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.


        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:


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}


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.



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