Editing Connection (principal bundle) (section)

==Induced covariant and exterior derivatives==
For any [[linear representation]] ''W'' of ''G'' there is an [[associated vector bundle]] <math> P\times^G W</math> over ''M'', and a principal connection induces a [[Connection (vector bundle)|covariant derivative]] on any such vector bundle. This covariant derivative can be defined using the fact that the space of sections of <math> P\times^G W</math> over ''M'' is isomorphic to the space of ''G''-equivariant ''W''-valued functions on ''P''. More generally, the space of ''k''-forms [[vector-valued differential form|with values in]] <math> P\times^G W</math>  is identified with the space of ''G''-equivariant and horizontal ''W''-valued ''k''-forms on ''P''. If ''α'' is such a ''k''-form, then its [[exterior derivative]] d''α'', although ''G''-equivariant, is no longer horizontal. However, the combination d''α''+''ω''Λ''α'' is. This defines an [[exterior covariant derivative]] d<sup>''ω''</sup> from <math> P\times^G W</math>-valued ''k''-forms on ''M'' to <math> P\times^G W</math>-valued (''k''+1)-forms on ''M''. In particular, when ''k''=0, we obtain a covariant derivative on <math> P\times^G W</math>.