Helmholtz theorem derivation question

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SUMMARY

The discussion centers on the derivation of the Helmholtz theorem, specifically addressing the application of the Dirac delta function and the gradient operator (∇') in the context of vector fields. Participants clarify that ∇' represents the gradient with respect to r', and that the volume integral of a vector field involves performing a triple integral on each component independently. The conversation highlights the use of vector identities to manipulate integrals, emphasizing the importance of understanding these mathematical tools in the derivation process.

PREREQUISITES
  • Understanding of Helmholtz decomposition theorem
  • Familiarity with vector calculus concepts, including gradients and integrals
  • Knowledge of the Dirac delta function and its applications
  • Proficiency in using vector identities in mathematical proofs
NEXT STEPS
  • Study the Helmholtz decomposition theorem in detail
  • Learn about the properties and applications of the Dirac delta function
  • Explore vector calculus identities and their proofs
  • Practice solving triple integrals of vector fields
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Mathematicians, physicists, and engineering students who are studying vector calculus and the Helmholtz theorem, as well as anyone interested in advanced mathematical derivations and applications in physics.

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http://en.wikipedia.org/wiki/Helmholtz_decomposition#Derivation
how do we go from
f8d3b3e44dfc58f31df26e57c121a22f.png

to

6f79acfa5e48004f3a3d8c262d7205db.png


also on the next step
f87a951aa4c1323f4886374808b14276.png

∇' just means the gradient with respect to r', right?
Also, why do i have to use the dirac delta function? i thought it was only used to deal with discontinuities?

also when we say
gif.gif

gif.gif

the volume integral of a vector field just means we do the triple integral on each component of the vector field independently right?
 
Last edited:
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image-jpg.jpg
del is an operator though? if i applied that to this case then id get stuff like
gif.gif
or
gif.gif


i don't know why theyd explain each step then leave one completely mising
 

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