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The helicity of a vector field is a measure of the average linking of pairsof integral curves of the field. Computed by a six-dimensional integral, it iswidely useful in the physics of fluids. For a divergence-free field tangent tothe boundary of a domain in 3-space, helicity is known to be invariant undervolume-preserving diffeomorphisms of the domain that are homotopic to theidentity. We give a new construction of helicity for closed $(k+1)$-forms on adomain in $(2k+1)$-space that vanish when pulled back to the boundary of thedomain. Our construction expresses helicity in terms of a cohomology classrepresented by the form when pulled back to the compactified configurationspace of pairs of points in the domain. We show that our definition isequivalent to the standard one. We use our construction to give a new formulafor computing helicity by a four-dimensional integral. We provide a Biot-Savartoperator that computes a primitive for such forms; utilizing it, we obtainanother formula for helicity. As a main result, we find a general formula forhow much the value of helicity changes when the form is pushed forward by adiffeomorphism of the domain; it relies upon understanding the effect of thediffeomorphism on the homology of the domain and the de Rham cohomology classrepresented by the form. Our formula allows us to classify thehelicity-preserving diffeomorphisms on a given domain, finding newhelicity-preserving diffeomorphisms on the two-holed solid torus, and provingthat there are no new helicity-preserving diffeomorphisms on the standard solidtorus. We conclude by defining helicities for forms on submanifolds ofEuclidean space. In addition, we provide a detailed exposition of some standard`folk' theorems about the cohomology of the boundary of domains in $\R^{2k+1}$.

A new cohomological formula for helicity in $\R^{2k+1}$ reveals the effect of a diffeomorphism on helicity