(1) Prove that S : = SijOp = 0 if Sij is a symmetric and j is an anti-symmetric tensor. (2) Prove that x E7) = 0 and 17 x u) 0 for any scalar o(x) and vector u(x) finetions, i.e., curl of a gradient field is zero and eurl of a veetor feld is divergence free (o solenoidal) (3) Prove that u : u-S : S--w2 where u is the fluid velocity vector, S is the rate of strain tensor and w is the fluid vorticity squared (or enstrophy) w2 = ww. (4) Prove that- x w = 2u for an incompressible flow (u = 0) where w and u are the fluid vorticity and velocity vectors, respectively. (5) Write the components of (u Vju in egliadrical cordinate system, where u li a vector. (1) Prove that S : = SijOp = 0 if Sij is a symmetric and j is an anti-symmetric tensor. (2) Prove that x E7) = 0 and 17 x u) 0 for any scalar o(x) and vector u(x) finetions, i.e., curl of a gradient field is zero and eurl of a veetor feld is divergence free (o solenoidal) (3) Prove that u : u-S : S--w2 where u is the fluid velocity vector, S is the rate of strain tensor and w is the fluid vorticity squared (or enstrophy) w2 = ww. (4) Prove that- x w = 2u for an incompressible flow (u = 0) where w and u are the fluid vorticity and velocity vectors, respectively. (5) Write the components of (u Vju in egliadrical cordinate system, where u li a vector