A two-peaked surface with three level sets shown

You can interpret this picture in two ways.
1. Thinking of the red object as a solid, then the level sets of the solid are the solid gray pictures.
2. Thinking of the red object as a surface (just the thin sheet which is the surface of the mountain), then the level curves of the surface are the edge of the gray pictures: an oval, an infinity symbol, and two separate ovals.

An animation showing the level curves of the function g( x,y) = x²+y²+2 for k = 2,3,4,5, and 6.

Solution: To begin with, g( x,y) = k is of the form x²+y²+2 = k, which simplifies to

x2+y2 = k-2

If k = 2, then x²+y² = 0 which is true only if ( x,y) = ( 0,0) . For k = 3,4,5,and 6, we have

k = 3: x2+y2 = 1 k = 5: x2+y2 = 3 k = 4: x2+y2 = 2 k = 6: x2+y2 = 4

These are circles centered at ( 0,0) with radii 1,Ö2,Ö3, and 2, respectively.

(Source: http://math.etsu.edu/MultiCalc/Chap2/Chap2-7/index.htm)