Tangent Space

Definition

Suppose that $\mathcal{M}$ is a Differentiable Manifold and that $x\in \mathcal{M}$. Pick a coordinate Chart $\phi :U\to {\mathbb{R}}^n,\phi (p)=(x_1(p),x_2(p),\dots ,x_n(p))$, where $U$ is an open subset of $\mathcal{M}$ containing $x$. The tangent space $T_p(\mathcal{M})$ is the set of all tangent vectors at $p$ on the manifold $\mathcal{M}$.

The basis of the tangent plane is given by the partial derivatives ${\partial }_i=\frac{\partial {\phi }^{-1}}{\partial x_i},i=1,\dots ,n$