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.
Finite-range separable pairing interaction in Cartesian coordinates
