The Tangent Bundle of a manifold is the union of all the tangent spaces at every point in the manifold.
Suppose <math>M</math> is a <math>C^k</math>
manifold, and <math>\phi : U \rightarrow \mathbb{R}^n </math>, where <math>U</math> is an
open subset of <math>M</math>, and <math>n</math> is the the dimension of the manifold, in the chart <math>\phi(\circ)</math>; furthermore suppose <math> T_{p}M </math> is the
tangent space at a point <math> p </math> in <math> M </math>. Then the tangent bundle,
<math>
{TM} = \bigcup_{p \in M} T_{p}M
</math>
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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