# Thread: Define a function z = f(x,y) by f(0,0) = 0 and otherwise

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

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.

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

Substitute$\displaystyle x=rcos\theta,y=rsin\theta$
Your function now becomes $\displaystyle f(r,\theta)=\frac{(rcos\theta)^2(rsin\theta)}{(rco s\theta)^2+(rsin\theta)^2}=rcos^2\theta sin\theta$
Since $\displaystyle cos^2\theta+sin^2\theta=1$
Also if $\displaystyle \theta$ is constant then $\displaystyle z=f(r)=mr$ where m is constant.

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

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.

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

Hi ignite,

Thanks for your help. I just need a path to follow and you provided that. I solved most of the parts. Thank you again

tc