What is the gradient of a divergence and is it always zero?

[James Essig]

Hi Folks,

Was just curious as to what is the gradient of a divergence is and is it always equal to the zero vector. I am doing some free lance research and find that I need to refresh my knowledge of vector calculus a bit. I am having some difficulty with finding web-based sources for the gradient of a divergence.

 

[fresh_42]

Here's a little discussion on the terms:
https://www.physicsforums.com/threads/laplacian-vs-gradient-of-divergence.491935/
and here are some nice examples:
https://www.maths.tcd.ie/~evd/2E2/Notes/vector,grad,div,curl.pdf

 

[James Essig]

Thanks for the info fresh_42. I found those two links very helpful and I solved my problem of the reason for my inquiry.

 

[Charles Link]

One very important vector identity, (it is used in showing Maxwell's equations result in an electromagnetic wave equation), is $\nabla \times \nabla \times \vec{A}=\nabla (\nabla \cdot \vec{A})-\nabla^2 \vec{A}$. For the case that is often shown to demonstrate the wave equation in a vacuum, $\nabla \cdot \vec{E}=0$, but in general, the first term on the right side of the vector identity equation is not equal to zero.