What is mathematical identity behind Charge continuum?

Question

Lorentz force (per unit 3-volume) f on a continuous charge distribution (charge density ρ) in motion. The 3-current density J corresponds to the motion of the charge element dq in volume element dV and varies throughout the continuum.

Answer 1

Divergence theorem is mathematical identity which says in calculus:

$$\frac{\partial{\rho}}{\partial{t}}+\nabla \cdot \textbf{J}$$

Yes? Is worked backwards:

$$I=\iint_{\partial\lambda}{\textbf{J}\cdot d \textbf{S}}=\frac{dq}{dt}$$

I hope is help. In fact, Maxwell there was no tensor knowledge, was ONLY calculus of numbers and vector! He work each number out for each vector! Amazing!