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 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.