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

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1. Nov 14, 2017

### 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.

2. Nov 14, 2017

### Staff: Mentor

3. Nov 14, 2017

### 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.

4. Nov 14, 2017

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.