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:<math>\phi = u^{-1}\pi^*\overline{\phi}</math>
where ''u'' is viewed as a linear isomorphism <math>V \overset{\simeq}\to E_{\pi(u)} = (\pi^*E)_u, v \mapsto [u, v]</math>. φ is then a tensorial form of type ρ. Conversely, given a tensorial form φ of type ρ, the same formula defines an ''E''-valued form <math>\overline{\phi}</math> on ''M'' (cf. the [[Chern–Weil homomorphism]].) In particular, there is a natural isomorphism of vector spaces
:<math>\Gamma(M, E) \
Example: Let ''E'' be the tangent bundle of ''M''. Then identity bundle map id<sub>''E''</sub>: ''E'' →''E'' is an ''E''-valued one form on ''M''. The [[tautological one-form]] is a unique one-form on the frame bundle of ''E'' that corresponds to id<sub>''E''</sub>. Denoted by θ, it is a tensorial form of standard type.<!--Mention this somewhere else: The [[exterior covariant derivative]] of θ, Θ = ''D''θ is called a [[torsion form]].-->
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