Zajecia 1. Krotkie wprowadzenie do srodowiska Mathematica 4^ D D 1. ?

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1 Zajecia Krotkie wprowadzenie do srodowiska Mathematica In[]:= Out[]= In[]:= Out[]= In[]:= Out[]= In[7]:= Out[7]= In[8]:= Out[8]= In[9]:= Out[9]= + ^ Sqrt@D Log@0, 00D Sin@Pi D In[0]:=! Out[0]= In[]:= Out[]= In[]:= Out[]= In[]:= Out[]= In[]:= Factorial@D Round@.D Mod@0, D? Mod Mod@m, nd gives the remainder on division of m by n. Mod@m, n, dd uses an offset d. In[]:= E Out[]= ã

2 lab_.nb In[]:= Out[]= In[7]:= Out[7]= In[8]:= Out[8]= In[98]:= ä 9 Out[98]= In[99]:= Out[99]= In[]:= N@ D 0. % Out[]= 0. In[00]:= %% Out[00]= Przypisywanie wartosci In[0]:= Out[0]= x = In[]:= x = y = 7; In[0]:= Clear@xD := Assign the value to x, but don' t do it right away; wait until x is actually used In[0]:= x := In[07]:= x == y Out[07]= False In[08]:= Solve@x ^ - x + 0, xd Solve::ivar : is not a valid variable. Out[08]= Solve@False, D In[09]:= Clear@xD In[0]:= Solve@x ^ - x + 0, xd Out[0]= In[]:= Out[]= 88x <, 8x << Clear@xD; Solve@x ^ - 7 x - 0, xd ::x I7 - M>, :x I7 + M>>

3 lab_.nb Transformacje wyrazen algebraicznych In[]:= Out[]= In[]:= Out[]= In[]:= x+x x Hx + L ^ H + xl Expand@H + xl ^ D Out[]= + x + x In[]:= Factor@%D Out[]= H + xl Definiowanie funkcji funkcja@x_d := x ^ + In[7]:= Out[7]= funkcja@d Define a new function f.note the underscore on the x on the left side of the statement! It must be there; it tells Mathematica to treat x as a pattern; thereafter, when you type something like f[a + b], Mathematica will then immediately consider that to be the same as (a + b)^ - (a + b) In[]:= funkcja@x_, y_d := x ^ + y ^ In[]:= funkcja@, D Out[]= In[]:=?# ð represents the first argument supplied to a pure function. ð n represents the nth argument. In[]:=?& Function@bodyD or body & is a pure function. The formal parameters are ð Hor ðl, ð, etc. Function@x, bodyd is a pure function with a single formal parameter x. Function@8x, x, <, bodyd is a pure function with a list of formal parameters. In[]:= f := ð ^ + & In[7]:= f@d Out[7]= In[]:= g := ð ^ + ð ^ + 0 &

4 lab_.nb In[7]:= Out[7]= D 9 Listy In[]:= In[7]:= lista = 8,,, 8<; lista@@dd Out[7]= In[]:=? Table Table@expr, 8imax <D generates a list of imax copies of expr. Table@expr, 8i, imax <D generates a list of the values of expr when i runs from to imax. Table@expr, 8i, imin, imax <D starts with i = imin. Table@expr, 8i, imin, imax, di<d uses steps di. Table@expr, 8i, 8i, i, <<D uses the successive values i, i,. In[]:= Out[]= In[7]:= Out[7]= In[8]:= Out[8]= In[9]:= In[0]:= Table@expr, 8i, imin, imax <, 8 j, jmin, jmax <, D gives a nested list. The list associated with i is outermost. Table@i ^, 8i, 0<D 8,, 9,,,, 9,, 8, 00< Table@i ^, 8i,, 0, <D 8,,,, 00< Table@Prime@iD, 8i, 0<D 8,,, 7,,, 7, 9,, 9,, 7,,, 7,, 9,, 7, 7, 7, 79, 8, 89, 97, 0, 0, 07, 09,, 7,, 7, 9, 9,, 7,, 7, 7, 79, 8, 9, 9, 97, 99,,, 7, 9< w := Table@i j, 8i, <, 8j, <D w MatrixForm Out[0]//MatrixForm= In[]:= Out[]= In[]:= Table@If@EvenQ@iD ÈÈ EvenQ@jD,, 0D, 8i, <, 8j, <D 880,, 0<, 8,, <, 80,, 0<< % MatrixForm Out[]//MatrixForm=

