If a vector field '''F''' with zero divergence is defined on a ball in '''R'''<sup>3</sup>, then there exists some vector field '''G''' on the ball with '''F''' = curl('''G'''). For regions in '''R'''<sup>3</sup> more complicated than this, the latter statement might be false (see [[Poincaré lemma]]). The degree of ''failure'' of the truth of the statement, measured by the [[homology (mathematics)|homology]] of the [[chain complex]]
:<math> \{\mbox{scalar fields on }U\} \;</math>
::<math> \to\{\mbox{vector fields on }U\} \;</math>
:::<math> \to\{\mbox{vector fields on }U\} \;</math>
::::<math> \to\{\mbox{scalar fields on }U\} \;</math>
(where the first map is the gradient, the second is the curl, the third is the divergence) serves as a nice quantification of the complicatedness of the underlying region ''U''. These are the beginnings and main motivations of [[de Rham cohomology]].
