In a few continuum classes I have seen indicial notation used to derive relations in Gibbs notation. However, Gibbs notation is valid for all coordinates while indicial notation is valid only for Cartesian coordinates. Is it safe to arrive at a relation in Gibbs notation based upon a proof in indicial notation? Is the result valid for all coordinate systems?
For example:
Showing
$\mathbf{u}\cdot \nabla \mathbf{u} = \frac{1}{2}\nabla(\mathbf{u}\cdot\mathbf{u})-\mathbf{u}\times (\nabla \times \mathbf{u})$
By using indicial/Einstein notation. I'm not at all concerned with showing the result above - just the generality of the approach of using Einstein notation in the middle.