Term Test 2   -   Solutions

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For the vector field defined by   ,

1. Find the divergence of    

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2. Hence find the total flux     of     through the sphere of radius 1, centre (0, 0, 0).

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   Using Gauss’ divergence theorem,
   

   

   [Note that the details of the sphere are irrelevant - for this source-free vector field, the net flux through any simple closed surface will be zero!]

In the x-y plane a vector field is defined by .

1. Use Green’s theorem to find the work done by     in one circuit around the unit square (vertices (0, 0), (1, 0), (1, 1) and (0, 1)).

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   Green’s theorem states that, in any simply connected domain,
   
   where   C   is a simple closed path that encloses the region   D.
   
   
   But   D   is a unit square, whose area is   A = 1.   Therefore

   

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2. Is the vector field     conservative?   Why or why not?

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   NO,   F   is not conservative.

The velocity field of a fluid is   .   Find the flux   Q   of the fluid through the hemisphere     in a direction outward from its centre.
[Hint:   use a spherical polar coordinate grid   ,   with  

But, everywhere on the hemisphere,     are all positive.
For an outward pointing normal vector (pointing away from the origin), we must select the '+' sign.




Therefore the flux is

  BONUS QUESTION

Find the potential function     for     such that     everywhere along the y-axis.

But     everywhere along the y-axis

Therefore the required potential function is