31. This is the tactic taken by the covariant derivative approach to connections : good behavior is equated with covariance. 32. Thus they quickly supplanted the classical notion of covariant derivative in many post-1950 treatments of the subject. 33. By definition, a covariant derivative of a scalar field is equal to the regular derivative of the field. 34. In differential geometry, the covariant derivative makes a choice for taking directional derivatives of vector fields along curves. 35. Thus one might want a derivative with some of the features of a functional derivative and the covariant derivative . 36. With the classical covariant derivatives , covariance is an " a posteriori " feature of the derivative. 37. Then decompose the ambient covariant derivative of ? along " X " into tangential and normal components: 38. Where we have taken advantage of the first covariant derivative of a function being the same as its ordinary derivative. 39. Cartan connections induce covariant derivatives and other differential operators on certain associated bundles, hence a notion of parallel transport. 40. Furthermore, many of the features of the covariant derivative still remain : parallel transport, curvature, and holonomy.
