Section Summary: 15.1

1. Definitions

   A function f of two variables is a rule that assigns to each ordered pairs of real numbers (x,y) in a set D a unique real number denoted by f(x,y). The set D is the domain of f and its range is the set of values that f takes on D, that is, . The variables x and y are the independent variables, whereas z=f(x,y) is the dependent variable.

   The graph of a function f of two variables is the set of all points (x,y,z) in such that z=f(x,y) and (x,y) is in D.

   A function f of n variables is a rule that assigns to each ordered n-tuple of real numbers in a set D a unique real number denoted by . The set D is the domain of f and its range is the set of values that f takes on D, that is, .

   level curves of a function f of two variables are curves with equations f(x,y)=k, where k is a constant (in the range of f). More generally, level surfaces of a function f of n variables are surfaces with equations , where k is a constant (in the range of f).

   f is a linear function if it is linear in each of its independent variables.

2. Theorems

   None.

3. Properties/Tricks/Hints/Etc.

   You need to learn to draw ``fake 3-d''! Practice, practice, practice....

4. Summary

   Most of the focus of this text will be on functions of two-variables, whose graphs are surfaces in three-dimensions. One of the primary problems is that of how to represent these surfaces in two-spaces (i.e. on the board, on your paper).

   Functions are defined in four ways:

   • verbally
   • numerically
   • algebraically
   • visually
   But we tend to work most in this course with functions defined algebraically.

   We may find it useful to use level curves, or level surfaces, to help us visualize multivariate functions. In the case of functions of two variables, we obtain level curves, which comprise a topographic map of the function.