In single-variable calculus we were concerned with functions that map the real numbers $\R$ to $\R$, sometimes called "real functions of one variable'', meaning the "input'' is a single real number and the "output'' is likewise a single real number. In the last chapter we considered functions taking a real number to a vector, which may also be viewed as functions $f\colon\R\to\R^3$, that is, for each input value we get a position in space. Now we turn to functions of several variables, meaning several input variables, functions $f\colon\R^n\to\R$. We will deal primarily with $n=2$ and to a lesser extent $n=3$; in fact many of the techniques we discuss can be applied to larger values of $n$ as well.

A function $f\colon\R^2\to\R$ maps a pair of values $(x,y)$ to a single real number. The three-dimensional coordinate system we have already used is a convenient way to visualize such functions: above each point $(x,y)$ in the $x$-$y$ plane we graph the point $(x,y,z)$, where of course $z=f(x,y)$.

Example 14.1.1 Consider $f(x,y)=3x+4y-5$. Writing this as $z=3x+4y-5$ and then $3x+4y-z=5$ we recognize the equation of a plane. In the form $f(x,y)=3x+4y-5$ the emphasis has shifted: we now think of $x$ and $y$ as independent variables and $z$ as a variable dependent on them, but the geometry is unchanged. $\square$

Example 14.1.2 We have seen that $x^2+y^2+z^2=4$ represents a sphere of radius 2. We cannot write this in the form $f(x,y)$, since for each $x$ and $y$ in the disk $x^2+y^2< 4$ there are two corresponding points on the sphere. As with the equation of a circle, we can resolve this equation into two functions, $\ds f(x,y)=\sqrt{4-x^2-y^2}$ and $\ds f(x,y)=-\sqrt{4-x^2-y^2}$, representing the upper and lower hemispheres. Each of these is an example of a function with a restricted domain: only certain values of $x$ and $y$ make sense (namely, those for which $x^2+y^2\le 4$) and the graphs of these functions are limited to a small region of the plane. $\square$

Example 14.1.3 Consider $f=\sqrt x+\sqrt y$. This function is defined only when both $x$ and $y$ are non-negative. When $y=0$ we get $f(x,y)=\sqrt x$, the familiar square root function in the $x$-$z$ plane, and when $x=0$ we get the same curve in the $y$-$z$ plane. Generally speaking, we see that starting from $f(0,0)=0$ this function gets larger in every direction in roughly the same way that the square root function gets larger. For example, if we restrict attention to the line $x=y$, we get $f(x,y)=2\sqrt x$ and along the line $y=2x$ we have $f(x,y)=\sqrt x+\sqrt{2x}=(1+\sqrt2)\sqrt x$. $\square$