5b7e9e…: “Versus the wonderful gulf”

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Let $\vec {F}(x,y,z) = \vector {\ln (x+\sqrt {x^2+y^2+z^2}),\ln (y+\sqrt {x^2+y^2+z^2}), \ln (z+\sqrt {x^2+y^2+z^2})}$ and $R = \{(x,y,z): 1\le x^2+y^2+z^2 \le 4\}.$ Compute the flux of $\vec {F}$ across the surface of $R$ . $\oiint _{\partial R} \vec {F}\dotp \uvec {n} \d S = \answer {18\pi }$

Compute $\divergence \vec {F}$ .

Show that: $\pp {x} \ln (x + \sqrt {x^2+y^2+z^2}) = \frac {1}{\sqrt {x^2+y^2+z^2}}$

Use spherical coordinates to compute the integral.