Question

A scalar function f : which is never zero has the properties

R? → R

$\left \| \nabla f \right \|^2 = 8f$ and $\nabla \cdot (f\nabla f ) =17f$

Evaluate the integral

$\iint_{S^2} \frac{\partial f}{\partial n} dS$

where $S^2$ is the surface of the unit sphere
$S^2 = \left \{ (x,y,z)\left | x^2+y^2+z^2=1 \right \}$

and $\frac{\partial f}{\partial n}$ means the directional derivative of f in the direction of the outward pointing unit normal on $S^2$ .

Answer 1

tion- x scalar Function fi R3 R which berer zero otll² = 8t and 7.(+ Of) = 17f thin and = (of.tf)+ f(794) = 174 llo fllt f1024) = 176 → 8f + f(p?f) = 174 7 v?t=9] Now we have to evaluate integral where 52= {(1,1,2)/ 27+9+2²=1} theorem gauss divery once Now using JA 4.ds = NSS 7.(**) av 2n Home Now of the of Il of ds = [ 0.194) dv = 1624) ar on - Jour =9far Now Soir = volyme y sphere = $77() Hence / Slat. ds = 9x411 - 127 An son 1 s2