Tangent bundle

Definition

Let $M$ be a smooth manifold of dimension $n$, and let $C^\infty(M)$ be the algebra of smooth functions $M \to \R$. For $p \in M$, a tangent vector at $p$ is a linear map $v \colon C^\infty(M) \to \R$ such that $v(fg) = f(p)\, v(g) + g(p)\, v(f)$ for all $f, g \in C^\infty(M)$; that is, a derivation at $p$. The tangent vectors at $p$ form a real vector space $T_pM$, the tangent space at $p$. The tangent bundle of $M$ is the disjoint union $$TM = \bigsqcup_{p \in M} T_pM = \set{(p, v) : p \in M, \ v \in T_pM},$$ with the projection $\pi \colon TM \to M$, $\pi(p, v) = p$. Let $(U, (x^1, \ldots, x^n))$ be a chart of $M$. $TM$ has a unique smooth structure in which each such chart gives a chart on $\pi^{-1}(U)$, namely $$(p, v) \mapsto \big(x^1(p), \ldots, x^n(p), v^1, \ldots, v^n\big), \qquad v = \sum_{i=1}^n v^i \left.\frac{\partial}{\partial x^i}\right|_p.$$