# Tangent vector in differential geometry: combining direction and derivative

Frederic Schuller explains the directional derivative operator at a point $p$ along a curve $\gamma$ on a manifold $M$ as the linear map $X_{\gamma\ p}: C^\infty (M) \to \mathbb R$ defined as the mapping of a smooth function $f\mapsto (f\circ\gamma)'(0),$ with $0$ being the parameter value at which the curve goes through the point $p.$ He goes on to state that in differential geometry, $X_{\gamma\ p}$ is usually called the tangent vector to the curve $\gamma$ at point $p.$

Immediately afterwards he points out that in differential geometry

There is no longer an independent notion of direction and an independent notion of derivative; there is only the combination of the two.

What does he mean by that?

• Don't worry about it... it will all make sense a bit later Sep 12, 2017 at 2:24
• @orangeskid I want to get a bit of additional insight into what it means to define a vector as $\frac{df}{dt}.$ Sep 12, 2017 at 2:26
• Wikipedia here en.m.wikipedia.org/wiki/Tangent_space does a pretty good job. Sometime just a mini idea from one place can enlighten another presentation. Sep 12, 2017 at 2:30
• Chapter 3 on Lee's smooth manifold will help alot. It will gives us reason why tangent vector defined that way. Sep 12, 2017 at 3:43