Multivariate functions

Multivariate functions: Basic notions

Multivariate functions

The definition of a function of two variables can be extended to more variables.

A real-valued function $f$ of $n$ variables $x_1,x_2,x_3,\ldots,x_n$ assigns a unique real number to each choice $\rv{a_1,a_2,\ldots,a_n}$ of real numbers for $\rv{x_1,\ldots,x_n}$ within a designated part of $\mathbb{R}^n$ .

This number is called the value of $f$ at $\rv{a_1,a_2,a_3,\ldots,a_n}$ and denoted by $f(a_1,a_2,a_3,\ldots,a_n)$ .
The expression $f(x_1,x_2,x_3,\ldots,x_n)$ usually indicates a formula describing how to compute the value of $f$ at $\rv{a_1,a_2,a_3,\ldots,a_n}$ by substituting $a_1$ for $x_1$ , $a_2$ for $x_2$ , and so on. This expression is called the functional rule of $f$ .
The designated part of $\mathbb{R}^n$ , that is, all points to which $f$ assigns a value, is called the domain of the function.
The set of all values of $f$ is the range of the function.

If $n=1$ , we are back at the case of a single variable, which is studied in several previous chapters.

If we do not wish to emphasize the number $n$ , we also refer to such a function as a multivariate function. For $n=2$ , the name function of two variables has already been introduced; occasionally, the name bivariate function will also be used.

In this chapter we will mostly study functions of $2$ or $3$ variables.

Mappings

You may wonder whether there is a notion corresponding to a little machine that produces a list of new numbers (instead of a single number) from a given list of numbers. In fact, there is: a mapping, also called a map.

In general, a mapping from a set $X$ to a set $Y$ assigns a unique element of $Y$ to each element of $X$ . Actually, in theory Functions, such a mapping was also called a function, but often, the notion of function is reserved for a mapping to a set of numbers.

The mapping that produces a list of $3$ numbers from a list of two numbers can be seen as a mapping from ${\mathbb R}^2$ to ${\mathbb R}^3$ . Another way to look at it is a list of three functions on ${\mathbb R}^2$ , one for each coordinate of ${\mathbb R}^3$ . The map that assigns $\rv{\frac{1}{\sqrt{x^2+y^2}},\e^{x},\e^{y}}$ to $\rv{x,y}$ is an example, whose coordinate functions are $\frac{1}{\sqrt{x^2+y^2}}$ , $\e^{x}$ , and $\e^{y}$ . Notice that its domain is ${\mathbb R}^2\setminus\{\rv{0,0}\}$ rather than ${\mathbb R}^2$ : the mapping is not defined at $\rv{0,0}$ .

Calculate the value of the function $f(x,y,z) = 12\frac{x^3}{y}+y^2\cdot z + 7x\cdot y$ at $\rv{x,y,z} = \rv{-1,3,2}$ .

Simplify your answer as much as possible.

$-7$

In order to calculate the value of the function $f$ at a given point $\rv{x,y,z}$ we substitute the values of $x$ , $y$ , and $z$ in the function rule as follows:
$f(-1,3,2) = \frac{12\cdot (-1)^3}{3}+1\cdot(3)^2\cdot {2} + 7\cdot(-1)\cdot(3)=-7\tiny.$

Thus, the value of the function $f$ at $\rv{-1,3,2}$ is $-7$ .

New example