2c1253…: “Upside a bitter water”

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Let $\vec {F}(x,y,z) = \vector {4x+y^2-yz,x^2-2y+xz,xy^3+z}$ and $R$ be the cone with a cap shown below:

Compute the flux of $\vec {F}$ across the surface of $R$ . $\oiint _{\partial R} \vec {F}\dotp \uvec {n} \d S = \answer {32(\sqrt {2}-1)\pi }$