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1. In the notation of our decomposition of local flow as a translation + deformation + rigid rotation, show explicitly that S.h = &xi x h / 2 .
2.
(a) Decompose the velocity field 
near the point [2,2,999] as a translation, a rigid rotation, and a deformation.
(b) What is the translation speed?
(c) What is the vorticity, and what is the rotation rate (radians per unit time)?
(d) Is the fluid locally expanding or contracting at this point? At what rate?
3. In the proof of Kelvin's circulation theorem Thursday we learned something that allows us to answer the following question. There are certainly infinitely many different isentropic inviscid flows possible. But can any field u(r,t) we dream up represent the velocity field of an isentropic inviscid flow (if we allow ourselves the freedom to choose the density and pressure fields at will)? For example, how about the velocity field u(r,t) = [xz,yt,2x]? Hint: Start by computing Du/Dt.
4. Grads only. Is there an incompressible, uniform-density, inviscid, steady, radial, non-zero, flow u(r) on the punctured plane R2-{[0,0]}? If so, find an example of such a field u(r), and the corresponding pressure field p(r). Hints: (i) for incompressible flow div u = 0 (follows from continuity and D&rho / Dt = 0), (ii) you may need to look up the form of grad in polar coordinates.