(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.4' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 234235, 4304] NotebookOptionsPosition[ 231131, 4201] NotebookOutlinePosition[ 231490, 4217] CellTagsIndexPosition[ 231447, 4214] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Functions", "Section", FontSize->36], Cell[TextData[{ StyleBox["Mathematica", FontSize->18, FontSlant->"Italic"], StyleBox[" has a large amount of functions already built in. The arguments \ of a function are put in between square brackets and separated by commas. \ Some basic functions are given below. Note that all built in functions and \ variables start with capital letters. To avoid potential conflicts it is a \ good idea to start your own functions and variables with lower case letters.", FontSize->18] }], "Text", CellChangeTimes->{{3.6215799409954243`*^9, 3.6215799498208237`*^9}}, FontSize->36], Cell["\<\ Sqrt[x] square root (Sqrt[x]) Exp[x] exponential (e^x) Log[x] natural logarithm (Subscript[log, e] x) Log[b, x] logarithm to base b (Subscript[log, b] x) Sin[x], Cos[x], Tan[x] trigonometric functions ArcSin[x], ArcTan[x] inverse trigonometric functions \ Abs[x] absolute value Round[x] closest integer to x \ \>", "Text", CellChangeTimes->{{3.6215797813073263`*^9, 3.6215798164369173`*^9}, { 3.621579867752907*^9, 3.621579920114786*^9}, {3.621580034963277*^9, 3.621580197674762*^9}, {3.682985639772854*^9, 3.682985771510921*^9}, { 3.682998864867455*^9, 3.6829988992268867`*^9}}, FontSize->18], Cell["\<\ Most built-in Mathematica functions act on lists element - by - element. That \ is, Mathematica typically maps functions across lists. See for example the \ trigonometric functions. \ \>", "ItemNumbered", CellChangeTimes->{{3.6944095019011307`*^9, 3.694409513044046*^9}, { 3.694409726222712*^9, 3.694409727373595*^9}, {3.6944133433209248`*^9, 3.694413348925702*^9}}, FontSize->16], Cell[BoxData[{ RowBox[{"Sin", "[", "Pi", "]"}], "\[IndentingNewLine]", RowBox[{"L", "=", RowBox[{"{", RowBox[{"Pi", ",", " ", RowBox[{ RowBox[{"1", "/", "2"}], " ", "Pi"}], ",", " ", RowBox[{ RowBox[{"1", "/", "3"}], "Pi"}], ",", " ", RowBox[{ RowBox[{"1", "/", "4"}], " ", "Pi"}]}], "}"}]}], "\[IndentingNewLine]", RowBox[{"Sin", "[", "L", "]"}]}], "Input", CellChangeTimes->{{3.694409543560973*^9, 3.694409690213665*^9}}, FontSize->16, Background->RGBColor[0.87, 0.94, 1]], Cell["\<\ We can compose any number of built-in Mathematica functions, as in the \ following example:\ \>", "ItemNumbered", CellChangeTimes->{{3.694408668045272*^9, 3.69440869775637*^9}, { 3.694409734296101*^9, 3.69440973722965*^9}}, FontSize->16], Cell[BoxData[ RowBox[{"Round", "[", RowBox[{"Sqrt", "[", RowBox[{"Abs", "[", RowBox[{"-", "5"}], "]"}], "]"}], "]"}]], "Input", CellChangeTimes->{{3.650078332584429*^9, 3.650078358957191*^9}, { 3.650079947288245*^9, 3.650079974511591*^9}, {3.682985789240891*^9, 3.682985810968217*^9}, {3.682985848090437*^9, 3.68298585556103*^9}, { 3.694408635152372*^9, 3.694408637005862*^9}}, FontSize->16, Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[{ "The symbol ", StyleBox["@", FontWeight->"Bold"], " ", StyleBox["(slash)", FontWeight->"Bold"], " maps a function (of one argument) to its argument. In place of writing \ f[x], one could simply write f @ x. With @ you do not have to move to the end \ of the expression to match the square bracket. You can also use the ", StyleBox["//", FontWeight->"Bold"], " postfix notation to accomplish the same thing." }], "ItemNumbered", CellChangeTimes->{{3.694408647300433*^9, 3.694408654118379*^9}}, FontSize->16], Cell[BoxData[ RowBox[{"Round", "@", RowBox[{"Sqrt", "@", RowBox[{"Abs", "@", RowBox[{"-", "5"}]}]}]}]], "Input", CellChangeTimes->{{3.650078332584429*^9, 3.650078358957191*^9}, { 3.650079947288245*^9, 3.650079974511591*^9}, {3.682985789240891*^9, 3.682985810968217*^9}, {3.682985848090437*^9, 3.68298585556103*^9}, { 3.694408635152372*^9, 3.694408637005862*^9}, {3.69440874850138*^9, 3.694408761612268*^9}}, FontSize->16, Background->RGBColor[0.87, 0.94, 1]], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "5"}], " ", "//", " ", "Abs"}], " ", "//", " ", "Sqrt"}], " ", "//", " ", "Round"}]], "Input", CellChangeTimes->{{3.694408876700135*^9, 3.6944089153416357`*^9}}, FontSize->16, Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[{ "The symbol ", StyleBox["@@", FontWeight->"Bold"], " ", StyleBox["(slash - slash)", FontWeight->"Bold"], " applies a function (of several arguments) to its arguments. For instance, \ Log @@ {2, 32} is the same as Log[2,32] and it computes the logarithm of 32 \ in base 2." }], "ItemNumbered", CellChangeTimes->{{3.6944089958952093`*^9, 3.694409008567635*^9}, { 3.694409100852066*^9, 3.694409160097211*^9}}, FontSize->16], Cell[BoxData[ RowBox[{"Log", " ", "@@", " ", RowBox[{"{", RowBox[{"2", ",", " ", "32"}], "}"}]}]], "Input", CellChangeTimes->{{3.69440902203893*^9, 3.694409040173744*^9}}, FontSize->14, Background->RGBColor[0.87, 0.94, 1]], Cell["\<\ Note that this is different from Log @ {2, 32}, which is equivalent to \ Log[{2, 32}] and returns a list of two elements, representing the natural \ logarithms (base e) of 2 and 32\ \>", "Text", CellChangeTimes->{{3.694409763172583*^9, 3.6944098968973427`*^9}}, FontSize->16] }, Open ]], Cell[CellGroupData[{ Cell["User defined functions", "Section", CellChangeTimes->{{3.650079899810685*^9, 3.6500799167444687`*^9}}, FontSize->36], Cell[CellGroupData[{ Cell["We can define our own functions as follows:", "Subsubsection", CellChangeTimes->{{3.621584593473613*^9, 3.62158460482414*^9}, { 3.650080001560597*^9, 3.6500800035757236`*^9}}, FontSize->24], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "x_", "]"}], ":=", RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"Sin", "[", "x", "]"}]}]}]], "Input", FontSize->24, Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[{ "In this expression the underscore represents a pattern. A single underscore \ will match a single expression (argument) and ", StyleBox["x", "Input", FontSize->24, FontSlant->"Italic"], StyleBox[" ", "Input"], "is the name of the pattern (it can be used to refer to the matched \ expression on the right hand side). The colon tells ", StyleBox["Mathematica", FontSlant->"Italic"], " not to evaluate the right hand side. It will only be evaluated when you \ invoke the function. This is why you get no output. Now you can use your \ function:" }], "Text", TextJustification->1., FontSize->18], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", "angle", "]"}], "+", RowBox[{"f", "[", RowBox[{"2", "Pi"}], "]"}]}]], "Input", CellChangeTimes->{{3.6215805170645037`*^9, 3.621580583696731*^9}, { 3.694406848453693*^9, 3.694406852692499*^9}}, FontSize->18, Background->RGBColor[0.87, 0.94, 1]], Cell["\<\ Try to assign some value to x, then compute f[2Pi]. What do you notice? Does \ f use the global or the local values of x when computing f[2Pi]?\ \>", "Text", CellChangeTimes->{{3.649646638131113*^9, 3.649646697210732*^9}, { 3.6496468180871058`*^9, 3.649646871198804*^9}, {3.6501079434264193`*^9, 3.650107979546966*^9}}, FontSize->18], Cell[BoxData[{ RowBox[{ RowBox[{"x", "=", "5"}], ";"}], "\[IndentingNewLine]", RowBox[{"f", "[", RowBox[{"2", "Pi"}], "]"}]}], "Input", CellChangeTimes->{{3.649646701167595*^9, 3.649646773958033*^9}, { 3.6501079859246073`*^9, 3.650108012036789*^9}}, FontSize->18, Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[{ StyleBox["Warning:", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Do not omit the colon when defining functions! 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You can look at the definition of a function by \ using a question mark:\ \>", "Subsubsection", FontSize->24], Cell[BoxData[{ RowBox[{"?", "f"}], "\[IndentingNewLine]", RowBox[{"?", "g"}]}], "Input", CellChangeTimes->{{3.694412893212604*^9, 3.694412900721884*^9}}, FontSize->18, Background->RGBColor[0.87, 0.94, 1]], Cell["To clear the function or variable definitions, use Clear[..]", "Text", CellChangeTimes->{{3.650078445974368*^9, 3.650078490880578*^9}}, FontSize->18], Cell[BoxData[ RowBox[{"Clear", "[", RowBox[{"x", ",", "f", ",", "g"}], "]"}]], "Input", CellChangeTimes->{{3.682986666648357*^9, 3.6829866681349688`*^9}, { 3.694406905318964*^9, 3.694406905877679*^9}}, FontSize->18, Background->RGBColor[0.87, 0.94, 1]], Cell["Now we can do a new definition for the function f.", "Text", CellChangeTimes->{{3.650078445974368*^9, 3.650078490880578*^9}, { 3.694406925470685*^9, 3.69440694875178*^9}}, FontSize->18], Cell[BoxData[ RowBox[{ RowBox[{"f", "[", RowBox[{"i_", ",", "j_"}], "]"}], ":=", RowBox[{"i", "+", "j"}]}]], "Input", CellChangeTimes->{{3.618757270156877*^9, 3.618757325612507*^9}}, FontSize->18, Background->RGBColor[0.87, 0.94, 1]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ The function f gives the sum of two variables. 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