I am really struggling with one concept in my study of differential geometry where there seems to be a conflict among different textbooks. To set up the question, let M be a manifold and let (U, φ) be a chart. Now suppose we have a curve γ:(-ε,+ε) → M such that γ(t)=0 at a ∈ M. Suppose further that we have a function ƒ defined on M. Now my question: Consider the directional derivative Dv. Some authors (e.g. J.M. Lee) imply that Dva acts at point a∈M. Other authors say that the derivative makes no sense on a manifold and that Dvφ(a) acts at φ(a)∈ℝn. Which is the correct understanding? Thanks in advance.