Can Curl V Determine V? A Method for Finding Vector Fields

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Discussion Overview

The discussion revolves around the question of whether a vector field V can be determined from its curl, specifically exploring methods or algorithms for finding V given Curl V. The scope includes theoretical considerations, boundary conditions, and the implications of uniqueness in solutions to partial differential equations (PDEs).

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about reasonable algorithms to find a vector field V given Curl V.
  • Another participant asks for examples of functions that satisfy Curl V = 0, suggesting multiple functions could meet this condition.
  • A different participant introduces the idea of imposing boundary conditions on a surface and questions the uniqueness and regularity conditions for PDE solutions.
  • It is mentioned that if the domain is suitable, a conventional approach is to find V such that Curl V equals a given vector field T with divergence equal to zero.
  • One participant asserts that V can be determined from its curl, but only up to an arbitrary gradient field.

Areas of Agreement / Disagreement

Participants express varying views on the ability to determine V from Curl V, with some suggesting it is possible under certain conditions while others highlight the limitations and conditions that must be met. The discussion remains unresolved regarding the specifics of the methods and conditions required.

Contextual Notes

There are limitations regarding the assumptions about the domain and boundary conditions, as well as the dependence on definitions related to uniqueness and regularity in PDE solutions.

Jerbearrrrrr
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Say V is a vector field.
Is there a way (or rather a reasonable algorithm) to find V, given Curl V?

Thanks
 
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If I told you curlV=0, can you give me a bunch of different functions that satisfy this requirement?
 
Okay, let's impose V satisfies boundary conditions on a (hyper)surface or something. (Though I don't remember the conditions for uniqueness and regularity and so on of PDE solutions)

I guess I asked the wrong question.
Is there a way to find V within a transformation in what we might call the Kernel of the curl operator?

(Like when someone asks what differentiates to f, it's understood that we can add a constant...)
 
Yes, if the domain is good enough. The conventional way to phrase the problem is: given a vector field T with div T = 0, find V so that curl V = T. This should be in all multivariable calculus textbooks. But perhaps only after the discussion of vector integration.
 
You can determine V out of its curl, up to an arbitrary gradient field.
 

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