$$\int^{\infty}_{-\infty} \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} e^{-x^2-y^2-z^2}\mathrm{d} x \mathrm{d} y \mathrm{d} z=\pi^2$$ $$e^{i\pi}=-1$$ Gauss's law $$\varepsilon_0 \oint_{\partial \Omega} \mathbf{E} \cdot \mathrm{d} \mathbf{S}=\iiint_{\Omega} \rho \, \mathrm{d} V$$ Gauss's law for magnetism $$\oint_{\partial \Omega} \mathbf{B} \cdot \mathrm{d} \mathbf{S}=0$$ Maxwell-Faraday equation (Faraday's law of induction) $$\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} = - \frac{\partial}{\partial t} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S}$$ Ampere's circuital law (with Maxwell's addition) $$\oint_{\partial \Sigma} \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 \iint_{\Sigma} \mathbf{J} \cdot \mathrm{d}\mathbf{S} + \mu_0 \varepsilon_0 \frac{\partial}{\partial t} \iint_{\Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{S}$$ Gauss's law $$\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}$$ Gauss's law for magnetism $$\nabla \cdot \mathbf{B} = 0$$ Maxwell-Faraday equation (Faraday's law of induction) $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$ Ampere's circuital law (with Maxwell's addition) $$\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right)$$

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