In electromagnetism we have a charge conservation law that says $$\int_V \frac{\partial \rho}{\partial t} dV + \oint_{S=dV} \boldsymbol j \cdot \hat{\boldsymbol n}\,\,dS = 0$$ From here I want to prove the continuity equation $\frac{\partial \rho}{\partial t}+ \nabla \cdot \boldsymbol j=0$.
easy way
Apply divergence theorem on the second term and joining the integrals (by linearity of integration) $$\int_V \frac{\partial \rho}{\partial t}+ \nabla \cdot \boldsymbol j \,\,dV = 0$$ from which continuity equation easily follows
reverse
I wanted to not solve it like this, but the other way around. So I nabla-ed both side. Since the volume and its boundary are fixed, the nabla can go inside the integral. $$\int_V\nabla \cdot \frac{\partial \rho}{\partial t} dV+ \int_{S=dV}\nabla\cdot(\boldsymbol j \cdot \hat{\boldsymbol n}) dS = 0$$
The first term can become a surface integral like the second one and we can join the together.
Does this approach bring somewhere? I can't seem to go any further