Mathematics 70 Online OpenStudy (anonymous):

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 OpenStudy (anonymous):

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

=0

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