This chapter defines the mathematical spaces to which the geometrical quantities discussed in the previous chapter—scalars, vectors, and the metric—belong. Its goal is to go from the concept of a vector as an object whose components transform as Tⁱ → 𝓡ⱼ ⁱTj under a change of frame to the ‘intrinsic’ concept of a vector, T. These concepts are also generalized to ‘tensors’. The chapter also briefly remarks on how to deal with non-Cartesian coordinates. The velocity vector v is defined as a ‘free’ vector belonging to the vector space ε3 which subtends ε3. As such, it is not bound to the point P at which it is evaluated. It is, however, possible to attach it to that point and to interpret it as the tangent to the trajectory at P.
Explicitly computing geodetic coordinates from Cartesian coordinates
