Diese Formelsammlung fasst Formeln und Definitionen der Analysis mit Vektor- und Tensorfeldern zweiter Stufe in der Kontinuumsmechanik zusammen.
Formelsammlung Tensoralgebra
Operatoren wie „
g
r
a
d
{\displaystyle \mathrm {grad} }
“ werden nicht kursiv geschrieben.
Buchstaben in der Mitte des Alphabets werden als Indizes benutzt:
i
,
j
,
k
,
l
∈
{
1
,
2
,
3
}
{\displaystyle i,j,k,l\in \{1,2,3\}}
Es gilt die Einsteinsche Summenkonvention ohne Beachtung der Indexstellung.
Kommt in einer Formel in einem Produkt ein Index doppelt vor wie in
c
=
a
i
b
i
{\displaystyle c=a_{i}b^{i}}
wird über diesen Index von eins bis drei summiert:
c
=
a
i
b
i
=
∑
i
=
1
3
a
i
b
i
{\displaystyle c=a_{i}b^{i}=\sum _{i=1}^{3}a_{i}b^{i}}
.
Kommen mehrere Indizes doppelt vor wie in
c
=
A
i
j
B
j
i
{\displaystyle c=A_{ij}B_{j}^{i}}
wird über diese summiert:
c
=
A
i
j
B
j
i
=
∑
i
=
1
3
∑
j
=
1
3
A
i
j
B
j
i
{\displaystyle c=A_{ij}B_{j}^{i}=\sum _{i=1}^{3}\sum _{j=1}^{3}A_{ij}B_{j}^{i}}
.
Ein Index, der nur einfach vorkommt wie
i
{\displaystyle i}
in
v
i
=
A
i
j
b
j
{\displaystyle v_{i}=A_{ij}b_{j}}
, ist ein freier Index. Die Formel gilt dann für alle Werte der freien Indizes:
v
i
=
A
i
j
b
j
↔
v
i
=
∑
j
=
1
3
A
i
j
b
j
∀
i
∈
{
1
,
2
,
3
}
{\displaystyle v_{i}=A_{ij}b_{j}\quad \leftrightarrow \quad v_{i}=\sum _{j=1}^{3}A_{ij}b_{j}\quad \forall \;i\in \{1,2,3\}}
.
Vektoren:
Alle hier verwendeten Vektoren sind geometrische Vektoren im dreidimensionalen euklidischen Vektorraum 𝕍={ℝ3 ,+,·}.
Vektoren werden mit Kleinbuchstaben bezeichnet.
Einheitsvektoren mit Länge eins werden wie in ê mit einem Hut versehen.
Vektoren mit unbestimmter Länge werden wie in
a
→
{\displaystyle {\vec {a}}}
mit einem Pfeil versehen.
Standardbasis
e
^
1
,
e
^
2
,
e
^
3
{\displaystyle {\hat {e}}_{1},{\hat {e}}_{2},{\hat {e}}_{3}}
Beliebige Basis
b
→
1
,
b
→
2
,
b
→
3
{\displaystyle {\vec {b}}_{1},{\vec {b}}_{2},{\vec {b}}_{3}}
mit dualer Basis
b
→
1
,
b
→
2
,
b
→
3
{\displaystyle {\vec {b}}^{1},{\vec {b}}^{2},{\vec {b}}^{3}}
Der Vektor
x
→
=
x
i
e
^
i
{\displaystyle {\vec {x}}=x_{i}{\hat {e}}_{i}}
wird durchgängig Ortsvektor genannt.
Tensoren zweiter Stufe werden wie in T mit fetten Großbuchstaben notiert. Insbesondere Einheitstensor 1 .
