After spending hours trying to understand the basic points of index notation, I have failed miserably. I will be getting a library book tomorrow, but as I have a few questions in for tomorrow, any help would be very appreciated.
1) Write the following in index notation:
grad(u.u)
grad x (fu) [u vector, f scalar]
2) Use index notation to prove the following identities:
grad . (u x v) = (grad x u) . v - (grad x v) . u
grad x (u x v) = (grad . v)u - (grad . u)v + (v . grad)u - (u . grad)v
grad x (grad f) = 0
grad(u . v) = u x (grad x v) + v x (grad x u) + (u . grad)v + (v . grad)u
3) Show using index notation that for an arbitrary vector field u:
u x (grad x u) = grad (1/2 u^2) - (u . grad)u [where u^2 = u.u]
Hence, or otherwise show that for a vector field u for which grad x u is parallel to u that:
grad x ((u . grad)u) = 0
Hence show that:
grad x ((G . grad)G) = 0 where G = (cosy+sinz, sinx+cosz, cosx+siny)
Cheers