\([ \boldsymbol{V}_{\boldsymbol{1}}, \boldsymbol{V}_{\boldsymbol{2}}]\) | \(\boldsymbol{X}_{\boldsymbol{1}}\) | \(\boldsymbol{X}_{\boldsymbol{2}}\) | \(\boldsymbol{X}_{\boldsymbol{3}}\) | \(\boldsymbol{X}_{\boldsymbol{4}}\) |
---|---|---|---|---|
\(\boldsymbol{X}_{\boldsymbol{1}}\) | 0 | \(f_{3} ' \frac{\partial }{\partial z} + f_{3}^{\prime \prime } \frac{\partial }{\partial u} + a_{1} \frac{\partial }{\partial p}\) | \(\frac{\partial }{\partial t} + a_{2} \frac{\partial }{\partial z} + a_{3} \frac{\partial }{\partial u} + a_{4} \frac{\partial }{\partial p}\) | \(a_{5} \frac{\partial }{\partial z} + a_{6} \frac{\partial }{\partial u} + a_{7} \frac{\partial }{\partial p}\) |
\(\boldsymbol{X}_{\boldsymbol{2}}\) | \(-( f_{3} ' \frac{\partial }{\partial z} + f_{3}^{\prime \prime } \frac{\partial }{\partial u} + a_{1} \frac{\partial }{\partial p} )\) | 0 | \(-t f_{3} ' \frac{\partial }{\partial z} + a_{8} \frac{\partial }{\partial u} - \frac{2}{r^{2} v} \frac{\partial }{\partial v} + a_{9} \frac{\partial }{\partial p}\) | \(f_{3} \frac{\partial }{\partial z} + f_{3} ' \frac{\partial }{\partial u} + \frac{4}{r^{2} v} \frac{\partial }{\partial v} + a_{10} \frac{\partial }{\partial p}\) |
\(\boldsymbol{X}_{\boldsymbol{3}}\) | \(-( \frac{\partial }{\partial t} + a_{2} \frac{\partial }{\partial z} + a_{3} \frac{\partial }{\partial u} + a_{4} \frac{\partial }{\partial p} )\) | \(-(-t f_{3} ' \frac{\partial }{\partial z} + a_{8} \frac{\partial }{\partial u} - \frac{2}{r^{2} v} \frac{\partial }{\partial v} + a_{9} \frac{\partial }{\partial p} )\) | 0 | \(a_{11} \frac{\partial }{\partial z} + a_{12} \frac{\partial }{\partial u} + a_{13} \frac{\partial }{\partial p}\) |
\(\boldsymbol{X}_{\boldsymbol{4}}\) | \(-( a_{5} \frac{\partial }{\partial z} + a_{6} \frac{\partial }{\partial u} + a_{7} \frac{\partial }{\partial p} )\) | \(-( f_{3} \frac{\partial }{\partial z} + f_{3} ' \frac{\partial }{\partial u} + \frac{4}{r^{2} v} \frac{\partial }{\partial v} + a_{10} \frac{\partial }{\partial p} )\) | \(-( a_{11} \frac{\partial }{\partial z} + a_{12} \frac{\partial }{\partial u} + a_{13} \frac{\partial }{\partial p} )\) | 0 |