, 3.68753954551407*^9}},ExpressionUUID->"42695b25-670a-4984-a757-\ 68b10b137893"], Cell[BoxData[ RowBox[{"GraphicsGrid", "[", " ", RowBox[{"{", " ", RowBox[{ RowBox[{"Table", "[", " ", RowBox[{ RowBox[{"gtorus3", "[", RowBox[{"3", ",", "r"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"r", ",", "0", ",", "2"}], "}"}]}], "]"}], ",", "\[IndentingNewLine]", " ", RowBox[{"Table", "[", " ", RowBox[{ RowBox[{"gtorus3", "[", RowBox[{"3", ",", "r"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"r", ",", "3", ",", "5"}], "}"}]}], "]"}]}], " ", "}"}], " ", "]"}]], "Input", CellChangeTimes->{ 3.636459629418723*^9, {3.68753955268489*^9, 3.6875395861177626`*^9}, { 3.6875404995377617`*^9, 3.687540499976194*^9}, {3.687540646656373*^9, 3.687540647509116*^9}, {3.906778939505659*^9, 3.906778951566078*^9}, { 3.9067789834961014`*^9, 3.906779000554409*^9}}, CellLabel->"In[16]:=",ExpressionUUID->"b4004d45-deda-4da6-9233-61ab82284a24"], Cell[BoxData[ RowBox[{"GraphicsGrid", "[", " ", RowBox[{"Table", "[", " ", RowBox[{ RowBox[{"gtorus3", "[", " ", RowBox[{"3", ",", " ", RowBox[{"r", "+", "s"}]}], " ", "]"}], ",", " ", RowBox[{"{", RowBox[{"s", ",", "0", ",", "3", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"r", ",", "0", ",", "2"}], "}"}]}], " ", "]"}], " ", "]"}]], "Input", CellChangeTimes->{ 3.636459629418723*^9, {3.68753955268489*^9, 3.687539600248726*^9}, { 3.6875399241822977`*^9, 3.6875399275697107`*^9}, {3.6875399900397573`*^9, 3.6875400029337473`*^9}, {3.6875400605281315`*^9, 3.6875400943466063`*^9}, {3.687540329106613*^9, 3.6875403378244505`*^9}, 3.6875406524960756`*^9, {3.9067790130509453`*^9, 3.9067790350613785`*^9}}, CellLabel->"In[17]:=",ExpressionUUID->"c0d2b2b9-e41b-4504-b595-41fba11e198b"], Cell["\<\ Note: The most external cycle (here, r=0,1,2) is executed first. 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Animation etc ", StyleBox["(Manipulate, Animate, ListAnimate)", FontSize->18] }], "Chapter", CellChangeTimes->{{3.4500076661265*^9, 3.4500076743945*^9}, { 3.542529108003581*^9, 3.5425291082375813`*^9}, 3.5439146999804525`*^9, { 3.906725453737033*^9, 3.9067254958639235`*^9}, {3.9067255849068375`*^9, 3.9067256148240414`*^9}, {3.906730190579305*^9, 3.9067302280226007`*^9}, { 3.9067750428584385`*^9, 3.9067750653065767`*^9}, {3.9067795467578545`*^9, 3.9067795747037306`*^9}, {3.906783077144533*^9, 3.9067830825651264`*^9}}, Background->RGBColor[ 0., 1., 1.],ExpressionUUID->"e1b99763-fbd1-4174-a6de-15c02a10dd9b"], Cell[TextData[{ StyleBox[" Manipulate[ expression which contains parameters , {parameters} \ ]\n", FontWeight->"Bold"], "creates an interface that \n- shows the expression, an\n- gives the \ possibility of interactively modify it changing the valus of the parameters \ (sliders)" }], "Text", CellChangeTimes->{{3.688022937590395*^9, 3.688023148664059*^9}, { 3.906779607336684*^9, 3.9067797725182114`*^9}}, FontSize->16, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"2234c179-c25d-4a31-805d-cf4cbb0d82df"], Cell[BoxData[ RowBox[{"Manipulate", "[", " ", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "r"], "//", "Expand"}], " ", ",", " ", RowBox[{"{", RowBox[{"r", ",", "0", ",", "5", StyleBox[",", FontColor->RGBColor[1, 0, 0]], StyleBox["1", FontColor->RGBColor[1, 0, 0]]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.906793303986332*^9, 3.9067933728961253`*^9}, { 3.9067934745743494`*^9, 