Further unwinding the definitions, the pushforward F ∗ : TM p → TN F (p) of a vector field under a map F : M → N between manifolds is defined by F ∗ ∗ ( X ) f = X ( f ∘ ∘ F ) , where f is a function on N. When M = ℝ, N = ℝ the pushforward of F reduces to DF : ℝ → ℝ, the ordinary differential, which is given by the Jacobian matrix of partial derivatives of the component functions. The differential is the best linear approximation of a function F from ℝ to ℝ. The pushforward is the smooth manifold version of this. It acts between tangent spaces, and is in coordinates represented by the Jacobian matrix of the coordinate representation of the function.