In one-variable calculus, we studied functions like $f(x)=x^2$ with
one input and one output. When studying curves, we looked at
vector-valued functions like ${\bf r}(t) = \langle \cos(t), \sin(t)
\rangle$ with one input and several outputs, or equivalently with a
vector as the output. Next we are going to study functions like
$f(x,y) = x^2 + y^2$ that have several inputs and one output. We call
these functions of several variables.

You find functions of several variables throughout science,
mathematics, and day-to-day life. On a hot summer day, the heat index
is a function of temperature and humidity. On a cold winter day, the
wind chill factor is a function of temperature and wind speed. On
Earth, elevation is a function of latitude and longitude. Your blood
pressure is a function of your age and how much exercise you get (and
other factors, too). The fraction of Democratic (or Republican) voters
in a county can be modeled by a complicated function of population
density, average income, average education level, concentration of
minorities, and rate of church attendance. It's not much of an
exaggeration to say that all quantities in the real world are
functions of several variables.

For simplicity, we're going to start with functions of just two
variables. We'll usually call the variables $x$ and $y$, and call the
function $f(x,y)$. The domain is the set of pairs $(x,y)$ that are
valid inputs, and the range is the set of output values. The domain is
a subset of the $x$-$y$ plane, and the range is a subset of the real
line.

Example:Find the domain and range of the
function $f(x,y) = \sqrt{1-x^2-y^2}$.

Solution: The domain is the unit disk $x^2 + y^2 \le 1$,
since the square root only makes sense if $1-x^2-y^2 \ge 0$.
The range is the interval $[0,1]$.

The following video explores functions of several variables and
ways to understand them.