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- TL;DR Summary
- can flux tubes be represented as a pair of functions in R^3

I am trying to see if Vector fields(I am thinking of electric and magnetic fields) without sources(divergence less) can be represented by a pair of functions f and g such that the level surfaces of the functions represent flux lines. I am trying to solve this problem in ## R^3 ## with a euclidean metric. It seems there is a linear space generated by ##a f +b g ## preserving the flux lines, so these functions are not uniquely defined.

I have some queries related to questions of this type.

I have some queries related to questions of this type.

- Can it be done locally ?( it seems this is the case)
- Can I also represent the magnitude of the vector fields(probably as a dual vector associated with ## df^dg ## and euclidean metric)
- Are there any topological obstructions when you try to solve the local problem and extend to all of ## R^3 ##
- Can it also be done if we include sources (remove the divergence free condition)
- Is there a general theory dealing with questions of this type? In specific if I have a manifold M of dimension d with a metric g and p-form fluxes, can I find d-p functions that can be used to represent these fluxes.
- Does this problem reduce to other mathematical quantities/results? Are there any general readings useful to approach these kinds of problems ?