i am not at this level, but still trying my best. infact i saw the following worked for the first problem.
let me solve part b) divergence first -let me take omega along z axis=omega k where k is unit vector. taking position vector of a fluid particle as xi + yj +zk , i am finding the velocity using v=omega cross r=(-y omega) i +(x omega)j
now taking divergence, and omega = constant, d/dx(-y omega)+d/dy (x omega)=0 (please note that the derivatives are partial derivatives)=0
for part a)take curl of v=curl of(omega cross r)=0(it is coming zero, use the previous values, u will get terms like(omega- omega) in brackets
write the left hand side using levi civita permutation symbol(see m.l.boas page 511-3rd edition) and will get the proof.
for problem 2, please wait. i am trying to work out.
regarding books on fluid mechanics, i have read only one (off time reading) -feynman lecturess vol2 - u will surely enjoy it - esspecially real fluids where my hero explains turbulance and a fascinating topic called karman vortex street and second coefficient of viscosity.
however, my knowledge is really limited in this regard. i dont know any book on fluids that discusses such problems.  | 
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