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Math Help - Define a function z = f(x,y) by f(0,0) = 0 and otherwise

  1. #1
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    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.
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  2. #2
    Junior Member ignite's Avatar
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    Re: Define a function z = f(x,y) by f(0,0) = 0 and otherwise

    Substitute  x=rcos\theta,y=rsin\theta
    Your function now becomes f(r,\theta)=\frac{(rcos\theta)^2(rsin\theta)}{(rco  s\theta)^2+(rsin\theta)^2}=rcos^2\theta sin\theta
    Since cos^2\theta+sin^2\theta=1
    Also if \theta is constant then z=f(r)=mr where m is constant.
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  3. #3
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    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.
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  4. #4
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    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
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