Mathematically, we can write
$$\mathbf{F}\cdot\mathrm{d}\mathbf{l}=\pm\mathrm{d}E,\qquad\mathrm{if}\ \mathbf{F}\equiv\pm\mathbf{\nabla}E.$$
Here $\mathbf{F}$ is a vector field, $\mathbf{l}$ is a position vector, and $E$ is a scalar field.
In physics, the work done by a force $\mathbf{F}$ over the infinitesimal displacement $\mathrm{d}\mathbf{l}$ equals the left-hand side in the above relation. Thus, under the condition as given above, the right-hand side is also true in physics. The scalar field $E$ is called the energy corresponding to the given (conservative) force. When evaluating the above relationship and a minus sign appears, the corresponding energy is called potential energy (as for weight and the restoring force on a spring, for example).
For Newton's second law, $\mathbf{F}\cdot\mathrm{d}\mathbf{l}=+\mathrm{d}\left(\frac{1}{2}mv^2\right)$. This is an example of $\mathbf{F}=+\mathbf{\nabla}E$.