Substitute
Your function now becomes
Since
Also if is constant then where m is constant.
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.
Your inability to understand the rules of this forum, your inability to understand, in particular, what the "New Users" forum is for, and finally, your inability to copy any of these problems correctly does not auger well.