# Tangent space

One can attach to every point $x$ of a differentiable manifold a tangent space, a real vector space that intuitively contains the possible “directions” at which one can tangentially pass through $x$. The elements of the tangent space are called tangent vectors at$x$. All tangent spaces of a connected manifold have the same dimension, equal to the dimension of the manifold

### Definition as velocities of curves

Suppose $M$ is a $C^k$ manifold ($k\geq 1$) and $x$ is a point in $M$. Pick a chart $\varphi:U\to\mathbf{R}^n$ where $U$ is an open subset of $M$ containing $x$. Suppose two  curves $\gamma_1:(-1,1)\to M$ and $\gamma_2:(-1,1)\to M$ with $\gamma_1(0)=\gamma_2(0)=x$ are given such that $\varphi\circ\gamma_1$ and $\varphi\circ\gamma_2$ are both differentiable at $0$. Then $\gamma_1$ and $\gamma_2$ are called equivalent at $0$ if the ordinary derivatives of $\varphi\circ\gamma_1$ and $\varphi\circ\gamma_2$ coincide. This defines an equivalence relation on such curves, and the equivalence classes are known as the tangent vector of $M$ at $x$. The equivalence class of the curve $\gamma$ is written as $\gamma'(0)$. The tangent space of $M$ at $x$, denoted by $T_xM$, is defined as the set of all tangent vectors, it does not depened on the choice of chart $\varphi$.

To define the vector space operations on $T_xM$, we use a chart $\varphi:U\to\mathbf{R}^n$ and define the map $d\varphi_x(\gamma'(0))=\frac{d}{dt}(\varphi\circ\gamma)(0)$. It turns out that this map is bijective and can thus be used to transfer the vector space operation from $\mathbf{R}^n$ over $T_xM$, turning the latter into an $n$-dimensional real vector space. Again, one need to check that this construction does not depend on the particular char $\varphi$ chosen, and in fact is does not.

### Definition via derivations

Suppose that $M$ is a $C^\infty$ manifold. A real valued function $f:M\to\mathbf{R}$ belongs to $C^\infty(M)$ if $f\circ\varphi^{-1}$ is infinitely differentaible for every chart $\varphi:U\to \mathbf{R}$. $C^\infty(M)$ is a real associative algebra for the pointwise product and sum of functions and scalar multiplication.

Pick a point $x$ in $M$. A derivation at $x$ is a liner map $D:C^\infty(M)\to\mathbf{R}$ that has the property that for all $f,g$ in $C^\infty(M)$:

$\displaystyle D(fg)=D(f)\times g(x)+f(x)\times D(g)$

modeled on the product rule of calculus. If we define addition and scalar multiplication for such derivations by

$\displaystyle (D_1+D_2)(f)=D_1(f)+D_2(f)\ \text{and}\ (\lambda D)(f)=\lambda D(f)$

we get a real vector space which we define as the tangent space $T_xM$.

The relation between the  tangent vectors defined earlier and derivatoin is as follows: if $\gamma$ is a curve with tangent vector $\gamma'(0)$, then the corresponding derivation $D(f)=(f\circ\gamma)'(0)$.

$\displaystyle \gamma'(0)\mapsto D_\gamma \ \text{where}\ D_\gamma(f)=\frac{d}{dt}(f\circ \gamma)\Big|_{t=0}.$