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Maxwell Equations(Differential)

Gauss’s Law:
\[\nabla\cdot\mathbf{\vec{E}} =\frac{\rho}{\varepsilon_0} \\ \nabla\cdot \mathbf{\vec{B}}=0\]
Circuital Law:
\[\nabla\times\mathbf{\vec{E}}=-\frac{\partial\mathbf{\vec{B}}}{\partial t}\\ \nabla\times\mathbf{\vec{B}}=\mu_0\left (\mathbf{\vec{J}}+\varepsilon_0 \frac{\partial\mathbf{\vec{E}}}{\partial t}\right )\]

Maxwell Equations(Integral)

Gauss’s Law:
\[\iint_{\partial\Omega}\mathbf{\vec{E}}\cdot d\mathbf{\vec{S}}=\frac{1}{\varepsilon_0}\iiint_\Omega \rho dV \\ \iint_{\partial\Omega}\mathbf{\vec{B}}\cdot d\mathbf{\vec{S}}=0 \\\]
Circuital Law:
\[\oint_{\partial\Sigma}\mathbf{\vec{E}}\cdot d{\vec{l}}=-\frac{d}{dt}\iint_\Sigma \mathbf{\vec{B}}\cdot d\mathbf{\vec{S}} \\ \oint_{\partial\Sigma}\mathbf{\vec{B}}\cdot d\mathbf{\vec{l}}=\mu_0\left (\iint_\Sigma\mathbf{\vec{J}}\cdot d\mathbf{\vec{S}}+\varepsilon_0\frac{d}{dt}\iint_\Sigma\mathbf{\vec{E}}\cdot d\mathbf{\vec{S}} \right )\]