( D ) is the solid inside the cylinder ( x^{2}+y^{2}=9 ), between the planes ( z=7 ) and ( z=x+17 ). Use the Divergence Theorem to find the flux of the vector field ( mathbf{F}(x, y, z)=leftlangle x y-sqrt{y^{2}+z^{2}}, 5 y+e^{z^{3}}, 3 z-x^{2} cot yrightrangle ) over the surface of ( D ). Consider cylindrical coordinates. Write the exact answer. Do not round.

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Step:1 ：Define cylindrical coordinates transformation: ((x, y, z) = (r cos theta, r sin theta, z)) with (0 le r le 3), (0 le theta le 2 pi), (7 le z le r cos theta + 17 ).Step:2 ：Find the divergence of ( mathbf{F} ): (operatorname{div}(mathbf{F}) = nabla cdot mathbf{F} = frac{partial}{partial x}(xy-sqrt{y^{2}+z^{2}}) + frac{partial}{partial y}(5y+e^{z^{3}}) + frac{partial}{partial z}(3z-x^{2}cot y)).Step:3 ：Calculate the triple integral using Divergence Theorem: ( iiint_D operatorname{div}(mathbf{F}) , dV = int_0^{2pi} int_0^3 int_7^{rcostheta + 17} operatorname{div}(mathbf{F}) , r , dz , dr , dtheta ).

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Step: 1 ：Define cylindrical coordinates transformation: ((x, y, z) = (r cos theta, r sin theta, z)) with (0 le r le 3), (0 le theta le 2 pi), (7 le z le r cos theta + 17 ).Step: 2 ：Find the divergence of ( mathbf{F} ): (operatorname{div}(mathbf{F}) = nabla cdot mathbf{F} = frac{partial}{partial x}(xy-sqrt{y^{2}+z^{2}}) + frac{partial}{partial y}(5y+e^{z^{3}}) + frac{partial}{partial z}(3z-x^{2}cot y)).Step: 3 ：Calculate the triple integral using Divergence Theorem: ( iiint_D operatorname{div}(mathbf{F}) , dV = int_0^{2pi} int_0^3 int_7^{rcostheta + 17} operatorname{div}(mathbf{F}) , r , dz , dr , dtheta ).

D is the solid inside the cylinder x2+y2=9, between the planes z=7 and z=x+17. Use the Divergence Theorem to find the flux of the vector field F(x,y,z)=⟨xy−y2+z2,5y+ez3,3z−x2cot⁡y⟩ over the surface of D. Consider cylindrical coordinates. Write the exact answer. Do not round.

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