Definition:Local Frame

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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 $M$.