$$ {\large \text{Integral form of Maxwell-Heaviside Equations}} $$ \[ \begin{array}{ll} \text{Gauss's Law for Electric Fields} & \oint_S \mathbf{E} \cdot \hat{\mathbf{n}}\ da = \frac{1}{\epsilon_0} \int_V \rho\ dv \\ \text{Gauss's Law for Magnetic Fields} & \oint_S \mathbf{B} \cdot \hat{\mathbf{n}}\ da = 0 \\ \text{Faraday's Law} & \oint_C \mathbf{E} \cdot d\mathbf{l} = -\frac{\partial}{\partial t} \int_S \mathbf{B} \cdot \hat{\mathbf{n}}\ da \\ \text{Amp}\grave{\mathrm{e}}\text{re-Maxwell Law} & \oint_C \mathbf{B} \cdot d\mathbf{l} = \int_S \left( \mu_0 \mathbf{J} + \mu_0\epsilon_0 \frac{\partial}{\partial t} \mathbf{E} \right) \cdot \hat{\mathbf{n}}\ da \\ \end{array} \] $$ \\ \\ $$ $$ \text{Apply the Divergence Theorem: } \oint_S \mathbf{F} \cdot \hat{\mathbf{n}}\ da = \int_V \nabla \cdot \mathbf{F}\ dV $$ \[ \begin{array}{ll} \oint_S \mathbf{E} \cdot \hat{\mathbf{n}}\ da = \frac{1}{\epsilon_0} \int_V \rho\ dv \; &\Longrightarrow \; \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \\ \oint_S \mathbf{B} \cdot \hat{\mathbf{n}}\ da = 0 \; &\Longrightarrow \; \nabla \cdot \mathbf{B} = 0 \\ \end{array} \] $$ \\ \\ $$ $$ \text{Apply Stokes Theorem: } \oint_C \mathbf{F} \cdot d\mathbf{l} = \int_S (\nabla \times \mathbf{F}) \cdot \hat{\mathbf{n}}\ da $$ \[ \begin{array}{ll} \oint_C \mathbf{E} \cdot d\mathbf{l} = \int_S -\frac{\partial \mathbf{B}}{\partial t} \cdot \hat{\mathbf{n}}\ da \; &\Longrightarrow \; \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \\ \oint_C \mathbf{B} \cdot d\mathbf{l} = \int_S \left( \mu_0 \mathbf{J} + \mu_0\epsilon_0 \frac{\partial}{\partial t} \mathbf{E} \right) \cdot \hat{\mathbf{n}}\ da \; &\Longrightarrow \; \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \\ \end{array} \] $$ \\ \\ $$ $$ {\large \text{Differential form of Maxwell-Heaviside Equations}} $$ \[ \begin{array}{ll} \text{Gauss's Law for Electric Fields} & \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \\ \text{Gauss's Law for Magnetic Fields} & \nabla \cdot \mathbf{B} = 0 \\ \text{Faraday's Law} & \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \\ \text{Amp}\grave{\mathrm{e}}\text{re-Maxwell Law} & \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \\ \end{array} \] $$ \\ \\ $$ $$ \text{Set } \rho \text{ and } \mathbf{J} \text{ to zero, as there is no free charge or current in a vacuum:} $$ \[ \begin{aligned} \nabla \cdot \mathbf{E} &= 0 \\ \nabla \cdot \mathbf{B} &= 0 \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \times \mathbf{B} &= \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \\ \end{aligned} \]