Divergence-Free Fields: Energy and Asymptotic Crossing Number

In more than one context the equations of classical physics lead to a closed 2-form co on 3-dimensional space which is a conserved quantity in the sense that it pulls back under time evolution. In the physics literature co is written as a divergence-free vector field W. Appendix A describes two settings in which a divergence-free W is conserved or frozen. One is the magnetic field in ideal magneto-hydrodynamics and the other is curl (velocity) in incompressible, inviscid fluid flow. If ht denotes the transport diffeomorphisms of a moving fluid in which the field is imbedded with ho = id, Wt denotes the field in the fluid at time t and x0 is the space variable at time t = 0, then