Component definition in curvilinear coordinates

[Shirish]

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?

Second question: 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 as far as I think:

Would appreciate any help in clarifying my concepts!

 

[pat_connell]

For the first question, yes, the "distance representing each coordinate separation" refers to the distance between two points with the same value for all coordinates except one. For example, if you have a Cartesian coordinate system, the distance between (1,2,3) and (1,2,4) would be the distance representing the coordinate separation for the third axis. For the second question, the definition of projection and component would still be the same as in the affine transformation case. The projection is still defined as the dot product of the vector with the corresponding basis vector and the component is still defined as the dot product of the vector with itself. However, in a curvilinear coordinate system, the distances between two points with the same values for all coordinates except one will not be the same. Therefore, the components of the vector will no longer correspond directly to the projections since the distances are not equal.