Multivariate Functions

(1) Definition (Multivariate Functions) 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.

(2) Definition (Operations with Multivariate Functions) If and are f unctions of the variables then so are

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

(3) 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

(4) 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

(5) Definition (Graph of Multivariate Functions) The graph of a function of several variables is the collection of all ordered -tuples such that is in the domain of and In three dimensions, the graph of is a surface in whose projection onto the -plane is the domain .

(6) Example (Graphs of Multivariate Functions) With a minimal learning curve the software packages Mathematica, Maple, MATLAB, and Derive all have the capability of producing great three dimensional graphics. Here are some examples of computer generated 3D graphs:  

(7) Definition (Trace) When the plane intersects the surface the result is the curve with the equation and such an intersection is called the trace of the graph of in the plane

(8) Definition (Level Curves) The set of points in the -plane that satisfies is called the level curve of at and an entire family of level curves is generated as varies over the range of . Level curves are obtained by projecting a trace onto the -plane. Because level curves are used to show the shape of a surface, they are sometimes called contour curves.

(9) Example (Level Curves) Sketch some level curves for the function with for the given function

    Solution. Graphing the lines we have the family of level curves corresponding to The level curves show that the function is a plane in




(10) Example (Level Curves) Sketch some level curves for the function with for the given function
    
    Solution. Graphing the ellipses in we have the family of level curves corresponding to The level curves show that the function is a elliptic paraboloid in

(11) Example (Level Curves) Sketch some level curves for the function Find the domain and range for the function and sketch some level curves for with and sketch the domain.
    
    Solution. The domain of is . To sketch some level curves let's square both sides and so and some level curves are  



Here's a graph of from two different viewpoints.

• print this page
• pages linking here
• related pages

Cite this as:
Multivariate Functions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/multivariate-functions.html