Definition:Tangent Vector

Definition
Let $X$ be a topological space.

Let $\phi: \R^k \to X$ be a mapping such that $\phi \left({0}\right) = x$.

Let the derivative of $\phi$ at $x$ be $d \phi_x$.

Then the tangent space of $X$ at $x$ is defined as the image of $d \phi_0 \left({\R^k}\right)$, and is denoted $T_x \left({X}\right)$.