Editing Moving frame (section)

===Coframes===

A moving frame determines a '''dual frame''' or '''[[coframe]]''' of the [[cotangent bundle]] over ''U'', which is sometimes also called a moving frame. This is a ''n''-tuple of smooth ''1''-forms
:''θ''<sup>1</sup>, ''θ''<sup>2</sup>, …, ''θ''<sup>''n''</sup>

which are linearly independent at each point ''q'' in ''U''. Conversely, given such a coframe, there is a unique moving frame ''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n''</sub> which is dual to it, i.e., satisfies the duality relation ''θ''<sup>''i''</sup>(''e''<sub>''j''</sub>) = ''δ''<sup>''i''</sup><sub>''j''</sub>, where ''δ''<sup>''i''</sup><sub>''j''</sub> is the [[Kronecker delta]] function on ''U''.

If <math>(x^1,x^2,\dots,x^n)</math> is a coordinate system on ''U'', as in the preceding section, then each covector field ''θ''<sup>i</sup> can be expressed as a linear combination of the coordinate covector fields <math>dx^i</math>:<math display="block">\theta^i = \sum_{j=1}^n B^i_j dx^j,</math>where each <math>B^i_j</math> is a function on ''U.'' Since <math display="inline">dx^i \left(\frac{\partial}{\partial x^j}\right) = \delta^i_j</math>, the two coordinate expressions above combine to yield <math display="inline"> \sum_{k=1}^n B^i_k A^k_j = \delta^i_j </math>; in terms of matrices, this just says that <math>A</math> and <math>B</math> are [[Matrix inverse|inverses]] of each other.

In the setting of [[classical mechanics]], when working with [[canonical coordinates]], the canonical coframe is given by the [[tautological one-form]]. Intuitively, it relates the velocities of a mechanical system (given by vector fields on the tangent bundle of the coordinates) to the corresponding momenta of the system (given by vector fields in the cotangent bundle; i.e. given by forms). The tautological one-form is a special case of the more general [[solder form]], which provides a (co-)frame field on a general [[fiber bundle]].