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Unformatted text preview: 4. The Euler equation is ∂ u ∂t + ( u · ∇ ) u =1 ρ ∇ p, where u = ( u,v,w ) is the velocity and p the pressure. Write out the second component of the Euler equation in components. 5. Simplify (i) ± ijk ± ijm ; (ii) ± ijk ± ijk . 6. Demonstrate that for φ ( x ), a scalar function of position ∇ × ∇ φ = 0 7. Prove the vector identity, for vector ﬁeld u ( x ) and v ( x ), ∇ × ( u × v ) = u ( ∇ · v )v ( ∇ · u ) + ( v · ∇ ) u( u · ∇ ) v 8. Show that for a vector ﬁeld u ( x ) u · ∇ (  u  2 / 2) = u · ( u · ∇ ) u 9. For each of the following functions, φ ( x ), ﬁnd ∇ φ and ∇ · ∇ φ ≡ ∇ 2 φ ≡ 4 φ using suﬃx notation and the summation convention. (i) φ = f ( r ); f is an arbitrary scalar function (ii) φ = a · x /r 3 , where a is a constant vector and r 2 = x · x ....
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 Fall '11
 Eggers
 Cartesian Coordinate System, Leonhard Euler, Eulerian velocity field, Lagrangian velocity ﬁeld

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