3.906793475042923*^9}}, CellLabel->"In[91]:=",ExpressionUUID->"5af9190b-5d54-4b4e-80a0-5b656812f096"], Cell[CellGroupData[{ Cell[" ", "Subsubsection", CellChangeTimes->{{3.687546582197356*^9, 3.68754659022091*^9}}, FontSize->16,ExpressionUUID->"1ee8efc6-d807-41c8-85b7-d981a0a80e9f"], Cell[BoxData[ RowBox[{"Manipulate", "[", " ", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "r"], "//", "Expand"}], " ", ",", " ", RowBox[{"{", RowBox[{"r", ",", "0", ",", "5"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.698133649797861*^9, 3.6981336611760497`*^9}, { 3.698133711604218*^9, 3.6981337155090322`*^9}, {3.7300967947457285`*^9, 3.7300967949483232`*^9}, {3.7300968747230673`*^9, 3.7300968962878*^9}, { 3.7615453873587804`*^9, 3.7615453875431757`*^9}, {3.761545444641413*^9, 3.761545445246071*^9}, {3.7936913091074944`*^9, 3.793691309259144*^9}, { 3.793691386172694*^9, 3.7936913867954698`*^9}, {3.824535137971838*^9, 3.824535139802397*^9}, {3.8245352604116893`*^9, 3.82453526149879*^9}}, CellLabel->"In[33]:=",ExpressionUUID->"d069a7e9-e8a2-434d-8cee-280dcc6c40ca"] }, Closed]], Cell[CellGroupData[{ Cell[" ", "Subsubsection", CellChangeTimes->{{3.6979961409938793`*^9, 3.697996144475996*^9}}, FontSize->16,ExpressionUUID->"8e92afaf-15ba-4dd8-ad4d-938cd241a51b"], Cell["\<\ ?? The syntax for the parameters in Manipulate look similar to that of iterators \ in Table, but there is an important difference: unless otherwise stated, parameters are incremented contnuously (as real \ numbers, not as integers)\ \>", "Text", CellChangeTimes->{{3.688022937590395*^9, 3.688023148664059*^9}, { 3.6880232278493023`*^9, 3.6880232506476765`*^9}, {3.6880233195686197`*^9, 3.6880234063214025`*^9}, {3.688023445437344*^9, 3.6880234551051455`*^9}, { 3.688023557046562*^9, 3.688023559980316*^9}, {3.7294842408259397`*^9, 3.729484255572812*^9}, {3.824468948844143*^9, 3.824468965850531*^9}, { 3.906780079842519*^9, 3.90678020381348*^9}}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"e6731386-16ef-406e-8c33-3c15bb860868"], Cell[BoxData[ RowBox[{"Manipulate", "[", " ", RowBox[{ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "r"], "//", "Expand"}], " ", ",", " ", RowBox[{"{", RowBox[{"r", ",", "0", ",", "5", StyleBox[",", FontColor->RGBColor[1, 0, 0]], StyleBox["1", 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RowBox[{ RowBox[{"gtorus3", "[", RowBox[{"R", ",", "r"}], "]"}], " ", ",", " ", RowBox[{"{", RowBox[{"r", ",", "0", ",", "5", ",", "1"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"R", ",", "0", ",", "5", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.6880231651251984`*^9, 3.688023176424839*^9}, { 3.688023271722515*^9, 3.6880232848239527`*^9}, {3.688023618536491*^9, 3.688023621910509*^9}, 3.6880240974252987`*^9, {3.9067798171640015`*^9, 3.906779818553565*^9}, 3.9067798739825974`*^9}, CellLabel->"In[32]:=",ExpressionUUID->"0c8a05df-dea8-4fcc-a361-0edc986a4c12"] }, Closed]], Cell[CellGroupData[{ Cell["", "Subsubsection",ExpressionUUID->"f1e53b88-d54a-4e27-ac8b-6a86e610353e"], Cell[TextData[{ "It is possible to \[OpenCurlyDoubleQuote]animate\[CloseCurlyDoubleQuote] \ the expression\n- inside the Manipulate, through the \ slider\[CloseCurlyQuote]s menu\n- with the command \n ", StyleBox["Animate[ expression which contains parameters , {parameters} ]\n \ ", FontWeight->"Bold"], 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