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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?

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  • $\begingroup$ Don't worry about it... it will all make sense a bit later $\endgroup$
    – orangeskid
    Sep 12, 2017 at 2:24
  • $\begingroup$ @orangeskid I want to get a bit of additional insight into what it means to define a vector as $\frac{df}{dt}.$ $\endgroup$ Sep 12, 2017 at 2:26
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    $\begingroup$ 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. $\endgroup$
    – orangeskid
    Sep 12, 2017 at 2:30
  • $\begingroup$ Chapter 3 on Lee's smooth manifold will help alot. It will gives us reason why tangent vector defined that way. $\endgroup$ Sep 12, 2017 at 3:43

2 Answers 2

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I think Schuller's statement is misleading. It makes perfectly good sense on a manifold to define a direction at a point to be an equivalence class of nonzero tangent vectors, where two vectors are equivalent if one is a positive scalar multiple of the other. So it is perfectly possible to talk about a "direction" without choosing a specific differentiation operator.

On the other hand, there are many natural differentiation operators on a smooth manifold that do not refer to any specific direction. The most common example is the differential of a smooth real-valued function.

However, what is true is that before you can talk about "directions" on a smooth manifold, you have to define tangent vectors, and the definition of a tangent vector typically entails some sort of differentiation.

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He has just defined direction (tangent vector) in terms of derivatives (the directional derivative operator). So the two are always going to be connected in this formulation of the theory. This is different from Euclidean space, where direction is independent of derivatives.

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