Question

prove that curl(A+B)=curlA+curlB​

Answer 1

Answer:

a vector field A, the curl of the curl is defined by

∇×(∇×A)=∇(∇⋅A)−∇2A

where ∇ is the usual del operator and ∇2 is the vector Laplacian.

How can I prove this relation? I tried the ugly/unefficient/brute-force method, by getting an expression for the LHS and the RHS for an arbitrary vector field

A=(a(x,y,z),b(x,y,z),c(x,y,z))

It does work (duh), but is there a more elegant way of doing this? Using matrix notation maybe?

EDIT: I got very good answers, from various perspectives. I would say @Spencer's derivation is the one I was looking for, using Einstein notation - and as a physics student, this was very helpful. However, @Vectornaut's solution not only is short and elegant, but it also introduced me to a whole new range of mathematics - and as a theoretical physics student, I appreciate learning new mathematical theories and trying to see how we can use them in physics.