## Encyclopedia > Tangent bundle

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# Tangent bundle

The Tangent Bundle of a manifold is the union of all the tangent spaces at every point in the manifold.

### Definition as directions of curves

Suppose $M$ is a $C^k$ manifold, and $\phi : U \rightarrow \mathbb{R}^n$, where $U$ is an open subset of $M$, and $n$ is the the dimension of the manifold, in the chart $\phi(\circ)$; furthermore suppose $T_{p}M$ is the tangent space at a point $p$ in $M$. Then the tangent bundle,
${TM} = \bigcup_{p \in M} T_{p}M$

It is useful, in distinguishing between the tangent space and bundle, to consider their dimensions, n and 2n respectively. That is, the tangent bundle accounts for dimensions in the positions in the manifold as well as directions tangent to it.

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