Mine is probably a very trivial question. I didn't get good training on vector calculus and I am not able to find the correct materials to learn to understand the following equation.
Given
$F = \int \frac{D^2}{8\pi\epsilon} dV$
where $\vec{D}$ is a vector, the book gives:
$\delta F=\int\frac{\vec{D}\cdot\delta\vec{ D}}{8\pi\epsilon}-\int\frac{D^2}{8\pi\epsilon^2}dV$
Can you explain why the 1st term is not $\int\frac{\vec{D}\cdot\delta\vec{ D}}{4\pi\epsilon}$? I would like to understand more about vector differential (is this what it call?). Can you give a suggestion on what this is called so that I can google the right keyword to learn more? Thanks a lot!
snapshot:
First of all, I have to say you're very brave for reading Landau and Lifshitz's Volume 8 about electrodynamics of continuous media $\ddot{\smile}$.
Anyway, you're right, it should be $4 \pi \epsilon$ in the denominator of the first term, and in my version of the book, that's how it is stated (so perhaps, yours is an older edition with typos?). So, we have: \begin{align} \delta\mathcal{F} &= \int\dfrac{\mathbf{D} \cdot \delta \mathbf{D}}{4 \pi \epsilon} \, dV - \int \dfrac{D^2}{8 \pi \epsilon^2} \delta \epsilon \, dV \end{align}
Anyway, this $\delta$ operation being done on the various quantities is called the "variation".