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Differentiability in R2

  1. Feb 18, 2015 #1
    Let U={(x,y) in R2:x2+y2<4}, and let f(x,y)=√.(4−x2−y2)
    Prove that f is differentiable, and find its derivative.

    I do know how to prove it is differentiable at a specific point in R2, but I could not generalize it to prove it differentiable on R2. Any hint?
     
  2. jcsd
  3. Feb 18, 2015 #2

    Svein

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    As far as I can see you are not asked to do that, you are asked to prove differentiability on U.
     
  4. Feb 18, 2015 #3

    HallsofIvy

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    As Svein said, you are not asked to prove differentiability on R2, you are asked to prove it on the given set, U. Now, what is the definition of "differentiable on a set A"?
     
  5. Feb 19, 2015 #4
    I know the definition of "differentiable at a point" , but i am not sure of the definition of differentiability on a set. Does it have to do with end points? i am stuck in this question and your help is much appreciated
     
  6. Feb 19, 2015 #5

    Svein

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    Find the derivative and decide where it is valid.
     
  7. Feb 19, 2015 #6
    Can i pick an arbitrary subset of U and prove that the function is differentiable there then conclude that it is differentiable on U ?
     
  8. Feb 19, 2015 #7

    LCKurtz

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    Why not just pick any (x,y) in the domain U and see if it works there? U is an open domain so there are no boundary points U to worry about.
     
  9. Feb 20, 2015 #8

    HallsofIvy

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    A function is differentiable "on a set U" if and only if it is differentiable at every point in U!
     
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