1.1  Graphs

Recall that a real valued function of several real variables is a map f: U R where R 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:

Graph(f) =

(x1,x2,,xn,y) Rn+1
(x1,,xn) U, y = f(x1,,xn)

1.2  Level Sets

To ease understanding of level sets, we will first start with some examples:

Example 1:   Recall the function f(x1,x2) = x12+x22 from example 1 above. The graph of f along with some of its level sets are shown in fig. 2. In this case, the level sets are defined implicitly by c = x12 + x22, c R+

 

 

 

 

 

 

 

 

 

Fig 2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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 R. The darker shading in the contour diagram indicates more negative values for c.
Fig. 3
Fig. 4

 

Example 3:   Consider the function f(x,y) = cosx + siny. Its graph and level sets are shown in fig. 5 and 6.

 
Fig. 5
Fig. 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.
Fig. 7
Fig. 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:

P-1(c) =

(x1,,xn) U
f(x1,,xn) = c

for each c R
It is important to remember that the level sets are projections down to Rn-1. Just like functions have graphs, so do their level sets. A common mistake is to think of the level sets as the cross-sections of the surface. The graphs of the level sets are not actually on the graph but are projected down one less dimension.

 

 

 

 

Example 5:   For a more interesting example, lets look at a function which takes three variables as input instead of two. Let f: U R be defined by f(x1,x2,x3) = [(x12)/( 12)]+[(x22)/( 22)]+[(x32)/( 32)]. Its level set graphs to a subset of R3 and are shown in fig. 9.

 

 

 

 

 

 

 
Fig. 9