---
tag: 0JPCHTXWG0
kind: concept
name: Function of several variables
aliases: [multivariable-function, function-of-several-real-variables]
msc: [26B99]
uses: [0EPAT51DXQ, 0XPR1E85W0]
sources:
  - url: https://en.wikipedia.org/wiki/Function_of_several_real_variables
    locator: lead section
same_as:
  wikidata: Q15284797
  wikipedia: Function of several real variables
---

## Summary

A function of several real variables is a [[function|0EPAT51DXQ]] $f \colon D \to \mathbb{R}$ whose domain $D$ is a subset of [[Euclidean space|0XPR1E85W0]] $\mathbb{R}^n$ for some $n \ge 2$.
Its value at a point $x = (x_1, \dots, x_n)$ of $D$ is written $f(x)$ or $f(x_1, \dots, x_n)$.
Examples are $f(x, y) = x^2 + y^2$ on $\mathbb{R}^2$ and $g(x, y, z) = xyz$ on $\mathbb{R}^3$.

## Remarks

When the values lie in $\mathbb{R}^m$ with $m \ge 2$, the function is a [[vector-valued function|018GTRS7A4]] of several variables.
The domain is often an open subset of $\mathbb{R}^n$, so that each point of the domain has a whole ball around it in the domain.

## Explanations

### L1 human

A function of two variables takes two numbers and gives one number. An example is the height of a hill above the point $(x, y)$ of a map.
Each input is now a point of the plane or of space, not a point of a line.