\begin{definition}[Definition:Local Frame]
Let $M$ be a smooth manifold.
Let $U \subseteq M$ be an open set.
Let $\sigma = \tuple {\sigma_1, \ldots \sigma_k}$ be an ordered $k$-tuple of local sections over $U$.
Suppose at each $p \in U$ the values of $\sigma$ constitute a basis for $M_p$.
Then $\sigma$ is known as a '''local frame''' for $M$.
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\end{definition}