Representing flux tubes as a pair of level surfaces in R^3

[Prathyush]

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.

1. Can it be done locally ?( it seems this is the case)
2. Can I also represent the magnitude of the vector fields(probably as a dual vector associated with $df^dg$ and euclidean metric)
3. Are there any topological obstructions when you try to solve the local problem and extend to all of $R^3$
4. Can it also be done if we include sources (remove the divergence free condition)
5. 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.
6. Does this problem reduce to other mathematical quantities/results? Are there any general readings useful to approach these kinds of problems ?

 

[jvicens]

Hello,

Thank you for your questions regarding vector fields and their representation using a pair of functions. Let me address each of your queries in turn:

1. Can it be done locally?

Yes, it is possible to represent vector fields locally using a pair of functions. This is known as the Helmholtz decomposition theorem, which states that any vector field can be decomposed into a solenoidal (divergence-free) and an irrotational (curl-free) component. The solenoidal component can be represented by a scalar potential function, while the irrotational component can be represented by a vector potential function.

2. Can I also represent the magnitude of the vector fields?

Yes, you can represent the magnitude of the vector fields by using the dual vector associated with the gradient of the scalar potential function and the curl of the vector potential function. This is known as the Poynting vector, which represents the magnitude and direction of energy flow in electromagnetic fields.

3. Are there any topological obstructions when solving the local problem and extending to all of $R^3$?

There are no topological obstructions when solving the local problem, as long as the vector field is well-behaved and the functions used to represent it are continuous. However, when extending to all of $R^3$, there may be topological obstructions depending on the specific vector field and functions used.

4. Can it also be done if we include sources (remove the divergence-free condition)?

Yes, it is possible to represent vector fields with sources using a pair of functions. In this case, the solenoidal component will not be purely divergence-free, but will also include a contribution from the sources.

5. Is there a general theory dealing with questions of this type?

Yes, there is a general theory for representing vector fields using a pair of functions, as discussed above with the Helmholtz decomposition theorem. This theory is widely used in electromagnetism and fluid mechanics.

6. Does this problem reduce to other mathematical quantities/results? Are there any general readings useful to approach these kinds of problems?

This problem is closely related to the theory of differential forms and their applications in vector analysis. A good resource for understanding this topic is the book "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach" by John Hubbard and Barbara Hubbard.

I hope this helps answer your questions and provides some guidance for further readings on this topic. Best of