31. This representation of a vector field depends on the coordinate system, and there is a well-defined tangent vector ). 32. *As indicated previously, the 1-jet of a curve through " p " is a tangent vector . 33. These basis vectors are by definition the tangent vectors of the curves obtained by varying one coordinate, keeping the others fixed: 34. At each point in M the timelike tangent vectors in the point's tangent space can be divided into two classes. 35. Notice that velocity always points in the direction of motion, in other words for a curved path it is the tangent vector . 36. The sum of two vectors is again a tangent vector to some other curve and the same holds for multiplying by a scalar. 37. The initial tangent vector is parallel transported to each tangent along the curve; thus the curve is, indeed, a geodesic. 38. This is an " n "-dimensional Euclidean space consisting of the tangent vectors of the curves through the point. 39. A four-velocity is thus the normalized future-directed timelike tangent vector to a world line, and is a contravariant vector. 40. So far a world line ( and the concept of tangent vectors ) has been described without a means of quantifying the interval between events.
