Assuming strong Newton's third law, why is $\nabla V(\vert {\bf r}_i-{\bf r}_j\vert)=({\bf r}_i-{\bf r}_j)f$? 
I don't understand how they come up with (1.34). All I know is that $\nabla V_{ij}=-{\bf F}_{ij}$, but I've never seen this scalar function $f$ appear. Has it something to do with the absolute value in the argument?
I can interpret it as that if we consider some distance between the particles, then $\nabla V$ can be given by this distance in vector-form times some scalar function... but how they came up with that, I have no clue.
 A: Mathematically,
$$\nabla V_{ij}(r) = \frac{\partial V}{\partial r}.\frac{\partial r}{\partial x} \hat i + \frac{\partial V}{\partial r}.\frac{\partial r}{\partial y} \hat j + \frac{\partial V}{\partial r}.\frac{\partial r}{\partial z} \hat k$$
Also, $$r^2 = x^2 + y^2 + z^2$$
(I have taken $r_{j} = 0 $ and $r_{i} = r$ for simplicity of calclulation). This will give 
$$\frac{\partial r}{\partial x_{i}} = \frac{x_{i}}{r}$$ for every i. 
And so,
$$\nabla V_{ij}(r) = \frac{\partial V}{\partial r}.\frac{1}{r}.(x\hat i + y\hat j +z\hat k)$$
$$=\frac{\partial V}{\partial r}.\frac{1}{r}.(\vec r)$$
Thus, $\frac{\partial V}{\partial r}.\frac{1}{r}$ is the $f$ mentioned.
Physically, the gradient of any potential is directed in the direction in which it is changing. And as $V$ depends only on $r$ it will change in the direction of $r$.
A: The force $\textbf{F}_{ij}$ must be pointing in the direction of $\textbf{r}_i-\textbf{r}_j$. I think what they want to mean there is that the vectorial part of the force must be in that direction. This can be written as $\textbf{r}_i-\textbf{r}_j$ times a scalar function $f$.
This example may help you. Gravitational potential between two masses can be written as $V(|\textbf{r}_i-\textbf{r}_j|)=-\frac{G m_i m_j}{|\textbf{r}_i-\textbf{r}_j|}$. On the other hand, gravitational force can be written as $\textbf{F}_{ij}=\frac{G m_i m_j}{|\textbf{r}_i-\textbf{r}_j|^2}(\textbf{r}_i-\textbf{r}_j)$ (you can calculate it explicitly). In this example the scalar function would be $f=\frac{G m_i m_j}{|\textbf{r}_i-\textbf{r}_j|^2}$.
Edit: I've seen Abhijeet Melkani's answer just now, you can see the general case there. My example is a particular situation.
