Things you must know about Mathematica

Here is a list of things you need to know about Mathematica to avoid frustration!

• Mathematica has very large inbuilt help pages. To access these, just click on the Help menu item at the top of the screen.

• Mathematica's inbuilt functions, (like Exp, Sin etc) are always capitalised. For this reason, it is always safest to use lower case names for your own variable and function names.

• All user defined variables and functions must begin with a letter, and must consists only of letters and numbers. So, e.g. underscores are not allowed!!

• You can define your own function (i.e. a user-defined function!) using the following syntax:

  f[x_]:=x^2
  

  Note the following:

  • you must have square brackets [ ] around the dummy argument of the function
  • you must have an underscore next to the dummy argument of the function
  • you must have a := to define the function. (The usual equals sign = won't work!)

  You can define more complicated function as follows:

  g[x_,y_]:= 6 Exp[x^3] Sin[y]
  

• To raise any function (in-built or user-defined) to a power, you use e.g. Sin[x]^2 and not Sin^2[x]

• Important numbers are pre-defined in Mathematica:

  Number	Mathematica symbol
  $\sqrt{-1}$	I
  $\pi$	Pi
  e	E
  $\infty$	Infinity

  Note that Mathematica can certainly handle complex arithmetic! e.g.(5 + 4I)/(2 - 7I) or Exp[I Pi] etc etc.

• To give a short description of a function or a variable (whether it be a Mathematica inbuilt one, or a user-defined one), just type ?function or ?variable. e.g. ?Sin gives you a short description of the sin function.

• To clear the definition of a user-defined function or variable, just type Clear[name]. e.g. if you have defined a=2, then Clear[a] will wipe this definition. It is always a good idea to Clear variables before you use them, in case you've already defined them earlier in your Mathematica session. If you want to Clear all variables, you can use:

  Clear["`*"]

  Note that the symbol just before * is the left-hand apostrophe.

• To avoid Mathematica printing to the screen after every time you hit Shift-Return, just add a semi-colon. i.e. a=2; will not echo 2 when you hit Shift-Return. You can use a semi-colon to put many Mathematica commands on the same line, e.g. if you type:

  3*8; a=2; 7a; Sin[Pi/2]; Exp[2]; Exp[5.]

  then Mathematica will only display 148.413 on the screen (once you hit Shift-Return) i.e. 148.413 = Exp[5.].

• You can add "comments" to your Mathematica content by typing in some text, and NOT typing control-enter ...!

• The multiplication sign is not always needed. e.g. the following are all equivalent:

  a b
  a*b

  i.e. Mathematica understands a space (as in the first example) as meaning multiplication as well as a * (as in the second example). Note that you must be careful because if you leave the space out, as in ab Mathematica understands this to be a new variable ab rather than a times b. Also note that when one of the factors is a number, or a bracketted expression, then no space is needed, e.g. 7a needs no space between the 7 and the a and 7a(b+c) requires no spaces around the a.

• There are four different types of brackets in Mathematica. It is crucial that you understand when each are used.

  Type	Example Usage	Description
  ( )	7a(b+c)	This has the same meaning as brackets in an ordinary
  		mathematical expression. i.e. it just groups together items
  [ ]	Sin[Pi/2]	This is always required for all functions
  { }	Integrate[f[x],{x,0,3} ]	This defines a list
  [[ ]]	vector[[2]]	This gives the access to a component of a list or array - see later.

• There are four different types of equal signs in Mathematica. It is crucial that you understand when each are used.

  Type	Example Usage	Description
  =	a=2	Assigns a value to the variable name
  :=	f[x_]:=x^2	Used only in the definition of user-defined functions
  ==	D[g[x,y],y] == D[h[x,y],x]	Used in equations and in relational expressions,
  		i.e. one whose answer is generally True or False.
  ->	Solve[3x==6,x]
  	{ x->2 }	This is how Mathematica displays the solution of an equation.

• The percentage key % can be used as a short-hand for the last command. e.g.

  a=2;
  %^3
  

  (with a Shift-Return at the end of both lines) will return 8.

• If you want Mathematica to stop performing a calculation (e.g. maybe you've made a mistake, and the calculation is taking too long to finish), then you press Alt-comma (i.e. the Alt and the , key at the same time!).