Show That the Vector Field $\vec { F } = - 3 x e ^ { x } \vec { i } - y ( 3 z - 5 ) e ^ { x } \vec{ j } + ( 3 z - 5 ) e ^ { x } \vec { k }$

question 55

Essay

Show that the vector field $\vec { F } = - 3 x e ^ { x } \vec { i } - y ( 3 z - 5 ) e ^ { x } \vec{ j } + ( 3 z - 5 ) e ^ { x } \vec { k }$ is a divergence free vector field.
Use this result to calculate $\hat { \mathrm {O } } _ {S}{ \vec { F } \times \vec { dA } }$ where S is the open surface which is the graph of $f ( x , y ) = - x ^ { 2 } - y ^ { 2 } + 4$ with f(x, y) $\ge$ 0.

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