Let Br be the open ball of radius r and center 0 in R³ and Sr be the sphere of radius r and center 0 in R³. Denote by r the volume form on Sr . Same notation with superscript x for the open ball of radius r and center x in R³, the sphere of radius r and center x in R³ and the volume form on Srx .

Let f : B1 R be a smooth function with f = 0

a. Prove that g(x):=x1 satisfies g=0 on B₁ \ {0}.

b. If g is a smooth function satisfying g=0 on B₁ \ {0}, prove that r1Sr(f(xg)g(xf))r is independent of r for 0