Definition:Tangent Space

Real Submanifold
Let $M$ be a real submanifold of $\R^n$ of dimension $d$.

Let $p\in M$.

Using Local Submersions
Let $U$ be a open neighborhood of $p$ in $\R^n$ and $\phi : U \to \R^{n-d}$ be a submersion such that:
 * $M \cap U = \phi^{-1}(0)$.

The tangent space of $M$ at $p$ is:
 * $T_pM = \ker d\phi(p)$

where $d\phi(p)$ is the differential of $\phi$ at $p$.

Using Local Embdeddings
Let $U$ be a open set of $\R^d$ and $\phi : U \to \R^{n}$ be an embedding such that:
 * $p\in \phi(U) \subset M$

The tangent space of $M$ at $p$ is:
 * $T_pM = \operatorname{im} (d\phi)(\phi^{-1}(p))$

where $(d\phi)(\phi^{-1}(p))$ is the differential of $\phi$ at $\phi^{-1}(p)$.

Differentiable Manifold
There are various ways to construct the tangent space of a differentiable manifold.

Let $M$ be a smooth manifold of dimension $m$.

Let $p \in M$.

Abstract Definition
Let $A$ be an atlas of $M$.

Let $B = \{(U,\phi) \in A : p\in U\}$.

The tangent space of $M$ at $p$ is the real vector space
 * $\displaystyle \left(\coprod_{b\in B}\{b\} \times \R^m\right)/\sim$

where $\sim$ is the equivalence relation defined by:
 * $(i,\xi) \sim (j,\eta) \iff d(\phi_j \circ \phi_i^{-1})(\phi_i(p)) (\xi) = \eta$

where $d(\phi_j \circ \phi_i^{-1})(\phi_i(p))$ is the differential of $\phi_j \circ \phi_i^{-1}$ at $\phi_i(p)$.

Addition is defined by:
 * $[(i,\xi)] + [(j,\eta)] = \left[\left( i, d(\phi_j \circ \phi_i^{-1})(\phi_i(p)) (\xi) + \eta \right)\right]$

and scalar multiplication by:
 * $\lambda \cdot [(i,\xi)] = [(i, \lambda\cdot\xi)]$

As a space of derivations
The tangent space at $p$, denoted by $T_p M$, is the set of all tangent vectors at $m$.

Equivalence of Definitions
While the above constructions are not the same, there are very closely related. See Equivalence of Definitions of Tangent Space.

Also see

 * Equivalence of Definitions of Tangent Space
 * Tangent Space is Vector Space
 * Definition:Tangent Vector
 * Definition:Tangent Bundle