The Clairaut-Schwarz theorem is the key fact needed to prove that for every C ∞ ∞ (or at least twice differentiable) differential form ω ω ∈ ∈ Ω Ω k ( M ) , the second exterior derivative vanishes: d 2 ω ω := d ( d ω ω ) = 0 . This implies that every differentiable exact form (i.e., a form α α such that α α = d ω ω for some form ω ω ) is closed (i.e., d α α = 0 ), since d α α = d ( d ω ω ) = 0 .