Define a function z = f(x,y) by f(0,0) = 0 and otherwise.

f(x,y) =(x^2 y) / (x^2+y^2 )

a. Show that in polar coordinates this function may be expressed (for r≠ 0) as z = r 〖cos〗^2 (θ)sin(θ)

b. Show that if θ is fixed then the graph is given by z = mr, a line of slope

m= 〖cos〗^2 (θ)sin(θ).

(Note that this says that the surface z = f(x,y) is what is called a ruled surface.)

c. Compute the directional derivatives of z in the θ direction. Does Df exist at the point (0,0)? Explain.