The derivative of your velocity, your acceleration vector, always points radially inward. A vector at a particular time t (for instance, the acceleration of the curve) is expressed in terms of ({\mathbf e}_r, {\mathbf e}_{\theta}), where {\mathbf e}_r and {\mathbf e}_{\theta} are unit tangent vectors for the polar coordinates, serving as a basis to decompose a vector in terms of radial and tangential components. This chapter examines the notion of the curvature of a covariant derivative or connection. Exterior Covariant Derivative. This coincides with the usual Lie derivative of f along the vector field v. A covariant derivative \nabla at a point p in a smooth manifold assigns a tangent vector (\nabla_{\mathbf v} {\mathbf u})_p to each pair ({\mathbf u},{\mathbf v}), consisting of a tangent vector v at p and vector field u defined in a neighborhood of p, such that the following properties hold (for any vectors v, x and y at p, vector fields u and w defined in a neighborhood of p, scalar values g and h at p, and scalar function f defined in a neighborhood of p): If u and v are both vector fields defined over a common domain, then \nabla_{\mathbf v}\mathbf u denotes the vector field whose value at each point p of the domain is the tangent vector (\nabla_{\mathbf v}\mathbf u)_p. Sign in. The covariant derivative of the r component in the q direction is the regular derivative plus another term.