Grad(div(V)) = 0: Why is this Vector Identity Dropped?

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In summary, the conversation discusses a vector identity used in deriving EM waves equation, where the grad(div(V)) part is dropped assuming it equals 0. The reason for this is because there are no charges present, making the divergence 0 according to Maxwell's equations. The person asking the question realizes this and thanks the other participant for the explanation.
  • #1
DoobleD
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This is closely related to this thread I posted yesterday, but the question is different so I created another thread. There is a vector identity often used when deriving EM waves equation :

d0e4740eaf9a820b14f267ae70cf9bca.png


Then the grad(div(V)) part of it is simply dropped, assuming it equals 0. And I wonder why.

Is it because, since there is no "sources" here (no charges), any divergence is 0 ? Can this be proven more formally ?
 
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  • #2
Isn't it because the identity is used for ##V=E## and ##V=H##, and according to Maxwell's equations (see your Wikipedia link):
##\nabla.{E}=0##
##\nabla.{H}=0##
?
 
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  • #3
Yes, that's the reason.
 
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  • #4
Samy_A said:
Isn't it because the identity is used for ##V=E## and ##V=H##, and according to Maxwell's equations (see your Wikipedia link):
##\nabla.{E}=0##
##\nabla.{H}=0##
?

Oh ! Of course ! Thank you. Well, I formulated that divergence without charges/sources is 0, that is indeed Gauss's law from Maxwell's in vacuum...There is the obvious formalism I was looking for, I should have seen it. -_-'
 

1. Why is this vector identity important in science?

The vector identity Grad(div(V)) = 0 is important in science because it helps to describe the behavior of vector fields, which are important in many fields of science such as physics, engineering, and fluid mechanics. It also plays a crucial role in the study of vector calculus.

2. Can you explain the meaning of this vector identity?

This vector identity states that the gradient of the divergence of a vector field is equal to zero. In simpler terms, it means that the change in the divergence of a vector field is zero, which has important implications in the study of fluid flow, electric fields, and other physical phenomena.

3. Why is this vector identity sometimes dropped in scientific calculations?

This vector identity is often dropped in scientific calculations because it is considered redundant. It can be derived from other vector identities, such as the product rule and the chain rule. In some cases, it is also dropped to simplify equations and make them easier to solve.

4. What are the applications of this vector identity in real-life situations?

This vector identity has numerous applications in real-life situations. For example, it is used in the study of fluid dynamics to describe the behavior of fluids in pipes, channels, and other systems. It is also used in the analysis of electric and magnetic fields, as well as in the study of heat transfer and diffusion.

5. Is this vector identity always true?

Yes, this vector identity is always true. It is a fundamental property of vector fields and is derived from the laws of vector calculus. However, it may not always be applicable in certain situations, such as when dealing with discontinuous or non-differentiable vector fields.

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