Definition:Smooth Vector Field

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Let $M$ be an $n$-dimensional smooth manifold with or without boundary.

Let $A \subseteq M$ be a subset.

Let $TM$ be the tangent bundle of $M$.

Suppose $TM$ is endowed with topology and smooth differentiable structure so that $TM$ could be treated as a $2n$-dimensional smooth manifold.

Let $T_p M$ be the tangent space at $p \in M$.

Then the smooth vector field is a smooth vector field $X : M \to TM$.