Let \(M\) be a differentiable manifold. The set. \[TM=\biguplus _{x\in M}T_{x}M,\]
together with the projection function
\[\pi \colon TM\longrightarrow M,\,(x,v)\longmapsto x\,,\]
is called the tangent bundle of \(M\). So, an element of \(TM\) can be thought of as a pair \((x,v)\), where \(x\) is a point in \(M\) and \(v\) is a tangent vector to \(M\) at \(x\).
