Finding Vector Fields that Satisfy Certain Curls

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SUMMARY

This discussion addresses the challenge of finding vector fields with specific curls, particularly focusing on the vector field whose curl is given by yi and xi. The user successfully identifies that the vector field F = <0, 0, y^2/2> yields a curl of yi. However, it is established that no vector field can have a curl of xi due to its divergence being non-zero (specifically, 1). The concept of vector potential is introduced as a relevant tool for further exploration.

PREREQUISITES
  • Understanding of vector calculus, specifically curl and divergence.
  • Familiarity with vector fields and their properties.
  • Knowledge of vector potential and its applications.
  • Basic proficiency in mathematical notation and operations involving vectors.
NEXT STEPS
  • Research the properties of curl and divergence in vector calculus.
  • Study the concept of vector potential and its derivation.
  • Explore examples of vector fields with specified curls.
  • Investigate the implications of non-zero divergence in vector field analysis.
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are working with vector fields and seeking to understand the conditions under which specific curls can be achieved.

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Homework Statement


Is it possible to find a vector field whose curl is yi? xi?

Homework Equations





The Attempt at a Solution


I found that if F = <0,0,y^2/2>, its curl will be yi. However, I cannot figure out a vector field whose curl will be xi. I tried using exponentials and just different combination of x's, y's and z's as components of F, and just building my function as I go through the 'curl determinant', but haven't been able to find anything. If someone could please help me out that would be greatly appreciated. Thank you
 
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Take the divergence of the vector field you're interested. If it's zero, then you can find the "vector potential" for the vector field you're interested in (for more info, see http://en.wikipedia.org/wiki/Vector_potential).

Since the divergence of xi is 1, you cannot find a vector field whose curl is xi.

Hope that helps!
 
Thanks for the help!
 

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