For the common vector identities (for example, those listed here), how is it proven that these are coordinate free formulae (at least in the sense of curvilinear coordinates)?
Most common proofs show the identities hold in a Cartesian coordinate system, however is this general enough to prove that they are also true for arbitrary curvilinear coordinates? Is it sufficient to say that since div, grad and curl have more general geometric definitions (that are coordinate free), if an identity is proven in a Cartesian basis it generalizes to all curvilinear coordinates? Is it something to do with classical vector calculus dealing with $\mathbb{R}^3$, for which there is already a Euclidean coordinate system and the cross-product?