Definition:Tangent Vector

Given a map $$\phi:\mathbb{R}^k \to X$$ such that $$\phi(0)=x$$ and its derivative at $$x$$, $$d\phi_x$$, define the tangent space of $$X$$ at $$x$$, $$T_x(X)$$ as the image of $$d \phi_0(\mathbb{R}^k)$$.