Functions of Several Variables
Definition of Functions of Several Variables
Finding the Domain
To find the domain of a function of several variables, we look for zero
denominators and negatives under square roots:
The graph to the right shows the domain as the shaded green region.
The topographical map shown below is of the Rubicon Trail. It represents the function that maps a longitude and latitude to an altitude.
Each curve represents a path where the z-coordinate (altitude) is a constant. Crossing many topo lines in a short distance represents a path that is very steep.
Now lets make our own contour map of the function.
f(x,y) = y - x2
We see that each topo line is a parabola and that the y-intercept gives the height. Below is a contour diagram of this function.
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.