User:Tohline/Appendix/Ramblings/ConcentricEllipsodalDaringAttack

Daring Attack

Background

Building on our general introduction to Direction Cosines in the context of orthogonal curvilinear coordinate systems, and on our previous development of the so-called T6 (concentric elliptic) coordinate system, here we take a somewhat daring attack on this problem, mixing our approach to identifying the expression for the third curvilinear coordinate. Broadly speaking, this entire study 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.

Direction Cosine Components for T6 Coordinates
<math>~n</math>	<math>~\lambda_n</math>	<math>~h_n</math>	<math>~\frac{\partial \lambda_n}{\partial x}</math>	<math>~\frac{\partial \lambda_n}{\partial y}</math>	<math>~\frac{\partial \lambda_n}{\partial z}</math>	<math>~\gamma_{n1}</math>	<math>~\gamma_{n2}</math>	<math>~\gamma_{n3}</math>
<math>~1</math>	<math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} </math>	<math>~\lambda_1 \ell_{3D}</math>	<math>~\frac{x}{\lambda_1}</math>	<math>~\frac{q^2 y}{\lambda_1}</math>	<math>~\frac{p^2 z}{\lambda_1}</math>	<math>~(x) \ell_{3D}</math>	<math>~(q^2 y)\ell_{3D}</math>	<math>~(p^2z) \ell_{3D}</math>
<math>~2</math>	---	---	---	---	---	<math>~\ell_q \ell_{3D} (xp^2z)</math>	<math>~\ell_q \ell_{3D} (q^2 y p^2z) </math>	<math>~- (x^2 + q^4y^2)\ell_q \ell_{3D}</math>
<math>~3</math>	<math>~\tan^{-1}\biggl( \frac{y^{1/q^2}}{x} \biggr)</math>	<math>~\frac{xq^2 y \ell_q}{\sin\lambda_3 \cos\lambda_3}</math>	<math>~-\frac{\sin\lambda_3 \cos\lambda_3}{x}</math>	<math>~+\frac{\sin\lambda_3 \cos\lambda_3}{q^2y}</math>	<math>~0</math>	<math>~-q^2 y \ell_q</math>	<math>~x\ell_q</math>	<math>~0</math>
<math>~\ell_{3D}</math> <math>~\equiv</math> <math>~[x^2 + q^4 y^2 + p^2 z^2]^{- 1/ 2 }</math> <math>~\ell_q</math> <math>~\equiv</math> <math>~[x^2 + q^4 y^2 ]^{- 1/ 2 }</math>

New Approach

As before, let's adopt the first-coordinate expression,

<math>~\lambda_1</math>

<math>~\equiv</math>

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

but for the third-coordinate expression we will abandon the trigonometric expression and instead simply use,

<math>~\lambda_3</math>

<math>~\equiv</math>

<math>~\frac{y^{1/q^2}}{x} \, .</math>

This modified third-coordinate expression means that the last row of the above table changes, as follows.

Daring Attack
<math>~n</math>	<math>~\lambda_n</math>	<math>~h_n</math>	<math>~\frac{\partial \lambda_n}{\partial x}</math>	<math>~\frac{\partial \lambda_n}{\partial y}</math>	<math>~\frac{\partial \lambda_n}{\partial z}</math>	<math>~\gamma_{n1}</math>	<math>~\gamma_{n2}</math>	<math>~\gamma_{n3}</math>
<math>~1</math>	<math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} </math>	<math>~\lambda_1 \ell_{3D}</math>	<math>~\frac{x}{\lambda_1}</math>	<math>~\frac{q^2 y}{\lambda_1}</math>	<math>~\frac{p^2 z}{\lambda_1}</math>	<math>~(x) \ell_{3D}</math>	<math>~(q^2 y)\ell_{3D}</math>	<math>~(p^2z) \ell_{3D}</math>
<math>~2</math>	---	---	---	---	---	<math>~\ell_q \ell_{3D} (xp^2z)</math>	<math>~\ell_q \ell_{3D} (q^2 y p^2z) </math>	<math>~- (x^2 + q^4y^2)\ell_q \ell_{3D}</math>
<math>~3</math>	<math>~\frac{y^{1/q^2}}{x} </math>	<math>~\frac{xq^2 y \ell_q}{\lambda_3}</math>	<math>~-\frac{\lambda_3}{x}</math>	<math>~+\frac{\lambda_3}{q^2y}</math>	<math>~0</math>	<math>~-q^2 y \ell_q</math>	<math>~x\ell_q</math>	<math>~0</math>

Notice that the direction cosine functions for the (as yet, unknown) second-coordinate function remain the same. This is because the direction-cosine functions associated with both <math>~\lambda_1</math> and <math>~\lambda_3</math> remain unchanged, so it must be true that the cross product of the first and third unit vectors leads to the same components for the second unit vector.

See Also

• Direction Cosines

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation