section_path: Definition/Multivariate functions
section_name: Multivariate functions
paragraph_text: Multivariate functions
  
  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 n variables is a function whose domain is a set of n -tuples. For example, multiplication of integers 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. Commonly, an n -tuple is denoted enclosed between parentheses, such as in (1, 2, …, n). {\displaystyle (1,2,\ldots,n).} When using functional notation, one usually omits the parentheses surrounding tuples, writing f (x 1, …, x n) {\displaystyle f(x_{1},\ldots,x_{n})} instead of f ((x 1, …, x n)). {\displaystyle f((x_{1},\ldots,x_{n})).}
  
  Given n sets X 1, …, X n, {\displaystyle X_{1},\ldots,X_{n},} the set of all n -tuples (x 1, …, x n) {\displaystyle (x_{1},\ldots,x_{n})} such that x 1 ∈ X 1, …, x n ∈ X n {\displaystyle x_{1}\in X_{1},\ldots,x_{n}\in X_{n}} is called the Cartesian product of X 1, …, X n, {\displaystyle X_{1},\ldots,X_{n},} and denoted X 1 × ⋯ × X n. {\displaystyle X_{1}\times \cdots \times X_{n}.}
  
  Therefore, a multivariate function is a function that has a Cartesian product or a proper subset of a Cartesian product as a domain.
  
  where the domain U has the form
  
  If all the X i {\displaystyle X_{i}} are equal to the set R {\displaystyle \mathbb {R} } of the real numbers or to the set C {\displaystyle \mathbb {C} } of the complex numbers, one talks respectively of a function of several real variables or of a function of several complex variables.