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Gradient and open ball

  1. Mar 9, 2006 #1
    Let f : Rn -> R.
    Suppose that grad(f(x)) = 0 for all x in some open ball B(a, r).
    Show that f is constant on B(a, r).
    [Hint: use part (a) to make this a problem about a function of one variable]

    part (a) is show that for any two points x, y in B
    there is a straight line starting at x and ending at y that is contained
    in B, which I got, but I don't understand what it has to do with anything. Isn't this just a property of the gradient?

    Any help would be greatly appreciated.
  2. jcsd
  3. Mar 9, 2006 #2


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    Well, try the one-variable case first. Suppose you have a differentiable function f:R->R and f'(x)=0 on some open interval (a,b). Show that f is constant on (a,b).

    Then, note that grad(f)(x)=0 are three equations and mix it with the result of part (a).
  4. Mar 9, 2006 #3


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    If the gradient is 0 at every point, then the derivative along any line is 0.
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