- #1
simba_lk
- 7
- 0
I need to prove the identity: [itex]\nabla[/itex]([itex]\vec{A} \times \vec{B}[/itex])=[itex]\vec{B} \bullet[/itex]([itex]\nabla \times \vec{A}[/itex]) - [itex]\vec{A} \bullet[/itex]( [itex]\nabla \times \vec{B}[/itex])
I need to prove for an arbitrary coordinate system, meaning I have scaling factors.
The proof should be quite straight forward if you use the levi chevita symbol but so far this is what i have:
[itex]\nabla[/itex]([itex]\vec{A} \times \vec{B}[/itex])=[itex]\frac{1}{h_{1}h_{2}h_{3}} \frac{\partial}{\partial x_{i}}[/itex]( [itex]\frac{h_{1}h_{2}h_{3}}{h_{i}} \epsilon_{ijk} A_{j} B_{k}[/itex]) = [itex]\frac{1}{h_{1}h_{2}h_{3}} \frac{\partial}{\partial x_{i}}[/itex]( [itex]\frac{h_{1}h_{2}h_{3}}{h_{i}} \epsilon_{ijk} A_{j} B_{k}[/itex]) [itex]\ast \hat{e_{i}} \bullet \hat{e_{i}}[/itex]
But I can't see how to take it into the desired format
I think I should squeeze in a kroncker delta somehow, but I'm not sure.
I need to prove for an arbitrary coordinate system, meaning I have scaling factors.
The proof should be quite straight forward if you use the levi chevita symbol but so far this is what i have:
[itex]\nabla[/itex]([itex]\vec{A} \times \vec{B}[/itex])=[itex]\frac{1}{h_{1}h_{2}h_{3}} \frac{\partial}{\partial x_{i}}[/itex]( [itex]\frac{h_{1}h_{2}h_{3}}{h_{i}} \epsilon_{ijk} A_{j} B_{k}[/itex]) = [itex]\frac{1}{h_{1}h_{2}h_{3}} \frac{\partial}{\partial x_{i}}[/itex]( [itex]\frac{h_{1}h_{2}h_{3}}{h_{i}} \epsilon_{ijk} A_{j} B_{k}[/itex]) [itex]\ast \hat{e_{i}} \bullet \hat{e_{i}}[/itex]
But I can't see how to take it into the desired format
I think I should squeeze in a kroncker delta somehow, but I'm not sure.