5 lab_.nb Grafika In[]:= 8x, 0, Pi<D.0 0. Out[]= In[]:= H + a xld, 8x, 0, <D, 8a, 0, <D a.0 0. Out[]= In[0]:= ListPlot@Table@Prime@iD, 8i, 0<DD 00 0 Out[0]=

6 lab_.nb In[70]:= Out[70]= In[7]:= + Sin@xD, 8x, 0, 0<DD 0 8 Out[7]= 8 0 Litery greckiego alfabetu Esc+litera+Esc, np. Θ, Α. Indeks dolny Ctrl + _ In[7]:=? Rectangle Rectangle@8xmin, ymin <, 8xmax, ymax <D is a two-dimensional graphics primitive that represents a filled rectangle, oriented parallel to the axes. In[7]:= Rectangle@8xmin, ymin <D corresponds to a unit square with its bottom-left corner at 8xmin, ymin <.? Graphics Graphics@ primitives, optionsd represents a two-dimensional graphical image.

7 lab_.nb In[7]:= Green, - <, 8, <D, Red, Blue, 0<D, Yellow, 0<, 8, <, 8, - <<D<D Out[7]= In[7]:=? GraphicsGrid GraphicsGrid@88g, g, <, <D generates a graphic in which the gij are laid out in a two-dimensional grid. In[7]:= GraphicsGrid@88Graphics@Rectangle@DD, Graphics@Disk@DD<, 8Graphics@Disk@DD, Graphics@Rectangle@DD<<D Out[7]= Obroty na okrêgu Zad. In[77]:= In[78]:= obrot@θ_, x_d := Mod@x + Θ, * PiD obrot@pi, Π D Π Out[78]= In[79]:= obrot@θ_, x_d := Mod@N@x + ΘD, * PiD 7

8 8 lab_.nb In[80]:= Out[80]= 0D 0.99 Zad. In[8]:= xd, 8x, 0, Pi<, AxesLabel -> 8"x", "fλ HxL"<, PlotStyle Red, AxesStyle LabelStyle BlackD, PlotLabel "Wykres funkcji obrotu"d fλ HxL Wykres funkcji obrotu Out[8]= 0 x Manipulate@ Plot@obrot@Θ, xd, 8x, 0, Pi<, AxesLabel -> 8"x", "fθ HxL"<, PlotStyle Red, AxesStyle Arrowheads@0.0D, LabelStyle Directive@, BlackD, PlotLabel "Wykres funkcji obrotu"d, 88Θ, Exp@D, "K¹t obrotu" <, 0, Pi<D K¹t obrotu. fλ HxL Wykres funkcji obrotu Out[8]= 0 x

9 lab_.nb Zad. In[8]:= Out[8]= In[8]:= Out[8]= In[8]:= In[88]:= x, D h@h@h@h@xdddd NestList@f, x, D :x, + x, + I + x M, + J + I + x M N, + K + J + I + x M N O > orbita@x0_, Θ_, n_d := NestList@obrot@Θ, ðd &, x0, n - D orbita@x0_, Θ_, n_d := Table@Mod@x0 + i * Θ, PiD, 8i, 0, n -, <D If you want to group several commands and output the last use the semicolon (;) between them, Just don' t use a for the last statement. In[89]:= In[90]:= Out[90]= In[9]:= fun@y_d := Hx = y + ; x ^ L fun@d 9 ManipulateAzp = orbita@x0, Θ, nd; ListPlotAzp, PlotStyle 8Red, PointSize@MediumD<, PlotRange 80, Pi<, AxesLabel 9"n", "fλ HnL Hx0 L"=, LabelStyle Directive@, BlackD, PlotLabel "wykres orbity"e, 88Θ, Exp@D, "K¹t obrotu"<, 0, Pi<, 88x0,, "x0 "<,<, 88n, 0, "Max. d³ugoœæ orbity"<, 80, 80, 0, 0<<E K¹t obrotu.09 x0 Max. d³ugoœæ orbity 0 80 fλ Hx0 L HnL Out[9]= 0 0 wykres orbity n 9

