Editing ThreeDimensionalConfigurations/Stability/RiemannEllipsoids (section)

===Strategy===

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"Let <math>\mathbf{u}(\mathbf{x}), p(\mathbf{x}), \rho(\mathbf{x})</math> represent the velocity field, pressure, and density, respectively, of an inviscid fluid mass in a steady state relative to a reference frame rotating with angular velocity <math>\boldsymbol{\omega} = \omega \mathbf{e}_3</math> about an axis fixed in space (the z-, or x<sub>3</sub>-, axis) &hellip; The stability of this steady state is determined, in linear approximation, by the solutions, with arbitrary initial data, of the &hellip; equation [governing the time-dependent behavior of] the Lagrangian displacement <math>\boldsymbol\xi</math>."
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— Drawn from the first paragraph of &sect;2 (p. 226) in {{ Lebovitz89b }}. 
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"This basic equation
</font>[is of the form],"  &nbsp; &nbsp; <math>\boldsymbol\xi_{tt} + A \boldsymbol\xi_t + B\boldsymbol\xi + \rho^{-1}\nabla \Delta p = 0 </math> &hellip; Eq. (10).
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— Drawn from the first paragraph of &sect;3.1 (p. 701) in {{ LL96 }}. 
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"We introduce for the solution space <math>\Sigma</math> a basis <math>\{\xi_i\}</math> the first <math>N</math> vectors <math>\{\xi\}_{i=1}^N</math> of which represent a basis for <math>\Sigma_n</math>, the space of solenoidal vector polynomials of degree not exceeding <math>n</math>, as in {{ Lebovitz89ahereafter }}, {{ Lebovitz89bhereafter }}. It is easily found (see {{ Lebovitz89ahereafter }}) that <math>N = N(n) = (n+1)(n+2)(2n+9)/6</math>.  Since <math>\Sigma_n</math> is invariant under the operators <math>A</math> and <math>B</math>, we seek solutions of Eq. (10) in this space:"</font><br />
<div align="center"><math>\boldsymbol\xi(\mathbf{x}, t) = \sum_{i=1}^{N} c_i(t) \xi_i</math> &hellip; Eq. (18)</div>
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— Drawn from the first paragraph of &sect;3.2 (p. 703) in {{ LL96 }}. 
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Here we will closely follow the derivation found in {{ Lebovitz89afull }}, hereafter {{ Lebovitz89ahereafter }}.