I am studying Tensors from Ref 1. After the introduction of the invariant expression for the gradient of a scalar field $F$ , I tried coming up with an expression for the Laplacian. (This was very early on, even before the introduction of elementary concepts like the covariant derivative)
$$ \left(\frac{\partial}{\partial Z^j}\left(\frac{\partial F}{\partial Z^i} \vec{Z}^{\,i} \right)\right)\cdot \vec{Z}^{\, j}$$
The argument I used to justify to myself that this works (at least in Euclidean spaces) is that since it is an invariant (between coordinate systems) and reduces to the correct form in Cartesian coordinates it should work in all coordinate systems that describe Euclidean space because of the tensor property.
However, I would like to know if this attempt (if valid) can be used for non-Euclidean spaces where I can't simply assume reduction to the correct form in Cartesian coordinates implies validity for all coordinate systems as Cartesian coordinates may not even exist.
References:
- Pavel Grinfeld - Introduction to Tensor Analysis and the Calculus of Moving Surfaces