| … | … | @@ -7261,20 +7261,66 @@ and solve it component by component. |
| 7261 | 7261 | \end_layout |
| 7262 | 7262 | |
| 7263 | 7263 | \begin_layout Standard |
| 7264 | | So |
| 7265 | | \begin_inset Formula $0=\omega_{\;b}^{1}\wedge e^{b}$ |
| 7264 | So then |
| 7265 | \begin_inset Formula |
| 7266 | \begin{align*} |
| 7267 | 0 & =de^{2}+\omega_{\;b}^{2}\wedge e^{b}\\ |
| 7268 | & =de^{2}+\omega_{\;1}^{2}\wedge e^{1}+\omega_{\;2}^{2}\wedge e^{2}+\omega_{\;3}^{2}\wedge e^{3}\\ |
| 7269 | & =\frac{1}{\lambda r}\left(\frac{1+r^{2}}{1-r^{2}}\right)e^{1}\wedge e^{2}+\omega_{\;1}^{2}\wedge e^{1}+\omega_{\;2}^{2}\wedge e^{2}+\omega_{\;3}^{2}\wedge e^{3} |
| 7270 | \end{align*} |
| 7271 | |
| 7266 | 7272 | \end_inset |
| 7267 | 7273 | |
| 7268 | | means that |
| 7274 | so conclude |
| 7275 | \begin_inset Formula |
| 7276 | \[ |
| 7277 | \omega_{\;1}^{2}=\frac{1}{\lambda r}\left(\frac{1+r^{2}}{1-r^{2}}\right)e^{2} |
| 7278 | \] |
| 7279 | |
| 7280 | \end_inset |
| 7281 | |
| 7282 | |
| 7283 | \end_layout |
| 7284 | |
| 7285 | \begin_layout Standard |
| 7286 | The next one is |
| 7269 | 7287 | \begin_inset Formula |
| 7270 | 7288 | \begin{align*} |
| 7271 | | 0 & =\omega_{\;1}^{1}\wedge e^{1}+\omega_{\;2}^{1}\wedge e^{2}+\omega_{\;3}^{1}\wedge e^{3}\\ |
| 7272 | | & = |
| 7289 | 0 & =de^{3}+\omega_{\;1}^{3}\wedge e^{1}+\omega_{\;2}^{3}\wedge e^{2}+\omega_{\;3}^{3}\wedge e^{3}\\ |
| 7290 | & =\frac{1}{\lambda r}\left[\left(\frac{1+r^{2}}{1-r^{2}}\right)\;e^{1}\wedge e^{3}+\frac{\cos\theta}{\sin\theta}\;e^{2}\wedge e^{3}\right]+\omega_{\;1}^{3}\wedge e^{1}+\omega_{\;2}^{3}\wedge e^{2}+\omega_{\;3}^{3}\wedge e^{3} |
| 7273 | 7291 | \end{align*} |
| 7274 | 7292 | |
| 7275 | 7293 | \end_inset |
| 7276 | 7294 | |
| 7295 | which gives |
| 7296 | \begin_inset Formula |
| 7297 | \[ |
| 7298 | \omega_{\;1}^{3}=\frac{1}{\lambda r}\left(\frac{1+r^{2}}{1-r^{2}}\right)e^{3} |
| 7299 | \] |
| 7300 | |
| 7301 | \end_inset |
| 7302 | |
| 7303 | and |
| 7304 | \begin_inset Formula |
| 7305 | \[ |
| 7306 | \omega_{\;2}^{3}=\frac{1}{\lambda r}\cdot\frac{\cos\theta}{\sin\theta}e^{3} |
| 7307 | \] |
| 7308 | |
| 7309 | \end_inset |
| 7310 | |
| 7311 | |
| 7312 | \end_layout |
| 7313 | |
| 7314 | \begin_layout Standard |
| 7315 | Finally |
| 7316 | \begin_inset Formula $0=\omega_{\;b}^{1}\wedge e^{b}$ |
| 7317 | \end_inset |
| 7318 | |
| 7319 | means that |
| 7320 | \begin_inset Formula $0=\omega_{\;1}^{1}\wedge e^{1}+\omega_{\;2}^{1}\wedge e^{2}+\omega_{\;3}^{1}\wedge e^{3}$ |
| 7321 | \end_inset |
| 7277 | 7322 | |
| 7323 | but this is trivially satisfied by the above results. |
| 7278 | 7324 | \end_layout |
| 7279 | 7325 | |
| 7280 | 7326 | \begin_layout Paragraph* |