Editing Connection form (section)

===Frames on a vector bundle===
{{main|Frame bundle}}
Let <math>E</math> be a [[vector bundle]] of fibre dimension <math>k</math> over a [[differentiable manifold]] <math>M</math>.  A '''local frame''' for <math>E</math> is an ordered [[basis of a vector space|basis]] of [[section (fiber bundle)|local sections]] of <math>E</math>.  It is always possible to construct a local frame, as vector bundles are always defined in terms of [[local trivialization]]s, in analogy to the [[atlas (topology)|atlas]] of a manifold. That is, given any point <math>x</math> on the base manifold <math>M</math>, there exists an open neighborhood <math>U \subseteq M</math> of <math>x</math> for which the vector bundle over <math>U</math> is locally trivial, that is isomorphic to <math>U \times \mathbb R^k</math> projecting to <math>U</math>. The vector space structure on <math>\mathbb R^k</math> can thereby be extended to the entire local trivialization, and a basis on <math>\mathbb R^k</math> can be extended as well; this defines the local frame. (Here the real numbers are used, although much of the development can be extended to modules over rings in general, and to vector spaces over complex numbers <math>\mathbb C</math> in particular.)

Let <math>\mathbf e = (e_\alpha)_{\alpha = 1, 2, \dots, k}</math> be a local frame on <math>E</math>.  This frame can be used to express locally any section of <math>E</math>.  For example, suppose that <math>\xi</math> is a local section, defined over the same open set as the frame <math>\mathbb e</math>. Then 
:<math>\xi = \sum_{\alpha=1}^k e_\alpha \xi^\alpha(\mathbf e)</math>
where <math>\xi^\alpha(\mathbf e)</math> denotes the ''components'' of <math>\xi</math> in the frame <math>\mathbf e</math>.  As a matrix equation, this reads
:<math>\xi = {\mathbf e}
\begin{bmatrix}
\xi^1(\mathbf e)\\
\xi^2(\mathbf e)\\
\vdots\\
\xi^k(\mathbf e)
\end{bmatrix}=
{\mathbf e}\, \xi(\mathbf e)
</math>

In [[general relativity]], such frame fields are referred to as [[Tetrad formalism|tetrads]]. The tetrad specifically relates the local frame to an explicit coordinate system on the base manifold <math>M</math> (the coordinate system on <math>M</math> being established by the atlas).