# Tangent space and tangent vectors

As I have heard, tangent vector to a smooth manifold $M$ in $p \in M$ is the operator $D_{\xi}$:$f \to D_{\xi}f$, where $f$ is a smooth function $f: M \to R$, with the following properties:

$D_{\xi}(f+g)=D_{\xi}f+D_{\xi}g$

$D_{\xi}(fh)=(D_{\xi}f)h(p)+f(p)(D_{\xi}h)$

So, the basis of the tangent space is:

${\frac{\partial }{\partial x^{1}}}, \frac{\partial }{\partial x^{1}}, ..., \frac{\partial }{\partial x^{n}}$, where ${x^{1},...,x^{n}}$ are coordinates of the local map $(U_{p}, \phi_{p})$.

and any vector in tangent space has the following form:

$\xi^{1}\frac{\partial }{\partial x^{1}}+...+\xi^{n}\frac{\partial }{\partial x^{n}} \in TM_{p}$, where the set $\xi^{1},...,\xi^{n}$ is called the coordinates of tangent vector.

How does this definition relate to intuitive graphical interpretation?

Tangent vectors to $\mathbb{R^n}$ (and likewise tangent vectors to embedded surfaces in $\mathbb{R^n}$) can be thought of in this way by identifying a vector $v$ with the directional derivative operator (or "derivation") $D_v = v^i \frac\partial {\partial x^i}$. In fact the directional derivatives are the only operators on functions that satisfy the derivation axioms; so talking about vectors is the same thing as talking about derivations - in a sense derivations are vectors.
Exactly as you've set it up. $\partial/\partial x^j$ should be thought of as the tangent vector to the $x^j$ parameter curve in $M$, and $\xi=\sum x^j \partial/\partial x^j$ as the corresponding linear combination in $T_pM$.
• I do not understand your point. By definition, $\partial/\partial x^j$ is partial derivative and therefore is a number. How can a set of real numbers forms a basis for a vector space in $\mathbb R^n$. – LaTeXFan May 10 '14 at 1:56