Koordinaten:
#Kartesische Koordinaten
x
1
,
x
2
,
x
3
∈
R
{\displaystyle x_{1},x_{2},x_{3}\in \mathbb {R} }
#Zylinderkoordinaten :
ρ
,
φ
,
z
{\displaystyle \rho ,\varphi ,z}
#Kugelkoordinaten :
r
,
ϑ
,
φ
{\displaystyle r,\vartheta ,\varphi }
Krummlinige Koordinaten
y
1
,
y
2
,
y
3
∈
R
{\displaystyle y_{1},y_{2},y_{3}\in \mathbb {R} }
Konstanten:
c
,
c
→
,
C
{\displaystyle c,{\vec {c}},\mathbf {C} }
Zeit t ∈ ℝ
Variablen: skalar r,s ∈ ℝ oder vektorwertig
r
→
,
s
→
∈
V
3
{\displaystyle {\vec {r}},{\vec {s}}\in \mathbb {V} ^{3}}
Feldfunktionen abhängig von
x
→
,
t
{\displaystyle {\vec {x}},t}
oder
y
→
,
t
{\displaystyle {\vec {y}},t}
:
Skalar
f
,
g
∈
R
{\displaystyle f,g\in \mathbb {R} }
oder vektorwertig
f
→
,
g
→
∈
V
3
{\displaystyle {\vec {f}},{\vec {g}}\in \mathbb {V} ^{3}}
Tensorwertig: S , T
Operatoren:
Differentialoperatoren :
#Nabla-Operator : 𝜵
#Gradient : grad
#Divergenz : div
#Rotation : rot
#Laplace-Operator : Δ
Ein Index hinter einem Komma bezeichnet die Ableitung nach einer Koordinate:
f
,
i
:=
∂
f
∂
x
i
,
f
i
,
j
k
=
∂
2
f
i
∂
x
j
∂
x
k
,
f
r
,
ϑ
=
∂
f
r
∂
ϑ
{\displaystyle f_{,i}:={\frac {\partial f}{\partial x_{i}}}\,,\quad f_{i,jk}={\frac {\partial ^{2}f_{i}}{\partial x_{j}\partial x_{k}}}\,,\quad f_{r,\vartheta }={\frac {\partial f_{r}}{\partial \vartheta }}}
Zeitableitung mit Überpunkt :
f
˙
=
d
f
d
t
,
f
→
˙
=
d
f
→
d
t
,
T
˙
=
d
d
t
T
{\displaystyle {\dot {f}}={\frac {\mathrm {d} f}{\mathrm {d} t}},{\dot {\vec {f}}}={\frac {\mathrm {d} {\vec {f}}}{\mathrm {d} t}},{\dot {\mathbf {T} }}={\frac {\mathrm {d} }{\mathrm {d} t}}\mathbf {T} }
Landau-Symbole : f = 𝓞(x): f wächst langsamer als x.
Kontinuumsmechanik:
δ
i
j
=
δ
i
j
=
δ
i
j
=
δ
j
i
=
{
1
falls
i
=
j
0
sonst
{\displaystyle \delta _{ij}=\delta ^{ij}=\delta _{i}^{j}=\delta _{j}^{i}=\left\{{\begin{array}{ll}1&{\text{falls}}\ i=j\\0&{\text{sonst}}\end{array}}\right.}
ϵ
i
j
k
=
e
^
i
⋅
(
e
^
j
×
e
^
k
)
=
{
1
falls
(
i
,
j
,
k
)
∈
{
(
1
,
2
,
3
)
,
(
2
,
3
,
1
)
,
(
3
,
1
,
2
)
}
−
1
falls
(
i
,
j
,
k
)
∈
{
(
1
,
3
,
2
)
,
(
2
,
1
,
3
)
,
(
3
,
2
,
1
)
}
0
sonst, d. h. bei doppeltem Index
{\displaystyle \epsilon _{ijk}={\hat {e}}_{i}\cdot ({\hat {e}}_{j}\times {\hat {e}}_{k})={\begin{cases}1&{\text{falls}}\;(i,j,k)\in \{(1,2,3),(2,3,1),(3,1,2)\}\\-1&{\text{falls}}\;(i,j,k)\in \{(1,3,2),(2,1,3),(3,2,1)\}\\0&{\text{sonst, d. h. bei doppeltem Index}}\end{cases}}}
Kreuzprodukt :
a
i
e
^
i
×
b
j
e
^
j
=
ϵ
i
j
k
a
i
b
j
e
^
k
{\displaystyle a_{i}{\hat {e}}_{i}\times b_{j}{\hat {e}}_{j}=\epsilon _{ijk}a_{i}b_{j}{\hat {e}}_{k}}
ϵ
