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

  1. Apr 24, 2012 #1
    please help me with this. At least some thing to start solving.....

    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. jcsd
  3. Apr 24, 2012 #2


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    welcome to pf!

    hi matthew! welcome to pf! :smile:

    (try using the X2 button just above the Reply box :wink:)

    show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:

    start with "a."
  4. Apr 24, 2012 #3

    Start for noting that in polar coordinates, [itex]x=r\cos\theta\,\,,\,\,y=r\sin\theta[/itex] , so substitute and get (a) at least . Now show some self effort.

  5. Apr 25, 2012 #4
    Thanks DonAntonio,

    thanks for showing me the path. I almost completed the part a and b. have a confusion about the part c. here Df means the derivative of the f. Do you have any idea about that one?

    Thanks again for helping me. I appreciate what you did. As you gave me before I just need a hint to proceed, not the whole proof. thanks again.
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