\[\mathbf{F}=q (\mathbf{E} + \mathbf{v} \times \mathbf{H})\]

\[\begin{equation} \begin{aligned} \mathbf{\nabla} \cdot \mathbf{F} &=q \mathbf{\nabla} \cdot(\mathbf{E} + \mathbf{v} \times \mathbf{H} ) \\ &= q \mathbf{\nabla} \cdot\mathbf{E} +q \mathbf{\nabla} \cdot (\mathbf{v} \times \mathbf{H} ) \\ &= q \frac{\rho}{\epsilon_0} + q \mathbf{\nabla} \cdot (\mathbf{v} \times \mathbf{H} \; Using \; Maxwell's \; Laws \\ &= q \frac{\rho}{\epsilon_0} + q \mathbf{H} \cdot \mathbf{\nabla} \times \mathbf{v} -q \mathbf{v} \cdot (\mathbf{\nabla} \times \mathbf{H}) \; Using \; a \; vector \; identity \\ &= q \frac{\rho}{\epsilon_0} + q \mathbf{H} \cdot \mathbf{\nabla} \times \mathbf{v} -q \mathbf{v} \cdot (\epsilon_0 \frac{\mathbf{E}}{\partial t} + \mathbf{J}) U \end{aligned} \end{equation} \]