The water from a bathtub flows out of the bath with velocity vector F= -(y+xz/(z²+1)²)i – (yz-x/(z²+1)²)j – (1/z²+1)k (a)The bath drain is a disc of radius 1 in the xy plane centered at the origin. Find the flow t which water comes out of the bathtub. (b)Find the divergence of F (c)Find the flux of water going through a hemisphere of radius 1, centered at the origin which is underneath the xy-plane and oriented downwards. (d)Find ∮G∙dr, where C is the side of the drain oriented counterclockwise when looked at from the top, and G=1/2((y/z²+1)i – (x/z²+1)j – (x²+y²/(z²+1)²)k) e)Compute

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The water from a bathtub flows out of the bath with velocity vector F= -(y+xz/(z²+1)²)i – (yz-x/(z²+1)²)j – (1/z²+1)k (a)The bath drain is a disc of radius 1 in the xy plane centered at the origin. Find the flow t which water comes out of the bathtub. (b)Find the divergence of F (c)Find the flux of water going through a hemisphere of radius 1, centered at the origin which is underneath the xy-plane and oriented downwards. (d)Find ∮G∙dr, where C is the side of the drain oriented counterclockwise when looked at from the top, and G=1/2((y/z²+1)i – (x/z²+1)j – (x²+y²/(z²+1)²)k) e)Compute

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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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b) \[divergence(f)(x) = \sum_{i=1}^{n} df_i/dx_i \] so divergence of F = \[d(-(y+xz)/(z^2+1)^2)/dx + d(-(yz-x)/(z^2+1)^2)/dy+d(-1/(z^2+1))/dz\] = \[-z/(z^2+1)^2 -z/(z^2+1)^2 +2z/(z^2+1)^2\]
=0

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