Question

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

Answer 1

b) \[divergence(f)(x) = \sum_{i=1}^{n} df_i/dx_i \]

so divergence of F = \[d(-(y+xz)/(z^2+1)^2)/dx + d(-(yz-x)/(z^2+1)^2)/dy+d(-1/(z^2+1))/dz\]

= \[-z/(z^2+1)^2 -z/(z^2+1)^2 +2z/(z^2+1)^2\]

Answer 2

=0