Gradient of A*B: Adding and Subtracting Terms

[xaos]

grad(A*B)=(A*grad)B + (B*grad)A + A curl B + B curl A

i'm not sure how to read the RHS to begin to work out the index definition.i'm thinking if add and subtract terms this will work out. i think i can see the first two terms, but the last two maybe "A cross nabla" is what they mean acting on B componentwise, otherwise it doesn't mean anything to me.

 

[dx]

This is the correct identity:

∇(A⋅B) = A⋅∇B + B⋅∇A + A x (∇ x B) + B x (∇ x A)