Editing Function (mathematics) (section)

=== Multivariate functions <span class="anchor" id="MULTIVARIATE_FUNCTION"></span> ===
{{distinguish|Multivalued function}}
[[File:Binary operations as black box.svg|thumb|A binary operation is a typical example of a bivariate function which assigns to each pair <math>(x, y)</math> the result <math>x\circ y</math>.]]

A '''multivariate function''', '''multivariable function''', or '''function of several variables''' is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time travelled and its average speed.

Formally, a function of {{mvar|n}} variables is a function whose domain is a set of {{mvar|n}}-tuples.<ref group=note>{{mvar|n}} may also be 1, thus subsuming functions as defined above. For {{math|1=''n'' = 0}}, each [[constant (mathematics)|constant]] is a special case of a multivariate function, too.</ref> For example, multiplication of [[integer]]s is a function of two variables, or '''bivariate function''', whose domain is the set of all [[ordered pairs]] (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every [[binary operation]]. The graph of a bivariate surface over a two-dimensional real domain may be interpreted as defining a [[Surface (mathematics)#Graph of a bivariate function|parametric surface]], as used in, e.g., [[bivariate interpolation]].

Commonly, an {{mvar|n}}-tuple is denoted enclosed between parentheses, such as in <math>(1,2,\ldots, n).</math>  When using [[functional notation]], one usually omits the parentheses surrounding tuples, writing  <math>f(x_1,\ldots,x_n)</math> instead of <math>f((x_1,\ldots,x_n)).</math>

Given {{mvar|n}} sets <math>X_1,\ldots, X_n,</math> the set of all {{mvar|n}}-tuples <math>(x_1,\ldots,x_n)</math> such that <math>x_1\in X_1, \ldots, x_n\in X_n</math> is called the [[Cartesian product]] of <math>X_1,\ldots, X_n,</math> and denoted <math>X_1\times\cdots\times X_n.</math>

Therefore, a multivariate function is a function that has a Cartesian product or a [[proper subset]] of a Cartesian product as a domain.

<math display="block">f: U\to Y,</math>

where the domain {{mvar|U}} has the form

<math display="block">U\subseteq X_1\times\cdots\times X_n.</math>

If all the <math>X_i</math> are equal to the set <math>\R</math> of the [[real number]]s or to the set <math>\C</math> of the [[complex number]]s, one talks respectively of a [[function of several real variables]] or of a [[function of several complex variables]].