Элегантная система дифференциальных уравнений

Last modified by Alexey Popov on 2019/08/13 12:23

$\mathbf E = - \nabla \varphi - \frac{\partial \mathbf A }{\partial t}-\frac{1}{c^2}\nabla \times \boldsymbol{\mathcal M}$

$\mathbf B = \nabla \times \mathbf A - \frac{1}{c^4}\nabla \xi- \frac{1}{c^4}\frac{\partial \boldsymbol{\mathcal M}}{\partial t}$

$\nabla \times \mathbf E = -\frac{\partial \nabla \times \mathbf A}{\partial t}-\frac{1}{c^2}\nabla \times \nabla \times \boldsymbol{\mathcal M}$

$\nabla \times \mathbf B = \nabla \times \nabla \times \mathbf A - \frac{1}{c^4}\frac{\partial \nabla \times \boldsymbol{\mathcal M}}{\partial t}$

$\frac{\partial \mathbf E}{\partial t} = - \nabla \frac{\partial \varphi}{\partial t}-\frac{\partial^2 \mathbf A}{\partial t^2}-\frac{1}{c^2}\nabla \times \frac{\partial\boldsymbol{\mathcal M}}{\partial t}$

$\frac{\partial \mathbf B}{\partial t} = \nabla \times \frac{\partial \mathbf A }{\partial t}-\nabla \frac{\partial \mathbf A}{\partial t} - \frac{1}{c^4}\frac{\partial^2 \boldsymbol{\mathcal M}}{\partial t^2}$

$\nabla \times \mathbf E = -\frac{\partial \mathbf B}{\partial t} - \frac{1}{c^4}\nabla \frac{\partial \xi}{\partial t} - \frac{1}{c^4}\frac{\partial^2 \boldsymbol{\mathcal M}}{\partial t^2} -\frac{1}{c^2}\nabla \times \nabla \times \boldsymbol{\mathcal M}$

$\nabla \times \mathbf B = \nabla \times \nabla \times \mathbf A + \frac{1}{c^2}\frac{\partial \mathbf E}{\partial t} + \frac{1}{c^2}\nabla \frac{\partial \varphi}{\partial t}+\frac{1}{c^2}\frac{\partial^2 \mathbf A}{\partial t^2}$

$\nabla \cdot \mathbf A + \frac{1}{c^2}\frac{\partial \varphi}{\partial t} = \psi$

$\nabla \cdot \boldsymbol{\mathcal M} + \frac{1}{c^2}\frac{\partial \xi}{\partial t} = \tau$

$\mu_0 \mathbf M = - \square\boldsymbol{\mathcal M}$

$\mu_0 \mathbf J = - \square\mathbf A$

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Created by Alexey Popov on 2019/07/25 11:49