### 1.1  Graphs

Recall 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:

 Graph(f) = ìí î (x1,x2,¼,xn,y) Î Rn+1 êê (x1,¼,xn) Î U, y = f(x1,¼,xn) üý þ

### 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 Î 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 Î 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) = ì í î (x1,¼,xn) Î U ê ê f(x1,¼,xn) = c ü ý þ for each c Î 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