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 R^{n} 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  x^{2}
by setting constant values
for z:
z 
Equation 
1 
y = x^{2} + 1 
2 
y = x^{2} + 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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