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.

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.


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

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


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