10 0 lab_.nb In[9]:= ManipulateAzp = orbita@x0, Θ, nd; ListPlotATable@8i -, zp@@idd<, 8i,, Length@zpD<D, PlotStyle 8Red, PointSize@MediumD<, PlotRange 80, Pi<, AxesLabel 9"n", "fλ HnL Hx0 L"=, LabelStyle Directive@, BlackD, PlotLabel "wykres orbity"e, 88Θ, Exp@D, "K¹t obrotu"<, 0, Pi<, 88x0,, "x0 "<,<, 88n, 0, "Max. d³ugoœæ orbity"<, 80, 80, 0, 0<<E K¹t obrotu.09 x0 Max. d³ugoœæ orbity Out[9]= 0 80 fλ HnL Hx0 L 0 0 wykres orbity n

11 lab_.nb In[7]:= = Θ, kdd; Circle@D, Black, Cos@zp@@iDDD<, 0.0D, 8i,, k<d<, PlotLabel "wizualzacja orbity", LabelStyle BlackDD, 88Θ, "K¹t obrotu"<, 0, Pi<, 88x0,, "x0 "<, 0, Pi<, 88n, 0, "Max. d³ugoœæ orbity"<, 80, 80, 0, 0, 00<<, 88k, n, "iloœæ kroków"<,, n, <D K¹t obrotu Π x0 Max. d³ugoœæ orbity iloœæ kroków wizualzacja orbity Out[7]=

12 lab_.nb In[9]:= D, 00DD 0 0 Out[9]= In[9]:= Histogram@orbita@Exp@D, Pi, 000D, 8 0<, "PDF", ChartStyle Green, PlotLabel "Histogram orbity", LabelStyle Directive@, BlackD, PlotRange 880, Pi<, All<D Histogram orbity Out[9]= 0. 0.

13 lab_.nb In[9]:= ManipulateAzp = orbita@x0, Θ, Max@n, kdd; GraphicsGridA9 9 Graphics@8Red, Circle@D, Black, Table@Disk@8Sin@zp@@iDDD, Cos@zp@@iDDD<, 0.0D, 8i,, k<d<, PlotLabel "wizualzacja orbity", LabelStyle Directive@, BlackDD, ListPlotATable@8i -, zp@@idd<, 8i,, k<d, PlotStyle 8Red, PointSize@MediumD<, PlotRange 880, n<, 80, Pi<<, AxesLabel 9"n", "fλ HnL Hx0 L"=, LabelStyle Directive@, BlackD, PlotLabel "wykres orbity"e =, 8 Plot@obrot@Θ, xd, 8x, 0, Pi<, AxesLabel 8"x", "fλ HxL"<, PlotStyle Red, AxesStyle Arrowheads@0.0D, LabelStyle Directive@, BlackD, PlotLabel "Wykres funkcji obrotu", ImageSize 0, Ticks 880, Pi, Pi, Pi, Pi<, 80, Pi, Pi, Pi <<, AxesOrigin 80, 0<D, = < Histogram@zp, 8 0<, "PDF", ChartStyle Green, PlotLabel "Histogram orbity", LabelStyle Directive@, BlackD, PlotRange 880, Pi<, All<D, ImageSize 0E, 88Θ, Exp@D, "K¹t obrotu"<, 0, Pi<, 88x0,, "x0 "<, 0, Pi<, 88n, 0, "Max. d³ugoœæ orbity"<, 80, 80, 0, 0, 0<<, 88k, n, "iloœæ kroków"<,, n, <E

14 lab_.nb K¹t obrotu x0 Max. d³ugoœæ orbity iloœæ kroków fλ Hx0 L HnL wizualzacja orbity 0 0 Out[9]= fλ HxL wykres orbity 0 Wykres funkcji obrotu Histogram orbity Π 0. Π 0. Π 0. Π Π Π x