aus Wikisource, der freien Quellensammlung
wo
T
=
[
E
M
]
+
[
E
[
u
E
]
]
+
[
M
[
u
M
]
]
f
1
=
−
[
E
⋅
P
[
u
M
]
]
+
[
M
⋅
P
[
u
E
]
]
w
=
1
2
(
E
⋅
E
+
M
⋅
M
)
+
ε
0
μ
0
u
⋅
[
E
M
]
.
{\displaystyle {\begin{aligned}T&={\mathsf {[EM]}}+{\bigl [}{\mathsf {E}}[u{\mathfrak {E}}]{\bigr ]}+{\bigl [}{\mathsf {M}}[u{\mathfrak {M}}]{\bigr ]}\\f_{1}&=-{\bigl [}{\mathfrak {E}}\cdot {\mathsf {P}}[u{\mathfrak {M}}]{\bigr ]}+{\bigl [}{\mathfrak {M}}\cdot {\mathsf {P}}[u{\mathfrak {E}}]{\bigr ]}\\w&={\tfrac {1}{2}}({\mathsf {E}}\cdot {\mathfrak {E}}+{\mathsf {M}}\cdot {\mathfrak {M}})+\varepsilon _{0}\mu _{0}u\cdot {\mathsf {[EM]}}.\end{aligned}}}
(33)
Den Ausdruck für
T
{\displaystyle T}
müssen wir umformen. Es ist nach (e ):
[
E
[
u
E
]
]
+
[
M
[
u
M
]
]
=
(
E
⋅
E
+
M
⋅
M
)
u
−
(
u
⋅
E
)
E
−
(
u
⋅
M
)
M
=
(
E
⋅
E
+
M
⋅
M
)
u
−
(
u
⋅
E
)
ε
E
−
(
u
⋅
M
)
μ
M
−
ε
0
μ
0
{
−
(
u
⋅
E
)
[
u
M
]
+
(
u
⋅
M
)
[
u
E
]
}
.
{\displaystyle {\begin{aligned}{\bigl [}{\mathsf {E}}[u{\mathfrak {E}}]{\bigr ]}+{\bigl [}{\mathsf {M}}[u{\mathfrak {M}}]{\bigr ]}&=({\mathsf {E}}\cdot {\mathfrak {E}}+{\mathsf {M}}\cdot {\mathfrak {M}})u-(u\cdot {\mathsf {E}}){\mathfrak {E}}-(u\cdot {\mathsf {M}}){\mathfrak {M}}\\&=({\mathsf {E}}\cdot {\mathfrak {E}}+{\mathsf {M}}\cdot {\mathfrak {M}})u-(u\cdot {\mathsf {E}})\varepsilon {\mathsf {E}}-(u\cdot {\mathsf {M}})\mu {\mathsf {M}}\\&\qquad -\varepsilon _{0}\mu _{0}\{-(u\cdot {\mathsf {E}})[u{\mathsf {M}}]+(u\cdot {\mathsf {M}})[u{\mathsf {E}}]\}.\end{aligned}}}
Aber
{
}
=
[
u
{
E
(
u
⋅
M
)
−
M
(
u
⋅
E
)
}
]
=
[
u
[
u
[
E
M
]
]
]
=
(
u
⋅
[
E
M
]
)
u
−
u
2
[
E
M
]
.
{\displaystyle {\begin{aligned}\{\quad \}&=[u\{{\mathsf {E}}(u\cdot {\mathsf {M}})-{\mathsf {M}}(u\cdot {\mathsf {E}})\}]={\Bigl [}u{\bigl [}u{\mathsf {[EM]}}{\bigr ]}{\Bigr ]}\\&=(u\cdot {\mathsf {[EM]}})u-u^{2}{\mathsf {[EM]}}.\end{aligned}}}
und
E
⋅
E
+
M
⋅
M
=
ε
E
2
+
μ
M
2
+
2
ε
0
μ
0
u
⋅
[
E
M
]
{\displaystyle {\mathsf {E}}\cdot {\mathfrak {E}}+{\mathsf {M}}\cdot {\mathfrak {M}}=\varepsilon {\mathsf {E}}^{2}+\mu {\mathsf {M}}^{2}+2\varepsilon _{0}\mu _{0}u\cdot {\mathsf {[EM]}}}
.
Also
T
=
(
1
+
ε
0
μ
0
u
2
)
[
E
M
]
+
1
2
(
E
⋅
E
+
M
⋅
M
)
u
+
1
2
(
ε
E
2
+
μ
M
2
)
u
−
(
u
⋅
E
)
ε
E
−
(
u
⋅
M
)
μ
M
.