i
j
k
e
^
k
=
e
^
i
×
e
^
j
{\displaystyle \epsilon _{ijk}{\hat {e}}_{k}={\hat {e}}_{i}\times {\hat {e}}_{j}}
Formelsammlung Tensoralgebra#Kreuzprodukt eines Vektors mit einem Tensor :
(
a
→
×
A
)
⋅
g
→
:=
a
→
×
(
A
⋅
g
→
)
{\displaystyle ({\vec {a}}\times \mathbf {A} )\cdot {\vec {g}}:={\vec {a}}\times (\mathbf {A} \cdot {\vec {g}})}
b
→
⋅
(
a
→
×
A
)
=
(
b
→
×
a
→
)
⋅
A
{\displaystyle {\vec {b}}\cdot ({\vec {a}}\times \mathbf {A} )=({\vec {b}}\times {\vec {a}})\cdot \mathbf {A} }
g
→
⋅
(
A
×
a
→
)
:=
(
g
→
⋅
A
)
×
a
→
{\displaystyle {\vec {g}}\cdot (\mathbf {A} \times {\vec {a}}):=({\vec {g}}\cdot \mathbf {A} )\times {\vec {a}}}
(
A
×
a
→
)
⋅
b
→
=
A
⋅
(
a
→
×
b
→
)
{\displaystyle (\mathbf {A} \times {\vec {a}})\cdot {\vec {b}}=\mathbf {A} \cdot ({\vec {a}}\times {\vec {b}})}
Kartesische Koordinaten
Bearbeiten
x
1
,
x
2
,
x
3
∈
R
{\displaystyle x_{1},x_{2},x_{3}\in \mathbb {R} }
mit Basisvektoren
e
^
1
=
(
1
0
0
)
,
e
^
2
=
(
0
1
0
)
,
e
^
3
=
(
0
0
1
)
{\displaystyle {\hat {e}}_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},\quad {\hat {e}}_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},\quad {\hat {e}}_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}
die Standardbasis oder allgemeiner eine beliebige Orthonormalbasis ist.
e
^
ρ
=
(
cos
(
φ
)
sin
(
φ
)
0
)
,
e
^
φ
=
(
−
sin
(
φ
)
cos
(
φ
)
0
)
,
e
^
z
=
(
0
0
1
)
{\displaystyle {\hat {e}}_{\rho }={\begin{pmatrix}\cos(\varphi )\\\sin(\varphi )\\0\end{pmatrix}},\quad {\hat {e}}_{\varphi }={\begin{pmatrix}-\sin(\varphi )\\\cos(\varphi )\\0\end{pmatrix}},\quad {\hat {e}}_{z}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}
e
^
ρ
,
φ
=
e
^
φ
,
e
^
φ
,
φ
=
−
e
^
ρ
e
^
z
,
φ
=
0
→
{\displaystyle {\hat {e}}_{\rho ,\varphi }={\hat {e}}_{\varphi },\quad {\hat {e}}_{\varphi ,\varphi }=-{\hat {e}}_{\rho }\quad {\hat {e}}_{z,\varphi }={\vec {0}}}
Winkelgeschwindigkeit#Zylinderkoordinaten :
ω
→
=
φ
˙
e
^
z
→
e
^
˙
ρ
/
φ
/
z
=
ω
→
×
e
^
ρ
/
φ
/
z
{\displaystyle {\vec {\omega }}={\dot {\varphi }}{\hat {e}}_{z}\;\rightarrow \;{\dot {\hat {e}}}_{\rho /\varphi /z}={\vec {\omega }}\times {\hat {e}}_{\rho /\varphi /z}}
e
^
r
=
(
sin
(
ϑ
)
cos
(
φ
)
sin
(
ϑ
)
sin
(
φ
)
cos
(
ϑ
)
)
,
e
^
ϑ
=
(
cos
(
ϑ
)
cos
(
φ
)
cos
(
ϑ
)
sin
(
φ
)
−
sin
(
ϑ
)
)
,
e
^
φ
=
(
−
sin
(
φ
)
cos
(
φ
)
0
)
{\displaystyle {\hat {e}}_{r}={\begin{pmatrix}\sin(\vartheta )\cos(\varphi )\\\sin(\vartheta )\sin(\varphi )\\\cos(\vartheta )\end{pmatrix}},\quad {\hat {e}}_{\vartheta }={\begin{pmatrix}\cos(\vartheta )\cos(\varphi )\\\cos(\vartheta )\sin(\varphi )\\-\sin(\vartheta )\end{pmatrix}},\quad {\hat {e}}_{\varphi }={\begin{pmatrix}-\sin(\varphi )\\\cos(\varphi )\\0\end{pmatrix}}}
