Proving Vector Integral Identity

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SUMMARY

The discussion centers on proving the vector integral identity using Stokes' theorem. The identity involves the line integral of the vector field A and its relationship to the surface integral of the curl of A. Participants emphasize the importance of selecting appropriate vector fields and their derivatives to apply Stokes' theorem effectively. The consensus is that while the proof is challenging, it is feasible without resorting to tensor notation.

PREREQUISITES
  • Understanding of Stokes' theorem
  • Familiarity with vector calculus
  • Knowledge of vector identities
  • Basic concepts of line and surface integrals
NEXT STEPS
  • Study the applications of Stokes' theorem in vector calculus
  • Explore common vector identities and their proofs
  • Investigate the relationship between line integrals and surface integrals
  • Learn about tensor notation and its applications in advanced calculus
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Mathematicians, physics students, and anyone interested in advanced vector calculus and integral identities.

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[tex]\oint {d{\bf{R}} \times {\bf{A}}} = \mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc} <br /> {\left( {d{\bf{S}} \times \nabla } \right) \times {\bf{A}}}[/tex]
 
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I would try Stokes' theorem. Just try to find apropriate fields and field derivatives instead of the original fields in the theorem. I'm almost sure it will work.
 
I am trying to prove this using the common vector identities and Stoke's theorem, without going into tensor notation at every step. It seems to be quite impossible.
 

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