Euclidean Tangent Space

Tangent Space

Statements

Definition. Let $V^n$ be an Euclidean space. The tangent space, denoted as $T_a({\mathbb{R}}^n)$, is the set of pair of points $(a,x)$ in ${\mathbb{R}}^n\times {\mathbb{R}}^n$, where $a$ are initial points and $x$ are terminal points $\overrightarrow{ax}$ having each pair is denoted as $X_a$ and as called as tangent vectors. Together with the injection

\varphi_a:&T_a(\mathbb{R}^n)&\to&V^n\\ &X_a&\mapsto&(x^1-a^1,\ldots,x^n-a^n)\;. \end{array}

satisfying the operations $X_a+Y_a={\phi }_a^{-1}({\phi }_a(X_a)+{\phi }_a(Y_a))$ and, for every $\alpha \in \mathbb{R}$, $\alpha X_a={\phi }_a^{-1}(\alpha {\phi }_a(X_a)).$ the tangent space is a vector space.

Remark. This definition is a particular case of tangent spaces to a manifold. More especifically, when the manifold is an Euclidean space.

References

@Boo75 pp. 30 @Tu11 pp. 11