Example 2.3.3: Gradient of 1/R

$1/R$ is constant on spheres centred at the source point, so its gradient is radial (along $a_R$). Since $1/R$ decreases as we move away from the source, the gradient points towards the source (the minus sign), and its magnitude $1/R^2$ is the inverse-square law (Figure 2.3.2). The same steps show that $\nabla R=a_R$. If instead we differentiate with respect to the source coordinates (written ${\nabla }^{\prime }$), each derivative of $x-x^{\prime }$ produces an extra factor $-1$, so ${\nabla }^{\prime }(1/R)=+R/R^3=-\nabla (1/R)$. These relations connect the field of a point charge, $q{a}_R/(4\pi {\epsilon }_0{R}^2)$, with the potential $q/(4\pi {\epsilon }_{0}R)$ (Sections 3.1 and 3.3).