Definition:Tangent Space/Geometric Tangent Space

Definition

Let $a \in \R^n$ be an element of the $n$-dimensional Euclidean space.

The geometric tangent space to $\R^n$ at $a$, denoted by $\R^n_a$, is the cartesian product of the singleton $\set a$ and $\R^n$:

$\R^n_a := \set a \times \R^n = \set {\tuple {a, v} : v \in \R^n}$

Sources

• 2013: John M. Lee: Introduction to Smooth Manifolds (2nd ed.): $\S 3$: Tangent Vectors. Tangent Vectors