# The tangent plane of a surface and the tangent space of a manifold (at one point)

The tangent plane of a surface
By a tangent vector to $S$ at a point $p \in S$, we mean the tangent vector $\alpha'(0)$ of a differentiable parametrized curve $\alpha: (-\epsilon, \epsilon) \to S$ with $\alpha(0) = p$.

What is the tangent plane of a surface? That is a plane which containes all of the tangent vectors of this surface at point $p \in S$.

Proposition
Let $\varphi: U \subset \mathbb{R}^2 \to S$ be a parametrization of a regular surface $S$ and let $q \in U$. The vector subspace of dim $2$,

$d\varphi_q(\mathbb{R}^2) \subset \mathbb{R}^3,$

coincides with the set of tangent vectors to $S$ at $\varphi(q)$

There is a similar situation in here. In the case of manifold, the tangent space is also built from the set of all of tangent vectors of a manifold.

Tangent spaces of a manifold

In Milnor’s book (\cite{Milnor}), the tangent space $TM_x$ at $x$ for arbitrary smooth manifold $M \subset \mathbb{R}^k$ is defined:

Choose a parametrization $g : U \to M \subset \mathbb{R}^k$ of a neighborhood $g(U)$ of $x$ in $M$, with $g(u) = x$. We have $dg_u: \mathbb{R}^m \to \mathbb{R}^k$. So the image $dg_u(\mathbb{R}^m)$ of $dg_u$ is equal to $TM_x$.

References

J. W. Milnor, Topology from differentiable viewpoint, 1965.

M. do Carmo, Differential geometry, curves and surfaces, 1976.