Difference between revisions of "User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates"
(→MF53) |
(→MF53) |
||
| Line 117: | Line 117: | ||
</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~\ | <math>~ (\xi_1^2 - d^2)^{- 1 } [ (\xi_1^2 - d^2)\xi_2^2 +\xi_1^2 (1-\xi_2^2) ]</math> | ||
</td> | |||
</tr> | |||
<tr> | |||
<td align="right"> | |||
| |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ \biggl[ \frac{ \xi_1^2 - d^2 \xi_2^2 }{\xi_1^2 - d^2} \biggr] \, ;</math> | |||
</td> | </td> | ||
</tr> | </tr> | ||
Revision as of 16:45, 15 October 2020
Elliptic Cylinder Coordinates
|
|---|
| | Tiled Menu | Tables of Content | Banner Video | Tohline Home Page | |
Building on our general introduction to Direction Cosines in the context of orthogonal curvilinear coordinate systems, here we detail the properties of Elliptic Cylinder Coordinates. First, we will present this coordinate system in the manner described by [MF53]; second, we will provide an alternate presentation, obtained from Wikipedia; then, third, we will investigate whether or not a related coordinate system based on concentric (rather than confocal) elliptic surfaces can be satisfactorily described.
MF53
From MF53's Table of Separable Coordinates in Three Dimensions (see their Chapter 5, beginning on p. 655), we find the following description of Elliptic Cylinder Coordinates (p. 657).
|
<math>~x</math> |
<math>~=</math> |
<math>~\xi_1 \xi_2 \, ;</math> |
|
<math>~y</math> |
<math>~=</math> |
<math>~\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2) \biggr]^{1 / 2} \, ;</math> |
|
<math>~z</math> |
<math>~=</math> |
<math>~\xi_3 \, .</math> |
Appreciating that,
|
<math>~\frac{\partial y}{\partial \xi_1}</math> |
<math>~=</math> |
<math>~ +\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2) \biggr]^{- 1 / 2}\xi_1(1-\xi_2^2) \, , </math> and that, |
|
<math>~\frac{\partial y}{\partial \xi_2}</math> |
<math>~=</math> |
<math>~ - \biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2) \biggr]^{- 1 / 2}\xi_2(\xi_1^2 - d^2) \, , </math> |
we find that the respective scale factors are given by the expressions,
|
<math>~ h_1^2</math> |
<math>~=</math> |
<math>~\biggl(\frac{\partial x}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial y}{\partial\xi_1} \biggr)^2 + \biggl(\frac{\partial z}{\partial\xi_1} \biggr)^2 </math> |
|
|
<math>~=</math> |
<math>~\xi_2^2 +\biggl[ (\xi_1^2 - d^2)(1 - \xi_2^2) \biggr]^{- 1 }\xi_1^2 (1-\xi_2^2)^2 </math> |
|
|
<math>~=</math> |
<math>~ (\xi_1^2 - d^2)^{- 1 } [ (\xi_1^2 - d^2)\xi_2^2 +\xi_1^2 (1-\xi_2^2) ]</math> |
|
|
<math>~=</math> |
<math>~ \biggl[ \frac{ \xi_1^2 - d^2 \xi_2^2 }{\xi_1^2 - d^2} \biggr] \, ;</math> |
See Also
|
|
|---|
|
© 2014 - 2021 by Joel E. Tohline |