3-Space Grapher

(a)

(b)

(c)

These are all graphical representations of the function f(x, y) = (xy)/(x^2 + y^2).

The (a) and (b) are produced by Wolframalpha and (c) is generated by Microsoft Mathematics.

Fig. (a) is called a contour map or level surface. It takes a 3-space surface and compresses it into a 2-space diagram. Areas of darker red/brown correspond to the negative z values of the function. A 3-space model on a contour map is the only stable way to represent a function of three variables w = F(x, y, z)

Fig. (b) and (c) are 3-space representations of the functions. If you had a piece of modeling clay and were told to mold it into the function f, you would generate this shape.