Hi guys, any help with the following questions will be very much appreciated!(adsbygoogle = window.adsbygoogle || []).push({});

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!

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# General derivatives and smooth functions

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