(* 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[ 177151, 4452] NotebookOptionsPosition[ 143311, 3953] NotebookOutlinePosition[ 144012, 3975] CellTagsIndexPosition[ 143969, 3972] 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 2 (October \ 13, 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}}, FontSize->12, Background->None,ExpressionUUID->"94be3d96-78df-4e05-b867-d3d278ae1776"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["1. A first look at ", FontColor->GrayLevel[0]], StyleBox["Mathematica", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[": What can you do? How?\n(part 2)", 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}}, Background->RGBColor[ 1., 0., 0.],ExpressionUUID->"328d4016-7c60-48ce-a8cd-126889825232"], Cell[CellGroupData[{ Cell[TextData[StyleBox["6. Definition of functions", FontSize->24, FontSlant->"Italic"]], "Section", CellChangeTimes->{{3.7603305394751625`*^9, 3.760330552624377*^9}, { 3.7603798490998344`*^9, 3.7603798499042997`*^9}, {3.7929893543089256`*^9, 3.7929893583080654`*^9}, {3.7931274848259177`*^9, 3.793127484904027*^9}, { 3.905514870952041*^9, 3.9055148984418287`*^9}, {3.9061673282389555`*^9, 3.9061673321128182`*^9}}, Background->RGBColor[ 0., 1., 1.],ExpressionUUID->"6170f93c-5736-45a1-a4cb-02629c6fdae8"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["6.1 Functions in ", FontSize->24], StyleBox["Mathematica", FontSize->24, FontSlant->"Italic"] }], "Subsection", CellChangeTimes->{{3.7931280204658203`*^9, 3.793128024004758*^9}, { 3.7931295911821847`*^9, 3.793129593322367*^9}, {3.9055683773938007`*^9, 3.905568386718421*^9}, {3.90556841798425*^9, 3.905568428967516*^9}}, FontColor->GrayLevel[0], Background->RGBColor[ 0., 1., 1.],ExpressionUUID->"8becf62f-7370-48ce-980b-0bf022c800ab"], Cell[TextData[{ "Extremely important: ", StyleBox["Mathematica", FontSlant->"Italic"], " is a functional language" }], "Text", CellChangeTimes->{{3.760381542983141*^9, 3.760381562558795*^9}, { 3.8237976484654245`*^9, 3.823797649114555*^9}, {3.9055683901528435`*^9, 3.905568411297187*^9}, {3.9055688750117817`*^9, 3.905568876114238*^9}},ExpressionUUID->"d4a2e060-819f-4fe7-a799-\ f3cedddfe041"], Cell[TextData[{ "Functions can be defined in different ways\nThe basic one is\n", StyleBox[" ", FontWeight->"Bold"], StyleBox[" f[x_] := expression", FontFamily->"Courier New", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["\n", FontWeight->"Bold"], "This specifies that \[OpenCurlyDoubleQuote]f\[CloseCurlyDoubleQuote] is the \ function that trasforms its argument into \[OpenCurlyDoubleQuote]expression\ \[CloseCurlyDoubleQuote]" }], "Text", CellChangeTimes->{{3.793127624472603*^9, 3.7931276266329093`*^9}, 3.7931278640235558`*^9, {3.823797654783281*^9, 3.823797655337716*^9}, { 3.8239292471823454`*^9, 3.8239292482702703`*^9}, {3.8239293028934803`*^9, 3.8239293040540757`*^9}, {3.905568880295766*^9, 3.9055688957688675`*^9}, { 3.9055692880872345`*^9, 3.9055693273977957`*^9}}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"91de3251-e7ec-4a9d-a52a-5d4fa863c1f7"], Cell[TextData[{ StyleBox["NB: ", FontWeight->"Bold"], "in the definition of a function", StyleBox["\n ", FontWeight->"Bold"], " the argument \[OpenCurlyDoubleQuote]x\[CloseCurlyDoubleQuote] is a \ symbol and must be followed by _ (\[OpenCurlyDoubleQuote]blank\ \[CloseCurlyDoubleQuote])\n ", StyleBox[" ", FontWeight->"Bold"], " use ", StyleBox[":= ", FontFamily->"Courier New", FontColor->RGBColor[1, 0, 0]], " ", StyleBox["\n", FontFamily->"Courier New", FontColor->RGBColor[1, 0, 0]], "(or else ", StyleBox["= ", FontFamily->"Courier New", FontColor->RGBColor[1, 0, 0]], "; the difference will be explained later)" }], "Text", CellChangeTimes->{{3.793127624472603*^9, 3.7931276266329093`*^9}, 3.793127885346733*^9, {3.8237976755831947`*^9, 3.823797676056276*^9}, { 3.905569406021552*^9, 3.9055694372323165`*^9}, {3.9055694918237514`*^9, 3.905569569324791*^9}, {3.905571631788954*^9, 3.905571661785116*^9}}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"d61104be-e8a8-431f-ae91-c04ae9d24e32"], Cell[TextData[{ "Functions of more variables:\n ", StyleBox[" f[x1_, x2_ , .... , xn_ ] ", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox[":= ....", FontWeight->"Bold"], " " }], "Text", CellChangeTimes->{{3.793127624472603*^9, 3.7931276266329093`*^9}, 3.7931278971436887`*^9, {3.82379768006703*^9, 3.823797680619447*^9}, { 3.9055695715583596`*^9, 3.9055695831318903`*^9}}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"576227af-900a-440f-9e6a-17725cb1da7b"], Cell["\<\ Functions are not only numerical. (Basic for functional programming)\ \>", "Text", CellChangeTimes->{{3.793127906610225*^9, 3.7931279120301085`*^9}, { 3.9055695860405607`*^9, 3.9055696206231985`*^9}}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"5a193c0d-29e8-422f-970e-db43cd9f18a4"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["6.2 Examples", FontSize->24]], "Subsection", CellChangeTimes->{{3.7931280204658203`*^9, 3.793128024004758*^9}, { 3.7931295911821847`*^9, 3.793129593322367*^9}, {3.9055683773938007`*^9, 3.905568386718421*^9}, {3.90556841798425*^9, 3.905568428967516*^9}, { 3.906167361271287*^9, 3.90616736484344*^9}}, FontColor->GrayLevel[0], Background->RGBColor[ 0., 1., 1.],ExpressionUUID->"ed07186d-e896-4ce5-b748-438db350947b"], Cell[TextData[{ StyleBox["Exercise: ", FontWeight->"Bold"], "write a function that takes as argument a function ", StyleBox["R", FontWeight->"Bold"], "\[Rule]", StyleBox["R", FontWeight->"Bold"], " (given as an expression containing the variable \[OpenCurlyDoubleQuote]x\ \[CloseCurlyDoubleQuote]) and plots its graph in the interval 0 < x < 1" }], "Text", CellChangeTimes->{{3.793128071767908*^9, 3.7931281008217955`*^9}, { 3.9061217056052017`*^9, 3.906121777810315*^9}, {3.9061218172029333`*^9, 3.9061218288394356`*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"0bafd984-1036-4cc1-a9d9-cc578d99f969"], Cell[CellGroupData[{ Cell["", "Subsubsection",ExpressionUUID->"67e856a3-2111-4202-8438-cdd811ba2bc7"], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.906182518155094*^9, 3.9061825306078796`*^9}}, CellLabel->"In[4]:=",ExpressionUUID->"d1647a79-0254-4b8d-9d08-d103ed4242ce"], Cell[BoxData[ RowBox[{ RowBox[{"pl", "[", "expr_", "]"}], " ", ":=", " ", RowBox[{"Plot", "[", RowBox[{"expr", ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "0", ",", " ", "1"}], "}"}]}], "]"}]}]], "Input", CellLabel->"In[5]:=",ExpressionUUID->"8ed4a36e-4e38-4777-ade1-14e1a0471857"], Cell["Test:", "Text", CellChangeTimes->{{3.7931281166618347`*^9, 3.7931281204612713`*^9}, { 3.906121790721326*^9, 3.90612179157953*^9}},ExpressionUUID->"2ad708e2-fb12-4c15-bcc9-\ 19f968933625"], Cell[BoxData[ RowBox[{"pl", "[", " ", RowBox[{"Sin", "[", SuperscriptBox["x", "2"], "]"}], " ", "]"}]], "Input", CellChangeTimes->{{3.823797730007721*^9, 3.823797765927846*^9}, { 3.823929879149024*^9, 3.823929897892555*^9}, {3.8239299503168044`*^9, 3.8239299686919355`*^9}, {3.9061825864719057`*^9, 3.9061825868408766`*^9}}, CellLabel->"In[7]:=",ExpressionUUID->"35913783-360a-4dfd-8552-8d6f530357fb"] }, Closed]], Cell[CellGroupData[{ Cell["", "Subsubsection",ExpressionUUID->"e565223c-d1db-4ef0-bfcc-bc4baea91326"], Cell[TextData[{ StyleBox["Exercise: ", FontWeight->"Bold"], "write a function", StyleBox["\n", FontWeight->"Bold"], " pl(expr,a,b)\nwhich does the saem thing but in \ the interval a < x < b with any a and b" }], "Text", CellChangeTimes->{{3.793128071767908*^9, 3.7931281008217955`*^9}, 3.793128146623452*^9, {3.8239300172440186`*^9, 3.823930049980094*^9}, { 3.9061218057990365`*^9, 3.906121806985854*^9}, {3.9061218389098153`*^9, 3.9061219167142*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"03fca7a8-9c3f-4d31-a0bd-89757f7b71fa"] }, Closed]], Cell[CellGroupData[{ Cell[" ", "Subsubsection", CellChangeTimes->{{3.8239300223102236`*^9, 3.8239300255565367`*^9}},ExpressionUUID->"c7927412-987f-45df-a465-\ 523c05b30aa3"], Cell[BoxData[ RowBox[{ RowBox[{"pl", "[", RowBox[{"expr_", ",", "a_", ",", "b_"}], "]"}], ":=", RowBox[{"Plot", "[", RowBox[{"expr", ",", RowBox[{"{", RowBox[{"x", ",", "a", ",", "b"}], "}"}]}], "]"}]}]], "Input", CellChangeTimes->{{3.823930282045566*^9, 3.8239303011473155`*^9}, 3.9061219259827733`*^9}, CellLabel->"In[8]:=",ExpressionUUID->"ead762ff-2e2e-4411-8b4a-5a039432abdb"], Cell[BoxData[ RowBox[{"pl", "[", RowBox[{ SuperscriptBox["E", RowBox[{"x", "-", "1"}]], ",", " ", RowBox[{"-", "3"}], ",", "1"}], "]"}]], "Input", CellChangeTimes->{{3.823930312876837*^9, 3.82393031303538*^9}, { 3.906121932637719*^9, 3.906121933512512*^9}}, CellLabel->"In[9]:=",ExpressionUUID->"3a675a98-7718-4753-9af5-c64351f0be3b"] }, Closed]], Cell[CellGroupData[{ Cell[" ", "Subsubsection", CellChangeTimes->{3.7931281630102386`*^9, 3.9061674044981747`*^9},ExpressionUUID->"910dc7f4-dc54-461a-8e55-\ c21d48436686"], Cell["\<\ Can we use the same symbol \ (\[OpenCurlyDoubleQuote]pl\[CloseCurlyDoubleQuote]) for the two functions? Yes, because they have a different number of arguments. \ \>", "Text", CellChangeTimes->{{3.793127624472603*^9, 3.7931276266329093`*^9}, 3.7931279940583453`*^9, {3.823797797417466*^9, 3.823797800775617*^9}, { 3.906121939932557*^9, 3.90612199552175*^9}}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"e30190e9-98d5-45c6-9ba9-80b31692cabc"], Cell["\<\ In fact, Mathematica has two definitions for the function \ \[OpenCurlyDoubleQuote]pl\[CloseCurlyDoubleQuote]: (see below for the command \[OpenCurlyDoubleQuote]?