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
x, replace it by 3:
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The variable
x has a value of 3.
Whenever you evaluate an expression, 3 is substituted for
x:
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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
x, replace it by
y^2. So far there are no rules associated with
y, so its value is itself.
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Now if you evaluate
x, the rule for
x says to replace
x by
y^2, and the rule for
y says to replace
y by 4, so the result is
4^2, or 16:
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If you change the value of
y, then the value of
x changes:
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Now assign a value to
z, like this:
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Since
y has already been assigned the value 3, the rule you have defined is "replace
z by 9", not "replace
z by
y^2". So
z does not depend on
y:
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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
y already has a value, this new rule says: whenever you see
z, replace it with
y^2. So in this case,
z depends on
y:
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Functions in
Mathematica are defined by rules that act on patterns. Here is a simple one:
f[x_] is a pattern in which
x_ stands for any expression (which we represent on the right-hand side by the name
x). The rule says: if you have
f 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
x is 9 and
y is 3. This rule says that anything matching the pattern
h[x_, y_] is replaced by 90:
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