Дополнительные спин электромагнитные уравнения

Last modified by Alexey Popov on 2019/07/07 10:34

$\nabla \cdot \mathbf E = \rho\\$

$\nabla \cdot \mathbf B = 0\\$

$\mathbf v = \frac{d\mathbf r}{dt}\\ ,\, \mathbf a = \frac{d\mathbf v}{dt}\\ ,\, \mathbf j = \frac{d\mathbf a}{dt}\\ ,\, \mathbf s = \frac{d\mathbf j}{dt}\\$

$\mathbf J = \rho \mathbf v$

$\mathbf M = \rho (\mathbf r \times \mathbf a)$

$\mathbf S = \rho \mathbf a$

$\mathbf T = \rho (\mathbf r \times \mathbf j) + \rho (\mathbf v \times \mathbf a)$

$\mathbf K = \rho \mathbf j$

$\mathbf L = \rho (\mathbf r \times \mathbf s) + \rho (\mathbf v \times \mathbf j)$

$\nabla \times \mathbf B = \mathbf J + \frac{\partial \mathbf E}{\partial t}\\$

$\nabla \times \mathbf E = - \mathbf M - \frac{\partial \mathbf B}{\partial t}\\$

$\nabla \cdot \mathbf J = - \frac{\partial \rho}{\partial t}\\$

$\nabla \cdot \mathbf M = 0\\$

$\nabla \times (\nabla \times \mathbf B) = \nabla \times \mathbf J + \frac{\partial \nabla \times \mathbf E}{\partial t}\\$

$\nabla \times (\nabla \times \mathbf E) = - \nabla \times \mathbf M - \frac{\partial \nabla \times \mathbf B}{\partial t}\\$

$\nabla \times \frac{\partial \mathbf B}{\partial t} = \frac{\partial\mathbf J}{\partial t} + \frac{\partial^2 \mathbf E}{\partial t^2}\\$

$\nabla \times \frac{\partial \mathbf E}{\partial t} = - \frac{\partial\mathbf M}{\partial t} - \frac{\partial^2 \mathbf B}{\partial t^2}\\$

$\nabla \times (\nabla \times \mathbf B) = \nabla \times \mathbf J - \frac{\partial\mathbf M}{\partial t} - \frac{\partial^2 \mathbf B}{\partial t^2}- \mathbf T\\$

$\nabla \times (\nabla \times \mathbf E) = - \nabla \times \mathbf M - \frac{\partial\mathbf J}{\partial t} - \frac{\partial^2 \mathbf E}{\partial t^2} - \mathbf S\\$

$\nabla \times (\nabla \times \mathbf B) + \frac{\partial^2 \mathbf B}{\partial t^2}= \nabla \times \mathbf J - \frac{\partial\mathbf M}{\partial t}- \mathbf T\\$

$\nabla \times (\nabla \times \mathbf E) +\frac{\partial^2 \mathbf E}{\partial t^2} = - \nabla \times \mathbf M - \frac{\partial\mathbf J}{\partial t} - \mathbf S\\$

$\nabla \times \square \mathbf B = \nabla \times (\nabla \times \mathbf J) - \frac{\partial\nabla \times \mathbf M}{\partial t}- \nabla \times \mathbf T\\$

$\nabla \times \square \mathbf E= - \nabla \times (\nabla \times \mathbf M) - \frac{\partial\nabla \times \mathbf J}{\partial t} - \nabla \times \mathbf S\\$

$\nabla \times \frac{\partial \mathbf J}{\partial t} = \frac{\partial \square \mathbf B}{\partial t} + \frac{\partial^2\mathbf M}{\partial t^2}+ \frac{\partial \mathbf T}{\partial t}\\$

$\nabla \times \frac{\partial \mathbf M }{\partial t} = -\frac{\partial \square \mathbf E}{\partial t} - \frac{\partial^2\mathbf J}{\partial t^2} - \frac{\partial \mathbf S}{\partial t}\\$

$\nabla \times \square \mathbf B = \nabla \times (\nabla \times \mathbf J) +\frac{\partial \square \mathbf E}{\partial t} + \frac{\partial^2\mathbf J}{\partial t^2} + \frac{\partial \mathbf S}{\partial t} - \nabla \times \mathbf T -\mathbf K\\$

$\nabla \times \square \mathbf E= - \nabla \times (\nabla \times \mathbf M) - \frac{\partial \square \mathbf B}{\partial t} - \frac{\partial^2\mathbf M}{\partial t^2}- \frac{\partial \mathbf T}{\partial t}- \nabla \times \mathbf S + \mathbf L\\$

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Created by Alexey Popov on 2019/07/01 14:54