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General derivatives and smooth functions

  1. Sep 5, 2011 #1
    Hi guys, any help with the following questions will be very much appreciated!

    1. Prove that u(x,y) = ln(x^2+y^2) has a generalised derivative on the unit ball in R^2 (Approximate by smooth functions?).

    2. Prove that g.u has a generalised derivative on A if:
    A is bounded and open in R^n
    g is a smooth function in R^n
    u is measurable, u is in L(1)loc(A) and has a generalised derivative on A.
    (Thinking of using the formula for the derivative of a product of smooth functions?)

    3. Construct a smooth function fn on R such that:
    fn(x) = 1 if x<n (includes equality)
    fn(x) = 0 if x>n+1 (includes equality)
    0<fn(x)<1 for all x in R (includes equality)
    the derivative of fn is bounded

    Thanks alot!
     
  2. jcsd
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