Difference between revisions of "User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates"

From VistrailsWiki
Jump to navigation Jump to search
Line 25: Line 25:
</tr>
</tr>
</table>
</table>
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>.)
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?
 
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,
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2x dx + 2q^2y dy + 2p^2z dz \, .</math>
  </td>
</tr>
</table>


=See Also=
=See Also=

Revision as of 15:51, 26 October 2020

Concentric Ellipsoidal (T6) Coordinates

Whitworth's (1981) Isothermal Free-Energy Surface
|   Tiled Menu   |   Tables of Content   |  Banner Video   |  Tohline Home Page   |

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,

<math>~\lambda_1</math>

<math>~\equiv</math>

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

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?

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,

<math>~0</math>

<math>~=</math>

<math>~2x dx + 2q^2y dy + 2p^2z dz \, .</math>

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 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