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24 May 2023
Let r: Rn âR be the function r(x) ⥠0 sothat
r2 = Σ x2i =||x||2 i=1 to n
(a) Show that âr = xË = 1/||x||*x = (1/r)*x.
(b) Suppose a vector field, F(x) = f(r)*xË, where f is acontinuousfunction.
Show that F = âg for some scalar valued function, g :Rnâ R. Give
a precise formula for an answer when f(r) = 1/r2
(c) Compute the Hessian for the function f(r) =1/r2 whenn = 3.
Let r: Rn âR be the function r(x) ⥠0 sothat
r2 = Σ x2i =||x||2 i=1 to n
(a) Show that âr = xË = 1/||x||*x = (1/r)*x.
(b) Suppose a vector field, F(x) = f(r)*xË, where f is acontinuousfunction.
Show that F = âg for some scalar valued function, g :Rnâ R. Give
a precise formula for an answer when f(r) = 1/r2
(c) Compute the Hessian for the function f(r) =1/r2 whenn = 3.
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