Editing Connection form (section)

====Change of frame====
In order to extend ''ω'' to a suitable global object, it is necessary to examine how it behaves when a different choice of basic sections of ''E'' is chosen.  Write ''ω''<sub>''α''</sub><sup>''β''</sup> = ''ω''<sub>''α''</sub><sup>''β''</sup>('''e''') to indicate the dependence on the choice of '''e'''.

Suppose that '''e'''{{prime}} is a different choice of local basis.  Then there is an invertible ''k'' × ''k'' matrix of functions ''g'' such that
:<math>{\mathbf e}' = {\mathbf e}\, g,\quad \text{i.e., }\,e'_\alpha = \sum_\beta e_\beta g^\beta_\alpha.</math>
Applying the exterior connection to both sides gives the transformation law for ''ω'':
:<math>\omega(\mathbf e\, g) = g^{-1}dg+g^{-1}\omega(\mathbf e)g.</math>
Note in particular that ''ω'' fails to transform in a [[tensor]]ial manner, since the rule for passing from one frame to another involves the derivatives of the transition matrix ''g''.