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:
 * $M \cap U = \phi^{-1} \sqbrk 0$.

The tangent space of $M$ at $p$ is:
 * $T_p M = \ker \d \map \phi p$

where $\d \map \phi p$ 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:
 * $p \in \phi \sqbrk U \subset M$

The tangent space of $M$ at $p$ is:
 * $T_p M = \Img {\map {\paren {\d \phi} } {\map {\phi^{-1} } p} }$

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

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 $B = \set {\struct {U, \phi} \in A : p \in U}$.

The tangent space of $M$ at $p$ is the real vector space
 * $\ds \paren {\coprod_{b \mathop \in B} \set b \times \R^m} / \sim$

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

where $\d \map {\map {\paren {\phi_j \circ \phi_i^{-1} } } {\map {\phi_i} p} } \xi$ is the differential of $\phi_j \circ \phi_i^{-1}$ at $\map {\phi_i} p$.

Addition is defined by:
 * $\sqbrk {\tuple {i, \xi} } + \sqbrk {\tuple {j, \eta} } = \sqbrk {\tuple {i, \d \map {\map {\paren {\phi_j \circ \phi_i^{-1} } } {\map {\phi_i} p} } \xi + \eta} }$

and scalar multiplication by:
 * $\lambda \cdot \sqbrk {\tuple {i, \xi} } = \sqbrk {\tuple {i, \lambda \cdot \xi} }$

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