マクスウェルの応力テンソル

${\displaystyle \mathrm {T} \equiv \left({\boldsymbol {D}}\otimes {\boldsymbol {E}}-\mathrm {I} \int {\boldsymbol {D}}\cdot d{\boldsymbol {E}}\right)+\left({\boldsymbol {B}}\otimes {\boldsymbol {H}}-\mathrm {I} \int {\boldsymbol {B}}\cdot d{\boldsymbol {H}}\right)}$

${\displaystyle \mathrm {T} =\epsilon _{0}\left({\boldsymbol {E}}\otimes {\boldsymbol {E}}-\mathrm {I} \,{\frac {{\boldsymbol {E}}^{2}}{2}}\right)+{\frac {1}{\mu _{0}}}\left({\boldsymbol {B}}\otimes {\boldsymbol {B}}-\mathrm {I} \,{\frac {{\boldsymbol {B}}^{2}}{2}}\right)}$

${\displaystyle {\begin{aligned}\nabla \cdot \mathrm {T} _{\text{e}}&=\partial _{i}(D_{i}{\boldsymbol {E}})-D_{i}\nabla E_{i}\\&=(\partial _{i}D_{i}){\boldsymbol {E}}+D_{i}\partial _{i}{\boldsymbol {E}}-D_{i}\nabla E_{i}\\&=(\nabla \cdot {\boldsymbol {D}}){\boldsymbol {E}}+({\boldsymbol {D}}\cdot \nabla _{E}){\boldsymbol {E}}-\nabla _{E}({\boldsymbol {D}}\cdot {\boldsymbol {E}})\\&=(\nabla \cdot {\boldsymbol {D}}){\boldsymbol {E}}-{\boldsymbol {D}}\times (\nabla \times {\boldsymbol {E}})\\\end{aligned}}}$

${\displaystyle {\boldsymbol {a}}\times ({\boldsymbol {b}}\times {\boldsymbol {c}})={\boldsymbol {b}}({\boldsymbol {a}}\cdot {\boldsymbol {c}})-({\boldsymbol {a}}\cdot {\boldsymbol {b}}){\boldsymbol {c}}}$

${\displaystyle {\begin{aligned}\nabla \cdot \mathrm {T} &=(\nabla \cdot {\boldsymbol {D}}){\boldsymbol {E}}-{\boldsymbol {D}}\times (\nabla \times {\boldsymbol {E}})+(\nabla \cdot {\boldsymbol {B}}){\boldsymbol {H}}-{\boldsymbol {B}}\times (\nabla \times {\boldsymbol {H}})\\&=\rho {\boldsymbol {E}}+{\boldsymbol {D}}\times {\frac {\partial {\boldsymbol {B}}}{\partial t}}-{\boldsymbol {B}}\times {\frac {\partial {\boldsymbol {D}}}{\partial t}}-{\boldsymbol {B}}\times {\boldsymbol {j}}\\&={\frac {\partial ({\boldsymbol {D}}\times {\boldsymbol {B}})}{\partial t}}+\rho {\boldsymbol {E}}+{\boldsymbol {j}}\times {\boldsymbol {B}}\\\end{aligned}}}$

${\displaystyle \oint _{\partial V}d{\boldsymbol {S}}\cdot \mathrm {T} ={\frac {\partial }{\partial t}}\int _{V}({\boldsymbol {D}}\times {\boldsymbol {B}})dV+\int _{V}(\rho {\boldsymbol {E}}+{\boldsymbol {j}}\times {\boldsymbol {B}})dV}$

${\displaystyle {\boldsymbol {g}}={\boldsymbol {D}}\times {\boldsymbol {B}}}$

${\displaystyle \{\lambda \}=\left\{-{\frac {\epsilon _{0}E^{2}+B^{2}/\mu _{0}}{2}},~\pm {\sqrt {\left({\frac {\epsilon _{0}E^{2}-B^{2}/\mu _{0}}{2}}\right)^{2}+\left({\frac {\epsilon _{0}}{\mu _{0}}}{\boldsymbol {E}}\cdot {\boldsymbol {B}}\right)^{2}}}\right\}}$

${\displaystyle \{\lambda \}=\left\{-{\frac {\epsilon _{0}E^{2}}{2}},~-{\frac {\epsilon _{0}E^{2}}{2}},~+{\frac {\epsilon _{0}E^{2}}{2}}\right\}}$

${\displaystyle \{{\boldsymbol {v}}\}=\left\{{\boldsymbol {E}}\times {\boldsymbol {E}}_{y},~-{\boldsymbol {E}}\times {\boldsymbol {E}}_{z},~{\boldsymbol {E}}E_{x}\right\}}$

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