Helmholtz-Hodge Decomposition of vector fields on 2-manifolds

A Morse-like Decomposition ? - Morse-Smale decomposition for gradient (of scalar) fields is an interesting way of decomposing the domain into regions of unidirectional flow (from a source to a sink ). - But works for gradient fields, which are conservative (irrotational), only. - Can such a decomposition and analysis be extended to generic (consisting rotational component) vector fields ? - Can we extract the rotational component out from generic vector fields ? Feature Identification ? - Analysis on the decomposed components of fields is simpler. eg Identification of critical points in the potentials of the two components is easy. Topological Consistency ? - Is there any relation between the topology of the components and the topology of the original field ? Limitation - So far, HH Decomposition exists only for piece-wise constant vector fields. Such a decomposition for piece-wise linear fields is desirable.