Sec 15.1 Functions of Several Variables

1. Sec 15.1 Functions of Several Variables Definitions: A function f of two variables is a rule that assigns to each ordered pair of real numbers (x, y) in a set D a unique real number denoted by f (x, y) [or z] . The set D is the domain of f . The range of f is {z = f (x, y)| (x, y) in D} x and y are independent variables. z is the dependent variable. The graph of f is the set of all points (x, y, z) in such That z = f (x, y) and (x, y) is in D.

2. A function of the form z = f (x, y) = ax + by + c is called a linear function and its graph is a plane. Definition: The level curves of a function f of two variables are the curves with equations f (x, y) = k , where k is a constant (in the range of f ). Definition: A function f of three variables is a rule that assigns to each ordered triple (x, y, z) in a set D in a unique real number denoted by f (x, y, z). Definition: The level surfaces of a function f of three variables are the surfaces with equations f (x, y, z) = k , where k is a constant

3. Sec 15.2 Limits and Continuity Definition: Let f be a function of two variables whose domain D includes point arbitrarily close to (a, b). We say “The limit of f (x, y) as (x, y) approaches (a, b) is L” and write if for every number ε> 0, there is a corresponding number δ > 0 such that if Other notations:

4. Note: does not exist if and where .

5. Definition: A function f of two variables is called continuous at (a, b) if We say f is continuous on Dis f is continuous at every point (a, b) in D.