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.

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