{\displaystyle T=(1+\varepsilon _{0}\mu _{0}u^{2}){\mathsf {[EM]}}+{\tfrac {1}{2}}({\mathsf {E}}\cdot {\mathfrak {E}}+{\mathsf {M}}\cdot {\mathfrak {M}})u+{\tfrac {1}{2}}(\varepsilon {\mathsf {E}}^{2}+\mu {\mathsf {M}}^{2})u-(u\cdot {\mathsf {E}})\varepsilon {\mathsf {E}}-(u\cdot {\mathsf {M}})\mu {\mathsf {M}}.}
Wir können daher (32) in folgender Form schreiben:
−
Γ
(
Σ
+
w
1
u
)
−
Γ
(
Y
)
=
∂
w
∂
t
+
E
⋅
Λ
+
u
⋅
(
f
0
+
f
1
)
oder
∫
(
Σ
n
+
w
1
u
n
)
d
S
+
∫
Y
n
d
S
=
∂
∂
t
∫
w
d
τ
+
∫
E
⋅
Λ
d
τ
+
∫
u
⋅
(
f
0
+
f
1
)
d
τ
,
}
{\displaystyle \left.{\begin{aligned}-{\mathsf {\Gamma }}({\mathsf {\Sigma }}+w_{1}u)-{\mathsf {\Gamma }}(Y)={\frac {\partial w}{\partial t}}+{\mathsf {E\cdot \Lambda }}+u\cdot (f_{0}+f_{1})\\{\text{oder}}\\\int ({\mathsf {\Sigma }}_{n}+w_{1}u_{n})dS+\int Y_{n}dS={\frac {\partial }{\partial t}}\int w\ d\tau +\int {\mathsf {E\cdot \Lambda }}\,d\tau +\int u\cdot (f_{0}+f_{1})d\tau ,\end{aligned}}\right\}}
(34)
wo
S
{\displaystyle S}
die Oberfläche von
τ
{\displaystyle \tau }
,
n
{\displaystyle n}
die innere Normale von
d
S
{\displaystyle dS}
,
Σ
=
(
1
+
ε
0
μ
0
u
2
)
[
E
M
]
{\displaystyle {\mathsf {\Sigma }}=(1+\varepsilon _{0}\mu _{0}u^{2}){\mathsf {[EM]}}}
(35)
w
1
=
1
2
(
E
⋅
E
+
M
⋅
M
)
w
=
1
2
(
E
⋅
E
+
M
⋅
M
)
+
ε
0
μ
0
u
⋅
[
E
M
]
}
{\displaystyle \left.{\begin{aligned}w_{1}&={\tfrac {1}{2}}({\mathsf {E}}\cdot {\mathfrak {E}}+{\mathsf {M}}\cdot {\mathfrak {M}})\\w&={\tfrac {1}{2}}({\mathsf {E}}\cdot {\mathfrak {E}}+{\mathsf {M}}\cdot {\mathfrak {M}})+\varepsilon _{0}\mu _{0}u\cdot {\mathsf {[EM]}}\end{aligned}}\right\}}
(36)
Y
n
=
1
2
(
ε
E
2
+
μ
M
2
)
u
n
−
(
u
⋅
E
)
ε
E
n
−
(
u
⋅
M
)
μ
M
n
{\displaystyle Y_{n}={\tfrac {1}{2}}(\varepsilon {\mathsf {E}}^{2}+\mu {\mathsf {M}}^{2})u_{n}-(u\cdot {\mathsf {E}})\varepsilon {\mathsf {E}}_{n}-(u\cdot {\mathsf {M}})\mu {\mathsf {M}}_{n}}
(37)
oder
Y
n
=
−
u
⋅
π
n
,
wo
π
n
ein Vector mit den Componenten
π
x
n
,
π
y
n
,
π
z
n
;
π
x
n
=
π
x
x
cos
(
n
x
)
+
π
x
y
cos
(
n
y
)
+
π
x
z
cos
(
n
z
)
;
π
x
x
=
−
1
2
(
ε
E
2
+
μ
M
2
)
+
ε
E
x
E
x
+
μ
M
x
M
x
π
x
y
=
π
y
x
=
ε
E
x
E
y
μ
M
+
x
M
y
etc.
}
{\displaystyle \left.{\begin{aligned}Y^{n}&=-u\cdot \pi ^{n},{\text{wo }}\pi ^{n}{\text{ ein Vector mit den Componenten }}\pi _{x}^{n},\pi _{y}^{n},\pi _{z}^{n};\\\pi _{x}^{n}&=\pi _{x}^{x}\cos(nx)+\pi _{x}^{y}\cos(ny)+\pi _{x}^{z}\cos(nz);\\\pi _{x}^{x}&=-{\tfrac {1}{2}}(\varepsilon {\mathsf {E}}^{2}+\mu {\mathsf {M}}^{2})+\varepsilon {\mathsf {E}}_{x}{\mathsf {E}}_{x}+\mu {\mathsf {M}}_{x}{\mathsf {M}}_{x}\\\pi _{x}^{y}&=\pi _{y}^{x}=\varepsilon {\mathsf {E}}_{x}{\mathsf {E}}_{y}\mu {\mathsf {M}}+{}_{x}{\mathsf {M}}_{y}\ {\text{etc.}}\end{aligned}}\right\}}
(37′)
f
0
{\displaystyle f_{0}}
und
f
1
{\displaystyle f_{1}}
aus (30) und (33).