Basis vectors at points other than the origin in a curvilinear coordinate

Question

I'm studying tensor recently with "Tensors, Differential Forms, and Variational Principles" by David Lovelock. When talking about curvilinear coordinates, I don't really understand why it is so important to have another coordinate system at points other than the origin. For example, if a vector (\vec{A}) whose point of application is at point p. Why is it important to define another set of basis of vectors at point p?

Answer 1

(Not sure I understand your question, so this may all be gibberish or general nonsense. Please specify your question further, so I can improve the answer)

Changing basis differently at each point $p$ of space allows you, for instance, to describe phenomena at $p$ (functions, geometry, etc) in an easier, more convenient, or "intrinsic" way. Take a plane curve. At each point $p$ in the curve you can describe the curve geometry in the basis $(t(p),n(p))$ (Frenet Frame), where $t$ is the tangent direction and $n$ is the normal direction, so that your description of the curve around $p$ is modulo an Euclidean change of basis from an arbitrary reference basis $(e_1,e_2)$. That means your equations at $p$ would be independent of, say, the absolute rotation and translation of the curve or the choice of reference. Indeed, the Euclidean shape of the curve is independent of such a coordinate change, yet you may need coordinates to compute. If you define the basis naturally in terms of some geometry, studying how the frame varies at each point reflects the geometry. That way, studying change of coordinates at each point can be an effective computational way to study geometry.

See also https://en.wikipedia.org/wiki/Moving_frame#Uses