Let v be a smooth vector field on a surface X in ℝ3 and let γ(t) be a parameterized curve in X. Then the covariant derivative Dv/dt is the orthogonal projection of d(v(γ(t))/dt into the tangent space at γ(t). The covariant derivative is in fact intrinsic to the surface and can be expressed in terms of Christoffel symbols. See parallel transport.