How can I prove that curl(curl F)=grad(div F)-div(grad Fi) in n dimensions?

[manehsan]

Hi
This is my first question in this forum.
How can I prove that curl(curl F)=grad(div F)-div(grad Fi)
in n dimensions?
I guess I should use Levi-Chivita but I don't know how.
Thanks.

 

[ksdhart2]

I do admit I'm perhaps a bit over my head, having only just completed a Calculus 4 class where we learned about Curl and Divergence. But, from what I remember, because the curl involves a cross product, it is only defined in 0, 1, 3, or 7 dimensions. Further, the curl only returns a vector field for 3 dimensions. Thus, it would appear as though this identity you're tasked with proving is only valid for 3 dimensions, which makes the proof that much easier. I could be very very wrong, but I'd consider sufficient to prove it for 3 dimensions.

 

[manehsan]

I do admit I'm perhaps a bit over my head, having only just completed a Calculus 4 class where we learned about Curl and Divergence. But, from what I remember, because the curl involves a cross product, it is only defined in 0, 1, 3, or 7 dimensions. Further, the curl only returns a vector field for 3 dimensions. Thus, it would appear as though this identity you're tasked with proving is only valid for 3 dimensions, which makes the proof that much easier. I could be very very wrong, but I'd consider sufficient to prove it for 3 dimensions.

Hi.
Thanks,
but you can generalize curl in n dimensions assuming that extra vector components are 0.