- #1

cianfa72

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- TL;DR Summary
- About the coordinate-free definition of tangent vector on manifold

I would ask for a clarification about the following definition of tangent vector from J. Lee - Introduction to Smooth Manifold. It applies to Euclidean space ##R^n## with associated tangent space ##R_a^n## at each point ##a \in R^n##.

$$D_v\left. \right|_a (f)=D_vf(a)=\left. \frac {df(a + tv)} {dt} \right|_{t=0}$$

From my understanding the above is actually a coordinate-free definition. In other words ##a## inside ##f(a + tv)## is a point in ##R^n## and it is

So for example ##a=

\begin{bmatrix}

1 \\

5 \\

3 \\

2

\end{bmatrix} ## is a point in ##R^4## and ##v=

\begin{pmatrix}

2 \\

1 \\

6 \\

4

\end{pmatrix} ## is a vector in ##R^4## with vector space structure.

$$D_v\left. \right|_a (f)=D_vf(a)=\left. \frac {df(a + tv)} {dt} \right|_{t=0}$$

From my understanding the above is actually a coordinate-free definition. In other words ##a## inside ##f(a + tv)## is a point in ##R^n## and it is

*not*the tuple of coordinates in some affine basis. The same for ##v##: it is a vector and is*not*the tuple of vector's components in some vector space basis.So for example ##a=

\begin{bmatrix}

1 \\

5 \\

3 \\

2

\end{bmatrix} ## is a point in ##R^4## and ##v=

\begin{pmatrix}

2 \\

1 \\

6 \\

4

\end{pmatrix} ## is a vector in ##R^4## with vector space structure.

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