Getting Started with MATHEMATICA

This short primer consists primarily of examples intended to illustrate basic features of the mathematical software package MATHEMATICA. For optimal use, it is suggested that you initiate a MATHEMATICA session and attempt to carry out the examples that are discussed.

1. Starting and Stopping.
   • To start: Sit down at a PC or MAC in any of the A&S Computer Labs and choose MATHEMATICA from the Windows Start menu or icon grouping.
   • To stop: select   Exit   from the File menu in the upper left corner of the MATHEMATICA window.
   • Getting help: There is a very extensive and well-organized Help facility built into MATHEMATICA. This may be accessed by choosing Help Browser... from the Help menu in the upper right corner of the MATHEMATICA window. An even better way to get help is to consult an experienced MATHEMATICA user.

2. Compiling and Editing.
   • Once you have typed in a command, Shift-Enter will send the command to MATHEMATICA. For example, try typing Expand[(x+y)^2] and Shift-Enter. Then try Expand[(x+y)^50].
     (Note: Shift-Enter means to simultaneously hold down the Shift and Enter keys.)
   • While MATHEMATICA is compiling your input, you will see "evaluating" written at the bottom of the screen. If "evaluating" remains there for a long time you have probably given MATHEMATICA a problem that involves too many calculations, often you can interrupt the calculation by choosing "Abort Evaluation" from the "Kernel" menu.
     
   • Return (or Enter) starts a new line on your screen but DOES NOT invoke MATHEMATICA. Use this to keep your lines from scrolling too far to the right, epecially if you are planning to make a print-out of your session.
   • For WINDOWS users: If previous text is blocked off using the mouse it can be copied using Control-c and then Control-v , after repositioning the cursor where you want to insert the blocked text.
   • To prepare a MATHEMATICA session for printing, you will likely want to delete any cells that were incorrect or irrelevant to the problem at hand. You will notice that each cell has an associated bracket shown on the right edge of the screen. If you single click the mouse on one brackets and then type Control-x this will delete the corresponding cells or groups of cells. If you double click on a bracket it will keep the cell but hide it (or if the cell is currently hidden this will redisplay it).
   • You can create titles or subtitles for your document by choosing the appropriate command from the Format menu. In this menu you'll find options for choosing fonts, font sizes, background colors and etc.

3. Basic Commands.
   • Commands and functions in MATHEMATICA always start with a capital letter. And arguments are always enclosed in square brackets " [ ]", not round parentheses. For example, try compiling the command Log[1] or ArcSin[1]. Then, for comparison, compile the commands Log(1) and ArcSin(1).
   • Parentheses (round brackets " ( )") should be used to make sure that operations are computed in the desired order. For example, compare the outcome of 1/2 * 2 with 1/(2 * 2).
   • The percent sign % can be used as a shorthand to denote the last result generated, and %% is the second-to-last result. On the other hand, %7 would represent the output result in the output line number 7, and so on.
   • A semi-colon " ; " at the end of a line will suppress the MATHEMATICA output. (So MATHEMATICA does the computation but doesn't print it to the screen.)

4. Elementary Functions.
   • Either x * y or x   y (with a blank space inserted between x and y) denotes multiplication of x by y. If you write xy without a space between the x and y then MATHEMATICA will attempt to interpret this as the name of a single variable (and it will NOT interpret it as multiplication). (However, if, in the above, x is replaced by an actual number then MATHEMATICA will interpret the juxtaposition as multiplication. For example, 5y means the same as 5 * y , but x5 will again be interpreted as a variable name.)
   • For exponents, write x^7 for x⁷, and so on.
   • The common constants and e are denoted by Pi and E respectively in MATHEMATICA.
   • Basic elementary functions are invoked by Sin[x], Cos[x], Sinh[x], Sec[x], Tan[x], ArcSin[x], ArcTan[x], Sqrt[x], Log[x], Exp[x], and etc. The interpretations of these functions should be obvious. Note that Log[x] is used for the natural logarithm function ln(x).)
   • Write N[ Sqrt[2] ] to get an approximation to , or N[ Sqrt[2] , 30 ] to get an approximation accurate to 30 decimal places.

5. Sums and Integers.
   •   Determine a finite sum: Sum[1/n,{n,1,10}]
   •   Sum an infinite series: Sum[3^(-n),{n,1,Infinity}]
   •   Find the divisors of 12 with Divisors[12].
     (NOTE: The output of this command is a list. The length of a list can be found with the Length command, so Length[Divisors[12]] will return the number of divisors of 12.)
   •   Print the 1000th prime with Prime[1000]
   •   Print out the first 100 primes with Prime[Range[100]]   (What does Range[100] give?)
   •   PrimePi[2500]   tells how many primes there are which are less than or equal to 2500.
   •   What does Prime[PrimePi[2500]]   return?
   •   Determine if 1301 or 1811921 are primes with PrimeQ[1301] and PrimeQ[1811921]
   •   Find that the prime factorization of 5863050 is 2¹3³5²43¹101¹ using FactorInteger[5863050]
   •   GCD[1677,1290,3225]   gives the greatest common divisor of 1677, 1290 and 3225, while   LCM[1677,1290,3225]   gives their least common multiple.

6. Algebra and Calculus.

   Try having MATHEMATICA compile each of the following to see what happens:

   •   Expand[ (x+y+z)^2 ]
   •   Factor[ x^3 + 2x^2 - 5x - 6 ]
   •   Simplify[ Sin[x]^2 + Cos[x]^2 ]
   •   Integrate[ x^5 * Sin[x] , x ]
   •   NIntegrate[ E^(x^2), {x, 0, 3} ]   (This numerically approximates the definite integral .)
   •   D[ Log[x] * Sin[x], x ]   (This computes the derivative with respect to x.)
   •   D[ Log[x] * Sin[x], {x,3} ]   (This computes the third derivative with respect to x.)
   •   Solve[ x^2 + x - 1 == 0, x ] (Note: == is a doubled equal sign, its always used for Logical Equality.)
   •   Limit[ Sin[x]/x, x -> 0 ]   (So MATHEMATICA knows L'Hospital's Rule!)

7. Fundamentals of Graphing.

   Try having MATHEMATICA compile each of the following to see what happens:

   •   Plot[ Cos[x^2] , {x,-Pi,Pi} ]
     This gives the graph of y=cos(x²) on the interval . You can get MATHEMATICA to enlarge the picture by first clicking on the graphic output and then dragging one of the little black rectangles while holding down the left mouse button.
   •   Plot[ Cos[ x^2 ] , {x, -Pi, Pi}, PlotRange -> { 0,1 } ]
   •   Plot[ Cos[ x^2 ] , {x, -Pi, Pi}, AxesLabel -> {"x-axis", "y-axis"},
         Frame -> True, PlotLabel -> {"My graph"} ]
   •   Plot3D[ x^2 - y^2 , {x, -3, 3}, {y,-3,3}]

   More details on controlling the output of the "Plot" command can be found on the 2-dimensional plotting page. Graphing functions of two variables is discussed on the "Plot3D" command page.