Functions of Several Variables

    Functions of one variable map a set of real numbers to another set of real numbers. A function assigns, to each number in the first set (domain), a unique real number in the second set (range). A function of several variables has as its domain a set of ordered pairs and its range is a set of real numbers. In general, a function of several variables assigns to each n-tuple a unique real number. For example, a function of two independent variables, assigns to each ordered pair (point in 2-space) a unique real number; and a function of three independent variables assigns to each ordered triple (point in 3-space) a unique real number, etc. In general, the number of coordinate axes needed to graph a function of several variables is the number of variables plus one. Techniques for graphing functions, including domain, range, level curves, and level surfaces are illustrated in detail.

Definition (Functions of Several Variables) A function of variables is a rule that assigns to each ordered pair in a set a unique number The set is called the domain of the function and the corresponding values constitute the range. When working with functions of two variables, we write where and are the independent variables and is the dependent variable. The domain is the largest set of points for which the functional formula is defined and real-valued.

Definition (Operations of Functions of Several Variables) If and aref unctions of the variables then so are

     (i)  
     
    (ii)  
    
    (iii)  
    
    (iv)   where
    

Example (Functions of Two Variables) Find the domain and range for the function
    
    Solution. The domain is because of the square root in the denominator of and the range is

Example (Functions of Two Variables) Find the domain and range for the function   

    Solution. The domain is because of the square root and the range is

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Functions Of Several Variables
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/functions-of-several-variables.html