For an incompressible fluid flow, two components of velocity (u, v, w) are given by u = -x^{2} - 2yz - 3z^{2}, v = x^{2} - y^{2} + zx. Determine the third component w so that they satisfy the equation of continuity. Also, find the z-component of acceleration.

3Medium10 marks

Prove that the function f(x, y) = x^{3} - 3xy^{2} - 3y^{2} is harmonic and find its harmonic conjugate and the corresponding analytic function f(z) in terms of z.

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