Let $\vec { F }$ Be a Vector Field Such That $\operatorname { div } { \vec { F } } = 4 - 2 x ^ { 2 } - 2 y ^ { 2 } - 2 z ^ { 2 }$

question 50

Essay

Let $\vec { F }$ be a vector field such that $\operatorname { div } { \vec { F } } = 4 - 2 x ^ { 2 } - 2 y ^ { 2 } - 2 z ^ { 2 }$ (a)Is $\vec { F }$ a curl field?
(b)Use spherical coordinates to evaluate $\int _ { W } \operatorname { div } \vec { F } d V$ where W is the solid ball of radius R centered at the origin.
(c)Use the result of part (b)to find the radius of a sphere centered at the origin, such that the flux of $\vec { F }$ out of this sphere is zero.

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