The curl transform of a linear force-free magnetic field

An expression is derived giving the curl transform of a linear force-free magnetic field in terms of the curl eigenfunction projection on the sum of two integrals over the field in physical space. The result is verified by direct substitution of the curl eigenfunction expansion of the field and exemplified by deriving the curl transform for the simplest force-free magnetic field. This provides the means to compare the curl transforms of previously-worked-out fields with those derived directly from the transform space representations, effectively completing the description of linear force-free magnetic fields in terms of the curl eigenfunction expansion begun previously. The direct and inverse curl transforms provide a new approach to the practical and theoretical study of linear force-free magnetic fields (as well as Trkalian-Beltrami flows) and their classification.