A function of the form \(y=f(x)\) is a function of a single variable; given a value of \(x\text{,}\) we can find a value \(y\text{.}\) Even the vector–valued functions of Chapter 11 are single–variable functions; the input is a single variable though the output is a vector.

There are many situations where a desired quantity is a function of two or more variables. For instance, wind chill is measured by knowing the temperature and wind speed; the volume of a gas can be computed knowing the pressure and temperature of the gas; to compute a baseball player's batting average, one needs to know the number of hits and the number of at–bats.

This chapter studies multivariable functions, that is, functions with more than one input.