This is a homework question. I just cannot wrap my head around on how to solve these kinds of questions. The question is as follows
Consider the following system of global conservation laws
$$ -\int_C \vec{H} \cdot d\vec{r} + \epsilon_0 \frac{d}{dt} \iint_{\Sigma} \vec{E} \cdot d\vec{S} = -\iint_{\Sigma} \vec{J} \cdot d\vec{S} $$
where $C$ is the boundary of the curve of the surface $\Sigma$ and all conditions of the Divergence and Stoke's Theorem are satisfied. Assuming that these global conservation laws hold for any $\Sigma$ , derive the corresponding system of partial differential equations.
So the question states that Stoke's Theorem can be used. This means that
$$\int_C \vec{F} \cdot d\vec{r} = \iint_{\Sigma} \nabla \cdot \vec{F} dS $$
Then the system of global conservation laws becomes
$$ - \iint_{\Sigma} \nabla \cdot \vec{H} dS + \epsilon_0 \frac{d}{dt} \iint_{\Sigma} \vec{E} \cdot d\vec{S} = - \iint_{\Sigma} \vec{J} \cdot d\vec{S} $$
but I do not know how to rewrite the second and the third term in order to get rid of the inner product between $d\vec{S}$ and the other term. I know that eventually, because of the mean-value theorem, the double integrals and $dS$ term of each part of the equation can be divided out.
What am I doing wrong? Am I missing something or am I interpreting Stoke's theorem incorrectly?