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1.31 Find all Mathematica commands beginning with Abs.
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1.32 Find all Mathematica commands beginning with Si and ending with al.
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1.33 Find all Mathematica commands beginning with Fi.
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1.34 Find all Mathematica commands beginning with Fi and ending with t.
SOLUTION
Fi*t
1.8 Packages
There are many specialized functions and procedures that are not loaded when Mathematica is initially invoked. Rather, they must be loaded separately from files in the Mathematica directory on the hard drive. These files are of the form filename.m.
EXAMPLE 26 A map of the world can be obtained from the command WorldPlot which is located in the package WorldPlot`. To load this command, simply type (note the ` at the end)
WorldPlot`
or Needs["WorldPlot`"]
The appropriate command can then be accessed. WorldPlot[World]
Once a package is loaded you can get a list of the functions it contains by using the Names command.
Getting Acquainted
EXAMPLE 27 (Continuation of Example 26)
Names["WorldPlot`*"] {Africa, Albers, Asia, ContiguousUSStates, Equirectangular, Europe, LambertAzimuthal, LambertCylindrical, Mercator, MiddleEast, Mollweide, NorthAmerica, Oceania, Orthographic, RandomColors, RandomGrays, ShowTooltips, Simple, Sinusoidal, SouthAmerica, ToMinutes, USData, USStates, World, WorldBackground, WorldBorders, WorldClipping, WorldCountries, WorldData, WorldDatabase, WorldFrame, WorldFrameParts, WorldGraphics, WorldGrid, WorldGridBehind, WorldGridStyle, WorldPlot, WorldPoints, WorldProjection, WorldRange, WorldRotatedRange, WorldRotation, WorldToGraphics}
EXAMPLE 28 The package Calendar` includes some interesting calendar functions.
Calendar` Names["Calendar`*"] {Calendar, CalendarChange, DateQ, DayOfWeek, DaysBetween, DaysPlus, EasterSunday, EasterSundayGreekOrthodox, Friday, Gregorian, Islamic, Jewish, JewishNewYear, Julian, Monday, Saturday, Sunday, Thursday, Tuesday, Wednesday} DaysBetween
DaysBetween[{year1, month1, day1}, {year2, month2, day2}] gives the number of days between the dates {year1, month1, day1} and {year2, month2, day2}. DaysBetween[{year1, month1, day1, hour1, minute1, second1}, {year2, month2, day2, hour2, minute2, second2}] gives the number of days between the given dates.
DaysBetween[{2007,8,3},{2008,12,5}] 490
SOLVED PROBLEMS
1.35 The function DayOfWeek appears in the package Calendar` and gives the day of the week of any date in the calendar. Load the package, obtain help to determine its syntax, and then determine which day of the week January 1, 2000, was.
SOLUTION
Calendar` DayOfWeek
DayOfWeek[{year, month, day}] gives the day of the week on which the given date {year, month, day} occurred. DayOfWeek[{year, month, day, hour, minute, second}] gives the day of theweek for the given date.
DayOfWeek[{2000, 1, 1}] Saturday
1.36 The package Combinatorica` contains functions in combinatorics and graph theory. One of these is KSubsets, which lists all subsets of size k of a given set. Load the package and execute Ksubsets[{1,2,3,4,5},3].
SOLUTION
Combinatorica` KSubsets[{1, 2, 3, 4, 5}, 3] {{1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {1, 4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}}
Getting Acquainted
1.9 A Preview of What Is to Come
If you have just purchased your copy of Mathematica, you probably cannot wait to give it a test run. The following examples are a collection of problems for you to try. What follows are some basic commands. To keep things simple, options have been omitted and Mathematica s defaults are used exclusively. We will discuss modifications to these commands in subsequent chapters, but for now, just have fun!
EXAMPLE 29 Obtain a 50 significant digit approximation to .
N[ , 50]
N[Sqrt[Pi], 50]
1.7724538509055160272981674833411451827975494561224
EXAMPLE 30 Solve the algebraic equation x3 2 x + 1 = 0.
Solve[x3 2 x + 1 0] or Solve[x^3 2 x + 1 0]
{x 1}, x 1 -1 - 5 , x 1 1 + 5 2 2
EXAMPLE 31 Express (x + 1)10 in traditional polynomial form.
10 Expand[(x + 1) ] //TraditionalForm
x 10 + 10 x 9 + 45 x 8 + 120 x 7 + 210 x 6 + 252 x 5 + 210 x 4 + 120 x 3 + 45x 2 + 10 x + 1
EXAMPLE 32 What is the 1000th prime
Prime[1000] 7919
EXAMPLE 33 The function ElementData gives values of chemical and physical properties of elements. Among
the properties included are AtomicWeight and AtomicNumber, whose definitions are self-explanatory. Compute the atomic weight and atomic number of titanium. (Note the quotation marks.) ElementData["Titanium","AtomicWeight"] 47.867 ElementData["Titanium","AtomicNumber"] 22
EXAMPLE 34 Plot the graph of y = sin x from 0 to 2 .
Plot[Sin[x],{x, 0, 2 o}]
1 0.5
1.0
Getting Acquainted
EXAMPLE 35 Sketch the graphs of y = sin x, y = sin 2x, and y = sin 3x, 0 x 2 , on one set of axes.
Plot[{Sin[x],Sin[2 x],Sin[3 x]},{x,0,2 o}]
0.5
1.0
EXAMPLE 36 Sketch the three-dimensional surface defined by z = ( x 2 + 3 y 2 )e ( x
2 + y2 )
Plot3D[(x2 + 3 y2) (x2 + y2),{x, 3,3},{y, 3,3}] or Plot3D[(x^2 + 3 y^2)* Exp[ (x^2 + y^2)],{x, 3,3},{y, 3,3}]
1.0 0.5 0.0 2 0 2 2 0 2
Click on the graph and drag the mouse to view the graph from any viewpoint.
C HA PTE R 2
Basic Concepts
2.1 Constants
Mathematica uses predefined symbols to represent built-in mathematical constants.
Pi or o is the ratio of the circumference of a circle to its diameter. E or is the base of the natural logarithm.
Both Pi and E are treated symbolically and do not have values, as such. However, they may be approximated to any degree of precision.
EXAMPLE 1 N[o, 500] will produce a 500 significant digit approximation to (499 decimal places).
N[o,500] 3.1415926535897932384626433832795028841971693993751058209749445923078164062 862089986280348253421170679821480865132823066470938446095505822317253594081 284811174502841027019385211055596446229489549303819644288109756659334461284 756482337867831652712019091456485669234603486104543266482133936072602491412 737245870066063155881748815209209628292540917153643678925903600113305305488 204665213841469519415116094330572703657595919530921861173819326117931051185 480744623799627495673518857527248912279381830119491
Degree is equal to Pi/180 and is used to convert degrees to radians. GoldenRatio has the value (1 + 5 ) / 2 and has a special significance with respect to Fibonacci series. It is used in Mathematica as the default width-to-height ratio of two-dimensional plots. Infinity or is a constant with special properties. For example, + 1 = . EulerGamma is Euler s constant and is approximately 0.577216. It has applications in integration and in asymptotic expansions. Catalan is Catalan s constant and is approximately 0.915966. It is used in the theory of combinatorial functions.
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