1.1 GraphsRecall that a real valued function of several real variables is a map f: U ® R where R Í Rn. This simply means that the function takes as input a given number of variables and provides a single number result.
Example 1: f(x1,x2) = x12
+ x22 is a function of two variables. Its graph
is found in fig. 1. It is important not to confuse the two. The function
is a map, while the graph is a subset of R3.
The graph of a function f: Rn ® R is the subset of Rn+1 defined by:
1.2 Level Sets
To ease understanding of level sets, we will first start with some examples:
Example 2: The graph and level sets of the function f(x1,x2) = x12 - x22 are shown in fig. 3 and fig. 4. Function f is called a saddle. The implicit definition of the limit sets is: c = x12 - x22, c Î R. The darker shading in the contour diagram indicates more negative values for c.
Example 3: Consider the function f(x,y) = cosx + siny. Its graph and level sets are shown in fig. 5 and 6.
Example 4: The function cleverly named a monkey saddle is given by f(x,y) = x3 - x y2. Its graph and level sets are given in fig. 7 and 8.
By now, the meaning of level sets should be intuitively clear. They can be thought of as cross-sections of the curve that are projected down to the x-y plane. The strict definition follows: