Question

Determine the intersection z = 0 of the monkey saddle M: z = f(x, y), f(x, y) = y3 – 3yx?, with the xy plane. On which regions of the plane is f> 0? f <0? How does this surface get its name? (Hint: see Fig. 5.19.)

Answer 1

The intersection of surface f=y3-3yx2 with xy plane is given by y3-3yx2=0 it implies y(y2-3x2)=0. This gives either y=0 or y2=3x2. Thus intersection of surface with xy plane are lines given by y=0, $y=\sqrt{3}x$ and $y=-\sqrt{3}x$ .

In the region $\{(x,y):y^2>3x^2\}$ , f>0. And

In the region $\{(x,y):y^2<3x^2\}$ , f<0

This surface has three depressions so one can think of it as two depressions are for the legs of monkey and one for tail. Thus this surface is called as monkey saddle.