User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates
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(→Concentric Ellipsoidal (T6) Coordinates) 
(→Concentric Ellipsoidal (T6) Coordinates) 

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==Background==  ==Background==  
Building on our [[User:Tohline/Appendix/Ramblings/DirectionCosinesgeneral introduction to ''Direction Cosines'']] in the context of orthogonal curvilinear coordinate systems, and on our previous development of [[User:Tohline/Appendix/Ramblings/T3IntegralsT3]] (concentric oblatespheroidal) and [[User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates#T5_CoordinatesT5]] (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T6) coordinate system. This is motivated by our [[User:Tohline/ThreeDimensionalConfigurations/Challenges#Trial_.232desire to construct a fully analytically prescribable model of a nonuniformdensity ellipsoidal configuration that is an analog to Riemann SType ellipsoids]].  Building on our [[User:Tohline/Appendix/Ramblings/DirectionCosinesgeneral introduction to ''Direction Cosines'']] in the context of orthogonal curvilinear coordinate systems, and on our previous development of [[User:Tohline/Appendix/Ramblings/T3IntegralsT3]] (concentric oblatespheroidal) and [[User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates#T5_CoordinatesT5]] (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T6) coordinate system. This is motivated by our [[User:Tohline/ThreeDimensionalConfigurations/Challenges#Trial_.232desire to construct a fully analytically prescribable model of a nonuniformdensity ellipsoidal configuration that is an analog to Riemann SType ellipsoids]].  
+  
+  ==Orthogonal Coordinates==  
+  
+  We start by defining a "radial" coordinate whose values identify various concentric ellipsoidal shells,  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~\lambda_1</math>  
+  </td>  
+  <td align="center">  
+  <math>~\equiv</math>  
+  </td>  
+  <td align="left">  
+  <math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} \, .</math>  
+  </td>  
+  </tr>  
+  </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>.)  
=See Also=  =See Also= 
Revision as of 08:17, 26 October 2020
Contents 
Concentric Ellipsoidal (T6) Coordinates
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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 oblatespheroidal) 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 nonuniformdensity ellipsoidal configuration that is an analog to Riemann SType ellipsoids.
Orthogonal Coordinates
We start by defining a "radial" coordinate whose values identify various concentric ellipsoidal shells,



When , we obtain the standard definition of an ellipsoidal surface, it being understood that, and . (We will assume that , that is, .)
See Also
© 2014  2020 by Joel E. Tohline 