A multivariable function is a function with several variables. Functions with more than one variable are needed in order to mathematically model complicated physical phenomena, engineering, chemical, agricultural ... systems.

Examples of Multivariable Functions

Example 1

A rectangle has a width \( W \) and a length \( L \). The area \( A \) of the rectangle is given by \( A = W \cdot L \). It is clear that if \( W \) and \( L \) vary, area \( A \) depends on two variables: width \( W \) and length \( L \). Area \( A \) is said to be a function of two variables \( W \) and \( L \).

Example 2

A rectangular solid has width \( W \), length \( L \), and height \( H \). The volume \( V \) of the rectangular solid is given by \( V = W \cdot L \cdot H \). If \( W \), \( L \), and \( H \) vary, volume \( V \) depends on 3 variables: width \( W \), length \( L \), and height \( H \).

Example 3

The volume \( V \) of a circular cylinder of radius \( r \) and height \( h \) is given by \( V = \pi r^2 h \). If \( r \) and \( h \) vary, we can say that volume \( V \) is a function of two variables \( r \) and \( h \).

Example 4

Let \( T \) be the temperature in a room. Using a rectangular coordinate system of axes (\( x, y, z \)), temperature \( T \) can be said to vary with \( x, y, z \), and time \( t \) and may be written as \( T(x,y,z,t) \) as a function of 4 variables.