Level curves.

It is generally hard to draw a picture of the graph of a function of two variables. Sometimes, we need to draw level curves; just remember the level curves on a map, in geography.

Take a function f of two variables x and y, defined on a domain D. For a real number k, the curve of level k, denoted $\mathcal{C}(k)$is the set of all the points in D such that f(x,y)=k.

Example 4.3   Let f(x,y)= x²+y².

• If k < 0, then $\mathcal{C}(k)$is empty.
• $\mathcal{C}(0)$contains a single point, namely the origin.
• If k >0, then $\mathcal{C}(k)$is a circle centered at the origin and whose radius is equal to $\sqrt{k}$.

 

   

Figure 2: Cylinders.