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

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</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>.) 
+  
+  A vector, <math>~\bold{\hat{n}}</math>, that is normal to the <math>~\lambda_1</math> = constant surface is given by the gradient of the function,  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~F(x, y, z)</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}  \lambda_1 \, .</math>  
+  </td>  
+  </tr>  
+  </table>  
+  
+  In Cartesian coordinates, this means,  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~\bold{\hat{n}}(x, y, z)</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  \hat\imath \biggl( \frac{\partial F}{\partial x} \biggr)  
+  + \hat\jmath \biggl( \frac{\partial F}{\partial y} \biggr)  
+  + \hat{k} \biggl( \frac{\partial F}{\partial z} \biggr)  
+  </math>  
+  </td>  
+  </tr>  
+  
+  <tr>  
+  <td align="right">  
+   
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  \hat\imath \biggl[ x(x^2 + q^2 y^2 + p^2 z^2)^{ 1 / 2} \biggr]  
+  + \hat\jmath \biggl[ q^2y(x^2 + q^2 y^2 + p^2 z^2)^{ 1 / 2} \biggr]  
+  + \hat\jmath \biggl[ p^2 z(x^2 + q^2 y^2 + p^2 z^2)^{ 1 / 2} \biggr]  
+  </math>  
+  </td>  
+  </tr>  
+  
+  <tr>  
+  <td align="right">  
+   
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  \hat\imath \biggl( \frac{x}{\lambda_1} \biggr)  
+  + \hat\jmath \biggl( \frac{q^2y}{\lambda_1} \biggr)  
+  + \hat\jmath \biggl(\frac{p^2 z}{\lambda_1} \biggr) \, ,  
+  </math>  
+  </td>  
+  </tr>  
+  </table>  
+  where it is understood that this expression is only to be evaluated at points, <math>~(x, y, z)</math>, that lie on the selected <math>~\lambda_1</math> surface — that is, at points for which the function, <math>~F(x,y,z) = 0</math>. The length of this normal vector is given by the expression,  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~[ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2}</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  \biggl[ \biggl( \frac{\partial F}{\partial x} \biggr)^2 + \biggl( \frac{\partial F}{\partial y} \biggr)^2 + \biggl( \frac{\partial F}{\partial z} \biggr)^2 \biggr]^{1 / 2}  
+  </math>  
+  </td>  
+  </tr>  
+  </table>  
+  
+  
+  
+  The properly normalized  
+  
+  Next, we appreciate that the vector that is normal to theWhat 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,  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, 
Revision as of 10:09, 27 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, .)
A vector, , that is normal to the = constant surface is given by the gradient of the function,



In Cartesian coordinates, this means,









where it is understood that this expression is only to be evaluated at points, , that lie on the selected surface — that is, at points for which the function, . The length of this normal vector is given by the expression,



The properly normalized
Next, we appreciate that the vector that is normal to theWhat is the expression for the unit vector normal to the surface at when written in terms of Cartesian unit vectors?
Well, to start with we know that is constant across the entire surface, so at any point on this specified surface we must find,



See Also
© 2014  2020 by Joel E. Tohline 