How to | Create Definitions for Variables and Functions
Mathematica has a very general notion of functions, as rules for arbitrary transformations. Values for variables are also assigned in this manner. When you set a value for a variable, the variable becomes a symbol for that value.
Here is a simple transformation rule. It says: whenever you see 
, replace it by 3:
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The variable 
has a value of 3.
Whenever you evaluate an expression, 3 is substituted for 
:
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You can remove the rule by defining a new one:
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The new rule says: whenever you see 
, replace it by 
. So far there are no rules associated with 
, so its value is itself.
Assign a value to 
:
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Now if you evaluate 
, the rule for 
says to replace 
by 
, and the rule for 
says to replace 
by 4, so the result is 
, or 16:
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If you change the value of 
, then the value of 
changes:
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Now assign a value to 
, like this:
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Since 
has already been assigned the value 3, the rule you have defined is "replace 
by 9", not "replace 
by 
". So 
does not depend on 
:
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This happened because when a rule is defined using 
(Set), the right-hand side is evaluated before the rule is defined.
You can also define rules using 
(SetDelayed), like this:
When a rule is defined with 
the right-hand side is not evaluated before the rule is defined. So even if 
already has a value, this new rule says: whenever you see 
, replace it with 
. So in this case, 
depends on 
:
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Functions in Mathematica are defined by rules that act on patterns. Here is a simple one:
is a pattern in which 
stands for any expression (which is represented on the right-hand side by the name 
). The rule says: if you have 
of any expression, replace it by that expression squared:
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Here is a function with two arguments:
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Always use 
to define functions, otherwise the variables on the right-hand side may not represent the associated expressions on the left-hand side, since they will be evaluated before the rule is defined:
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That happened because 
is 9 and 
is 3. This rule says that anything matching the pattern 
is replaced by 90:
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