Field Theory Representation and Angular Momentum Description of Vortex Structures

Based on field theory and vector analysis, this paper presents a rigorous derivation of the curl of the velocity field. Using the velocity definition of particle trajectories and the fundamental relations between momentum and angular momentum, a pure momentum expression and a full angular momentum expression with complete elimination of momentum are derived for the curl, highlighting the rotational nature of vortex motion. All derivations follow fundamental field theory identities and vector operation rules, with explicit identification of implicit physical constraints, achieving theoretical unification among the momentum field, angular momentum, and curl structure.