\[CloseCurlyDoubleQuote])\ \>", "Text", CellChangeTimes->{{3.9061674819485545`*^9, 3.906167507059391*^9}, { 3.906167562561507*^9, 3.9061675760439043`*^9}},ExpressionUUID->"5facc633-bd36-4adf-b957-\ 7c11b2ef8910"], Cell[BoxData[ RowBox[{"?", "pl"}]], "Input", CellChangeTimes->{{3.906167508449715*^9, 3.906167509090147*^9}}, CellLabel->"In[10]:=",ExpressionUUID->"0e51c72a-c159-427d-98b8-e2d95b7a35d4"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[StyleBox["6.3 Function evaluations", FontSize->24]], "Subsection", CellChangeTimes->{{3.7931280204658203`*^9, 3.793128024004758*^9}, { 3.7931295911821847`*^9, 3.793129593322367*^9}, {3.9055683773938007`*^9, 3.905568386718421*^9}, {3.90556841798425*^9, 3.905568428967516*^9}, { 3.906167361271287*^9, 3.90616736484344*^9}, {3.906167603742343*^9, 3.906167612999722*^9}, 3.906168027027893*^9}, FontColor->GrayLevel[0], Background->RGBColor[ 0., 1., 1.],ExpressionUUID->"9d432bfb-d497-4c6f-b73b-aa42d6c3e8cf"], Cell[TextData[{ "In evaluating a function, ", StyleBox["Mathematica", FontSlant->"Italic"], "\n\[FilledSmallCircle] substitutes the argument to the variable (here \ \[OpenCurlyDoubleQuote]x\[CloseCurlyDoubleQuote]) that appears in the \ expression at the left hand side of the definition ", StyleBox["f[x_]:=.... ", FontFamily->"Terminal", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], "\n\[FilledSmallCircle] and, possibly, makes some (obvious/simple) \ computations and simplifications" }], "Text", ShowGroupOpener->False, CellChangeTimes->{ 3.729056371346322*^9, {3.906122031190653*^9, 3.906122182962476*^9}}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"d0c2ca70-34f2-47bf-9c2a-4a2c652c6069"], Cell[TextData[{ StyleBox["Example:", FontWeight->"Bold"], " define the function\n ", StyleBox["f[u_] := ", FontFamily->"Courier New", FontWeight->"Bold"], Cell[BoxData[ SqrtBox[ SuperscriptBox["u", "2"]]], ShowGroupOpener->False, CellChangeTimes->{{3.5401791189040194`*^9, 3.5401791199960213`*^9}, { 3.5433074394620676`*^9, 3.543307439727268*^9}, {3.5904886338707423`*^9, 3.590488638925151*^9}},ExpressionUUID-> "047da164-bca2-4221-972c-a10f4171deb5"], " ( Sqrt[u^2] ) .\n ", StyleBox["\n", FontFamily->"Courier New", FontWeight->"Bold"], StyleBox["Which will be the ", FontFamily->"Courier New"], StyleBox["resulta of ", FontFamily->"Times New Roman", FontSize->16], StyleBox[" ", FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox["f[2] ", FontFamily->"Courier New", FontWeight->"Bold"], StyleBox["? And of ", FontFamily->"Times New Roman"], StyleBox["f[x] ", FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[" or ", FontFamily->"Times New Roman"], StyleBox["f[gianni] ?", FontFamily->"Courier New", FontWeight->"Bold"] }], "Text", CellChangeTimes->{{3.793128071767908*^9, 3.7931281008217955`*^9}, 3.793128146623452*^9, {3.793128675812248*^9, 3.7931286826861773`*^9}, { 3.8239304492989006`*^9, 