Let \(M\) be a smooth \(n\)-manifold. A tangent vector on \(p \in M\) is a linear map \(v\colon C^\infty (M)\to \mathbb{R}\) satisfying the product rule
\begin{equation*} v(fg)=f(p)v(g)+g(p)v(f) \end{equation*}for \(f,g \in C^\infty(M)\). The space of all tangent vectors on \(p\) is denoted by \(T_p(M)\), and it is called tangent space.
Remarks
- A tangent vector is also called contravariant vector .
- Several alternative definitions exist [@lee2013smooth_manifolds]:
- The derivative of \(\gamma\colon \mathbb{R}\to M\) can be seen as an element of \(T_pM\).
- \(T_p(M)\) is a vector space .