Functions of Several Variables

Definition of Functions of Several Variables

A function of several variables is a function where the domain is a subset of Rⁿ and range is R.

Example:  

        f(x,y) = x - y 

is a function of two variables

        g(x,y,z) = (x - y)/(y - z)

is a function of three variables.

Finding the Domain

To find the domain of a function of several variables, we look for zero denominators and negatives under square roots:

Example

Find the domain of 

       

First, the inside of the square root must be positive, that is 

        x - y > 0

second, the denominator must be nonzero, that is 

        x + y      0 

hence we need to stay off the line 

        y = -x

Putting this together gives 

        {(x,y)| x - y > 0 and y  -x}

Exercise

Find the domain of the function 

   

Level Curves

Topo maps such as that of the desolation wilderness that represents the function that maps a longitude and latitude to an altitude.  We will investigate what the contour lines mean.  We will make our own contour map of the function.

        f(x,y) =  y - x²  

by setting constant values for z:

z	Equation
1	y = x2 + 1
2	y = x2 + 2

Names for the curves drawn are level curves, isotherms (for temperature), isobars (for pressure), and equipotential lines (for electric potential fields) depending on what the two variable function represents.

Back to the Math 117 Home Page

Back to the Math Department Home

e-mail Questions and Suggestions