# Introduction to Multivariable Functions

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 1A rectangle has a width W and a length L. The area A of the rectangle is given by A = W 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 L 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 = π 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.

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