3.8239304715947294`*^9}, {3.9061222024188004`*^9, 3.906122258152427*^9}, {3.90616763889184*^9, 3.9061676451023483`*^9}, 3.906182904546816*^9}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"2fc837ad-0e84-4546-96c6-32b1f8099a00"], Cell[CellGroupData[{ Cell["", "Subsubsection",ExpressionUUID->"8e907b29-4b94-450c-ab6c-c72e6d7523e5"], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "u_", "]"}], ":=", " ", RowBox[{"Sqrt", "[", " ", SuperscriptBox["u", "2"], " ", "]"}]}]], "Input", CellLabel->"In[11]:=",ExpressionUUID->"e99ad195-5aa1-4165-9c6a-c3b44bc85868"], Cell[BoxData[ RowBox[{"f", "[", "2", "]"}]], "Input", CellLabel->"In[12]:=",ExpressionUUID->"fac8a332-3676-4f61-bc69-bd03d0bc34fd"], Cell[BoxData[ RowBox[{"f", "[", "x", "]"}]], "Input", CellLabel->"In[13]:=",ExpressionUUID->"24e3da9c-4726-4a23-a17c-5ce068d31926"] }, Closed]], Cell[CellGroupData[{ Cell["", "Subsubsection",ExpressionUUID->"263860b6-a1a5-4b17-a996-046990ebfb20"], Cell["\<\ Why no simplification in the second case? Can we force it?\ \>", "Text", CellChangeTimes->{{3.793127624472603*^9, 3.7931276266329093`*^9}, 3.793128767562643*^9, {3.90612227325515*^9, 3.90612232734908*^9}, { 3.906124166595804*^9, 3.9061241791397448`*^9}, 3.906167671686249*^9}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"7d14578a-6d29-4aa6-a889-8c7f9359fd00"], Cell[BoxData[ RowBox[{"FullSimplify", "@", "%"}]], "Input", CellChangeTimes->{{3.823930599362142*^9, 3.8239306072097926`*^9}}, CellLabel->"In[14]:=",ExpressionUUID->"01d396d1-5219-4f1c-844f-9130a7f41379"] }, Closed]], Cell[CellGroupData[{ Cell["", "Subsubsection",ExpressionUUID->"36bcb70f-b571-4c96-be68-dc31b0090265"], Cell["\<\ No. Does it make sense?\ \>", "Text", CellChangeTimes->{{3.793127624472603*^9, 3.7931276266329093`*^9}, 3.7931287944917364`*^9, {3.8237978370635767`*^9, 3.8237978378507357`*^9}, 3.9061223429952116`*^9}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"293609f8-6ea4-417a-9edb-172adc4eb5fc"], Cell[TextData[{ "Yes:\n ", Cell[BoxData[ FormBox[ 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", FontSlant->"Italic"], "do when evaluating a function?\n\n ", StyleBox["Trace[command] ", FontWeight->"Bold"], " gives a list of the expressions used in evaluating \ \[OpenCurlyDoubleQuote]command\[CloseCurlyDoubleQuote]" }], "Text", CellChangeTimes->{{3.6975193365951185`*^9, 3.697519418168228*^9}, { 3.728845056299037*^9, 3.728845077085026*^9}, 3.728845492902314*^9, 3.729056371346322*^9, {3.7603779986623545`*^9, 3.760378001052109*^9}, { 3.7603780514848385`*^9, 3.7603780581230817`*^9}, 3.7603785622804356`*^9, { 3.823930615513896*^9, 3.823930616665665*^9}, {3.9061223463901997`*^9, 3.9061223904028645`*^9}, {3.906168052743195*^9, 3.9061680788546553`*^9}, { 3.90616815652546*^9, 3.906168211367102*^9}}, Background->RGBColor[ 1, 1, 0],ExpressionUUID->"da4abf5f-6a38-4bce-8027-b7c75cd3bb47"], Cell[BoxData[ RowBox[{"Trace", "[", " ", RowBox[{"f", "[", "2", " ", "]"}], "]"}]], "Input", CellChangeTimes->{{3.906168082411353*^9, 3.906168089268374*^9}, { 3.9061831290190425`*^9, 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