Gradient wrt. a Position Vector

Question

I found the following equations in a dynamics textbook,

The gravitational potential energy for any two particles in a $n$-particle system is given by,

$$V_{ij} = - \frac {G m_i m_j}{r_{ij}}$$

where $r_{ij}$ is the distance between $m_i$ and $m_j$. The total potential energy of the system is,

$$V = \frac{1}{2} \sum_{i = 1}^{n} \sum_{j = 1}^{n}V_{ij} \qquad (i \neq j)$$

If $R_i$ is the postion vector of the $i^{th}$ particle, then

$$\frac{\partial{V}} {\partial {\vec{R_{i}}}} = - \frac{\partial{V}}{\partial{\vec{r_{ji}}}} + \frac{\partial{V}}{\partial{\vec{r_{ij}}}} = -2 \frac{\partial{V}}{\partial{\vec{r_{ij}}}}.$$

What does it mean to take the derivative of a Scalar Function$(V)$ with respect to a vector$(\vec{R_1})$? Is it directional derivative?

I've been trying all day to get the last equation. I would be very grateful if somebody could help me(or mention some reference perhaps). I don't really know which part of math is used to get the last equation.

Answer 1

$\frac{\partial V}{\partial \vec R}$ is the vector whose $i^{th}$ component is $\frac{\partial V}{\partial R_i}$. In other words, $$\frac{\partial V}{\partial \vec R} = \left<\frac{\partial V}{\partial R_1},\frac{\partial V}{\partial R_2}, \frac{\partial V}{\partial R_3}\right>$$