I'm watching this lecture that gives an introduction to tensors. If we have a coordinate system that's an affine transformation of the Cartesian coordinate system, then the projection of a vector $v$ (onto a particular axis) is defined as $v_m = v.e_m$ or the dot product of the vector with the corresponding basis vector (mentioned at this timestamp).
Here the prof states that if the "distance representing each coordinate separation" were the same, then the projections would correspond to components of the vector, or that $v_m=v.e_m=v^m$, where $v=v^me_m$ (Einstein summation convention).
First question: what's a precise way of defining "distance representing each coordinate separation"? Does it refer to the distance between $x^m=k$ and $x^m=k+1$ keeping all other coordinates fixed?
This is all fine, but what do we do if the surfaces $x^m=k$ were curved (curvilinear coordinate system)? And even if they weren't curved, what if the "distance representing each coordinate separation" varied? In either case, how would you define a projection and how would you define a component?
A natural intuitive way of defining the projection/component for a curvilinear coordinate system with equally spaced coordinate separations would be something like this. [Can't upload image for some reason].
Would appreciate any help in clarifying my concepts!