How to | Create and Use Rules
Transformation rules in Mathematica let you set local values for symbols, functions, and all other types of expressions. Using rules provides a powerful and extensible method to replace all or part of another expression with the value you specify.
The short form for a rule uses a right arrow, which you get by typing 
(with no space between - and >). The Mathematica front end automatically converts 
into 
upon further typing. Either symbol is a short form for Rule.
Create the following transformation rule, which can be thought of as "
goes to 
":
| Out[1]= |  |
By looking at the output for 
, you can see that this rule does not do anything: the output is simply the rule itself. This is because rules do not do anything when they are alone. You must use a rule with an expression for it to be of any use.
Rules can be applied to expressions by using 
(the short form for ReplaceAll). The general syntax for this is 
.
Use 
to use a rule with an expression:
| Out[57]= |  |
To use two or more rules with an expression, place them in a list 
:
| Out[58]= |  |
If you give two rules for the same variable, Mathematica will use only the first rule:
| Out[59]= |  |
You can replace variables with any expression, not just individual values.
Substitute 
for 
:
| Out[60]= |  |
You can also use a rule to replace larger parts of an expression:
| Out[62]= |  |
In fact, you can use rules with any expression, including functions.
Substitute 
for 
:
| Out[65]= |  |
Use a rule for 
. Note that this rule matches 
exactly and does not affect 
:
| Out[64]= |  |
To replace the function 
regardless of its argument, you must use a pattern in the rule.
The rule 
can be read as "
goes to 
":
| Out[18]= |  |
For more information on using patterns, see "Introduction to Patterns."
Rules that are set up using 
are immediate rules. That is, the right-hand side is evaluated at the same time as the rule:
| Out[1]= |  |
You may need to use delayed rules instead, which are not evaluated until they are used with an expression. Delayed rules are created by using RuleDelayed.
The short form for a delayed rule is 
(with no space between : and >). The Mathematica front end automatically converts 
into 
upon typing. Either represents the short form for RuleDelayed:
| Out[2]= |  |
Consider a problem where you want to use a rule to generate three random real numbers in the range of 0 to 1. Using an immediate rule results in the generation of the same three numbers:
| Out[3]= |  |
To generate three different numbers, use a delayed rule:
| Out[4]= |  |
Assignments set explicitly using 
have a global effect, while rules only affect the expression with which they are used.
Use 
to assign 
to 
, and then evaluate 
to see the value:
| Out[2]= |  |
Use a rule to assign a value for 
:
Evaluating 
, you can see that the value assigned by the rule was not saved:
| Out[4]= |  |
You must use a rule with an expression for it to work. However, you can explicitly assign a rule to a symbol and then use that symbol as you would the rule.
Use 
to assign the rule 
to 
, and then use 
with an expression:
| Out[69]= |  |
Since 
is now stored globally as the symbol 
, you can continue to use 
in place of 
.
Similarly, you can explicitly assign an expression to a symbol and then use a rule on the symbol:
| Out[6]= |  |
This is especially convenient if you plan to use the expression in more than one computation.
