### Multivariate functions: Basic notions

### Functions of two variables

Intuitive definition of a function of two variables

So far we have limited ourselves to functions of a single variable. These can be viewed as little machines producing a new number from a given number according to a given formula. But you can also think of little machines and formulas that produce a new number from two given numbers. In that case, we speak of a **function of two variables.**

Recall from the theory *Functions *that a function from a set #X# to a set #Y# assigns a unique element of #Y# to each element of #X#. "Functions" in this course are almost always understood to be "real functions". This means #Y=\mathbb R#, both for real functions of a single variable and for functions of two variables.

For real functions of a single variable, #X# is a subset of #\mathbb R#. In the case of a function of two variables, #X# is a subset of #{\mathbb R}^2#, the set of all pairs of real numbers, and #Y=\mathbb R#. Thus, an element of the domain belongs to #{\mathbb R}^2# and is usually denoted by its coordinates #\rv{x,y}#.

You may wonder whether there is a notion corresponding to a little machine that produces two numbers (instead of a single number) from two given numbers. In fact, there is: a function from #{\mathbb R}^2# to #{\mathbb R}^2#.

Three simple functions of two variables

- The area \(O\) of a triangle with base \(b\) and height \(h\): \[O(b,h)=\tfrac{1}{2}\cdot b\cdot h\tiny.\]
- The distance \(s\) covered by an object in uniform motion with velocity \(v\) and time \(t\): \[s(v,t) = v\cdot t\tiny.\]
- The milk consumption #x# as a function of the price #p# of milk and the average income #m# per family: \[x(m,p)=3\cdot \frac{m^{2.07}}{p^{1.4}}\tiny.\]

These examples are written in the form of a *function rule*, that is a description of the actual machine producing the new number from the given pair #\rv{b,h}# in the first example and #\rv{v,t}# in the second example.

In the second example, the **function rule** is the expression \(v\cdot t\). The dependent variable #s# is *explicitly* written down, isolated from the independent variables #v# and #t#.

The *terminology of functions which we already know* is also used for functions of two variables.

Functions of two variables

A relation between three variables \(x\) , \(y\), and \(z\), where \(x\) and \(y\) occur as independent variables, is a **function** \(z=z(x,y)\) if, for any admissible values \(x\) and \(y\), there is *exactly one* value for \(z\) corresponding to it. This value #z# is called the **value of the function at the point** #\rv{x,y}#.

The set of all pairs#\rv{x,y}# of admissible values \(x\) and \(y\) is called the **domain** of the function**. **If #D# is the domain, we speak of a** function on **#D#.

The value of \(z\) at a point #\rv{x,y}# of the domain is called the **function value** at #\rv{x,y}#.

The set of all values that the function can assume is called the **range** of the function.

The **graph** of a relation between three variables is the set of all points #\rv{x,y,z}# satisfying the relation. In particular, the **graph** of a function #f# of two variables is the set of all points #\rv{x,y,f(x,y)}# for #\rv{x,y}# ranging over the points of the domain of #f#.

The definition of graph is a straightforward generalization of the *notion of graph given in the case of a single variable*.

Substituting #r=-4# and # s=-2# in the expression # (3({r}\cdot{s}-2{s}))\cdot(2{r}^2+{s})-2{r}^2-{s}# gives \[ (3\cdot ((-4)\cdot (-2)-2\cdot (-2)))\cdot (2\cdot (-4)^2+(-2))-2\cdot (-4)^2-(-2)\tiny.\] Simplification of this expression gives the answer #1050#.

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