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(Concentric Ellipsoidal (T6) Coordinates)
(Concentric Ellipsoidal (T6) Coordinates)
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When <math>~\lambda_1 = a</math>, we obtain the standard definition of an ellipsoidal surface, it being understood that, <math>~q^2 = a^2/b^2</math> and <math>~p^2 = a^2/c^2</math>.  (We will assume that <math>~a > b > c</math>, that is, <math>~p^2 > q^2 > 1</math>.)
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When <math>~\lambda_1 = a</math>, we obtain the standard definition of an ellipsoidal surface, it being understood that, <math>~q^2 = a^2/b^2</math> and <math>~p^2 = a^2/c^2</math>.  (We will assume that <math>~a > b > c</math>, that is, <math>~p^2 > q^2 > 1</math>.) What is the expression for the unit vector normal to the surface at <math>~(x, y, z)</math> when written in terms of Cartesian unit vectors?
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Well, to start with we know that <math>~\lambda_1^2</math> is constant across the entire surface, so at any point on this specified surface we must find,
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<math>~0</math>
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<math>~=</math>
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<math>~2x dx + 2q^2y dy + 2p^2z dz \, .</math>
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=See Also=
=See Also=

Revision as of 08:51, 26 October 2020

Contents

Concentric Ellipsoidal (T6) Coordinates

Whitworth's (1981) Isothermal Free-Energy Surface
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Background

Building on our general introduction to Direction Cosines in the context of orthogonal curvilinear coordinate systems, and on our previous development of T3 (concentric oblate-spheroidal) and T5 (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T6) coordinate system. This is motivated by our desire to construct a fully analytically prescribable model of a nonuniform-density ellipsoidal configuration that is an analog to Riemann S-Type ellipsoids.

Orthogonal Coordinates

We start by defining a "radial" coordinate whose values identify various concentric ellipsoidal shells,

~\lambda_1

~\equiv

~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} \, .

When ~\lambda_1 = a, we obtain the standard definition of an ellipsoidal surface, it being understood that, ~q^2 = a^2/b^2 and ~p^2 = a^2/c^2. (We will assume that ~a > b > c, that is, ~p^2 > q^2 > 1.) What is the expression for the unit vector normal to the surface at ~(x, y, z) when written in terms of Cartesian unit vectors?

Well, to start with we know that ~\lambda_1^2 is constant across the entire surface, so at any point on this specified surface we must find,

~0

~=

~2x dx + 2q^2y dy + 2p^2z dz \, .

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

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