3Dconfigurations/RiemannEllipsoids - Revision history

Jet53man: /* First Three Governing Relation */

2022-02-24T18:49:41Z

First Three Governing Relation

← Older revision		Revision as of 14:49, 24 February 2022
Line 2,507:		Line 2,507:
	<tr>		<tr>
	<td align="center"><math>(A, B, C)</math></td>		<td align="center"><math>(A, B, C)</math></td>
	<td align="center"><math>(A_1, ~~B_1~~, ~~C_1~~)</math></td>		<td align="center"><math>(A_1, A_2, A_3)</math></td>
	</tr>		</tr>
	<tr>		<tr>

2021-06-27T01:58:45Z

v and w Sums and Differences

@@ Line 1,975: / Line 1,975: @@
 (4a^2  - b^2 + c^2)
 \pm [16a^4 - 8a^2b^2  + b^4 - 2b^2c^2 + c^4 - 8a^2c^2]^{1 / 2}\biggr\}^{1 / 2}
-  S^{1 / 2}
+<tr>
 </math>
    </td>
 </tr>
 </table>
 Similarly,

2021-06-27T01:26:54Z

v and w Sums and Differences

@@ Line 1,945: / Line 1,945: @@
 (4a^2  - b^2 + c^2)
 \pm [(4a^2  - b^2 + c^2) - 4ac]^{1 / 2}[(4a^2 - b^2 + c^2) + 4ac]^{1 / 2} \biggr\}^{1 / 2}
-  S^{1 / 2} \, .
+  S^{1 / 2}
 </math>
    </td>

2021-06-27T01:15:23Z

v and w Sums and Differences

@@ Line 2,144: / Line 2,144: @@
 \pm [ (4a^2 + b^2 - c^2) - 4ab ]^{1 / 2}[(4a^2 + b^2 -c^2)  + 4ab]^{1 / 2} \biggr\}^{1 / 2}
   \biggl[T^{1 / 2}\biggr] \, .
 </math>
    </td>

2021-06-27T01:08:10Z

v and w Sums and Differences

@@ Line 2,050: / Line 2,050: @@
    <td align="left">
 <math>
-- \biggl\{1 \mp \biggl[
+\biggl\{\pm \biggl[
 \frac{ (2a-b-c)(2a - b + c) }{ (2a + b + c)(2a + b - c) }
-\biggr]^{1 / 2}  \biggr\}
+\biggr]^{1 / 2} -1 \biggr\}
 (2a + b + c)^{1 / 2} (2a + b - c)^{1 / 2} \biggl[\pm T^{1 / 2}\biggr]
 </math>
@@ Line 2,065: / Line 2,065: @@
    <td align="left">
 <math>
-\biggl\{\pm \biggl[
+\biggl\{\biggl[
 \frac{ (2a-b-c)(2a - b + c) }{ (2a + b + c)(2a + b - c) }
-\biggr]^{1 / 2} -1 \biggr\}
+\biggr]^{1 / 2} \pm 1 \biggr\}
-(2a + b + c)^{1 / 2} (2a + b - c)^{1 / 2} \biggl[\pm T^{1 / 2}\biggr]
+(2a + b + c)^{1 / 2} (2a + b - c)^{1 / 2} \biggl[T^{1 / 2}\biggr]
 </math>
    </td>

2021-06-26T22:20:25Z

v and w Sums and Differences

@@ Line 2,007: / Line 2,007: @@
 <math>
 (v' - v) \, .
 </math>
    </td>

2021-06-26T21:40:16Z

v and w Sums and Differences

@@ Line 1,902: / Line 1,902: @@
 [(2a - c)^2 - b^2]^{1 / 2} \pm [(2a + c)^2 - b^2]^{1 / 2} \biggr\}
   S^{1 / 2}
 </math>
    </td>

2021-06-26T21:10:39Z

v and w Sums and Differences

@@ Line 1,787: / Line 1,787: @@
 ====''v'' and ''w'' Sums and Differences====
-We also find that,
 ====First Three Governing Relation====

2021-06-26T20:35:40Z

Case when u = u' = 0

← Older revision		Revision as of 16:35, 26 June 2021
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	</tr>		</tr>
	</table>		</table>

			====''v'' and ''w'' Ratios====

	From the first modification (mod.01) of equation <math>(\alpha.4)</math>, we find that,		From the first modification (mod.01) of equation <math>(\alpha.4)</math>, we find that,

Line 1,783:		Line 1,786:
	As Riemann (1961) points out — see the top of p. 184 of [http://www.kendrickpress.com/Riemann.htm BCO2004] — <font color="orange">"Taking this together with</font> the constant volume constraint <math>(\alpha.10)</math> <font color="orange">we see that <math>a, b, c</math> are constant, and consequently <math>v, v', w, w'</math> are constant."</font>		As Riemann (1961) points out — see the top of p. 184 of [http://www.kendrickpress.com/Riemann.htm BCO2004] — <font color="orange">"Taking this together with</font> the constant volume constraint <math>(\alpha.10)</math> <font color="orange">we see that <math>a, b, c</math> are constant, and consequently <math>v, v', w, w'</math> are constant."</font>

			====''v'' and ''w'' Sums and Differences====
			We also find that,

			====First Three Governing Relation====
	Given that <math>a, b, c</math> are constant and <math>u = u' = 0</math>, the first three governing equations become,		Given that <math>a, b, c</math> are constant and <math>u = u' = 0</math>, the first three governing equations become,

2021-06-26T16:46:56Z

Step 3

@@ Line 1,043: / Line 1,043: @@
 \biggl\{ [(2a + c)^2 - b^2]^{1 / 2} +[(2a - c)^2 - b^2]^{1 / 2}
 \biggr\}~S^{1 / 2}
 </math>
    </td>