### Multivariate functions: Basic notions

### Multivariate functions

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

Multivariate functions

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}#.

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:

\[f(-1,3,2) = \frac{9\cdot (-1)^3}{3}+3\cdot(3)^2\cdot {2} + 6\cdot(-1)\cdot(3)=33\tiny.\]

Thus, the value of the function #f# at #\rv{-1,3,2}# is #33#.

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