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.

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.

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.

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