Helmholtz theorem derivation question

In summary, the conversation discusses the derivation of the Helmholtz decomposition, specifically the use of the gradient operator and the Dirac delta function. The conversation also mentions the use of a vector identity to slide the gradient operator inside the triple integral, and the confusion about the usage of the Deldel operator.
  • #1
DivergentSpectrum
149
15
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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  • #3
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
 

1. What is the Helmholtz theorem?

The Helmholtz theorem is a mathematical concept in vector calculus that states that any vector field can be broken down into two components: a solenoidal (divergence-free) component and an irrotational (curl-free) component.

2. What is the significance of the Helmholtz theorem?

The Helmholtz theorem is important because it allows us to simplify and better understand complex vector fields by breaking them down into their fundamental components.

3. How is the Helmholtz theorem derived?

The Helmholtz theorem is derived using the vector identities of gradient, divergence, and curl as well as the fundamental theorem of calculus.

4. What is the mathematical formula for the Helmholtz theorem?

The mathematical formula for the Helmholtz theorem is: V = S + I, where V represents the original vector field, S represents the solenoidal component, and I represents the irrotational component.

5. In what areas of science is the Helmholtz theorem commonly used?

The Helmholtz theorem has applications in various fields of science, including physics, engineering, meteorology, and fluid dynamics. It is particularly useful in the study of electromagnetism, as it helps to simplify and analyze complex electric and magnetic fields.

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