Winkelgeschwindigkeit#Kugelkoordinaten :
ω
→
=
(
−
ϑ
˙
sin
(
φ
)
ϑ
˙
cos
(
φ
)
φ
˙
)
=
φ
˙
cos
(
ϑ
)
e
^
r
−
φ
˙
sin
(
ϑ
)
e
^
ϑ
+
ϑ
˙
e
^
φ
→
e
^
˙
r
/
ϑ
/
φ
=
ω
→
×
e
^
r
/
ϑ
/
φ
{\displaystyle {\begin{aligned}&{\vec {\omega }}={\begin{pmatrix}-{\dot {\vartheta }}\sin(\varphi )\\{\dot {\vartheta }}\cos(\varphi )\\{\dot {\varphi }}\end{pmatrix}}={\dot {\varphi }}\cos(\vartheta ){\hat {e}}_{r}-{\dot {\varphi }}\sin(\vartheta ){\hat {e}}_{\vartheta }+{\dot {\vartheta }}{\hat {e}}_{\varphi }\\&\rightarrow \;{\dot {\hat {e}}}_{r/\vartheta /\varphi }={\vec {\omega }}\times {\hat {e}}_{r/\vartheta /\varphi }\end{aligned}}}
Krummlinige Koordinaten
Bearbeiten
y
1
,
y
2
,
y
3
∈
R
{\displaystyle y_{1},y_{2},y_{3}\in \mathbb {R} }
b
→
i
=
∂
x
→
∂
y
i
,
b
→
i
=
grad
(
y
i
)
=
∂
y
i
∂
x
→
→
b
→
i
⋅
b
→
j
=
δ
i
j
{\displaystyle {\vec {b}}_{i}={\frac {\partial {\vec {x}}}{\partial y_{i}}},\quad {\vec {b}}^{i}=\operatorname {grad} (y_{i})={\frac {\partial y_{i}}{\partial {\vec {x}}}}\quad \rightarrow \quad {\vec {b}}_{i}\cdot {\vec {b}}^{j}=\delta _{i}^{j}}
Ableitung von Skalar-, Vektor- oder Tensorfunktionen
Bearbeiten
Gâteaux-Differential
Bearbeiten
D
f
(
x
)
[
h
]
:=
d
d
s
f
(
x
+
s
h
)
|
s
=
0
=
lim
s
→
0
f
(
x
+
s
h
)
−
f
(
x
)
s
{\displaystyle \,\mathrm {D} f(x)[h]:=\left.{\frac {\mathrm {d} }{\mathrm {d} s}}f(x+sh)\right|_{s=0}=\lim _{s\rightarrow 0}{\frac {f(x+sh)-f(x)}{s}}}
mit
s
∈
R
{\displaystyle s\in \mathbb {R} }
,
f
,
x
,
h
{\displaystyle f,x,h}
skalar-, vektor- oder tensorwertig aber
x
{\displaystyle x}
und
h
{\displaystyle h}
gleichartig.
Produktregel :
D
(
f
(
x
)
⋅
g
(
x
)
)
[
h
]
=
D
f
(
x
)
[
h
]
⋅
g
(
x
)
+
f
(
x
)
⋅
D
g
(
x
)
[
h
]
{\displaystyle \mathrm {D} (f(x)\cdot g(x))[h]=\mathrm {D} f(x)[h]\cdot g(x)+f(x)\cdot \mathrm {D} g(x)[h]}
Kettenregel :
D
f
(
g
(
x
)
)
[
h
]
=
D
f
(
g
)
[
D
g
(
x
)
[
h
]
]
{\displaystyle \mathrm {D} f{\big (}g(x){\big )}[h]=\mathrm {D} f(g)[Dg(x)[h]]}
Existiert ein beschränkter linearer Operator
A
{\displaystyle {\mathcal {A}}}
, sodass
A
[
h
]
=
D
f
(
x
)
[
h
]
∀
h
{\displaystyle {\mathcal {A}}[h]={Df}(x)[h]{\quad \forall \;}h}
gilt, so wird
A
{\displaystyle {\mathcal {A}}}
Fréchet-Ableitung von
f
{\displaystyle f}
nach
x
{\displaystyle x}
genannt. Man schreibt dann auch
∂
f
∂
x
=
A
{\displaystyle {\frac {\partial f}{\partial x}}={\mathcal {A}}}
.
Ableitung von Potenzen eines Tensors
Bearbeiten
(
T
−
1
)
˙
=
−
T
−
1
⋅
T
˙
⋅
T
−
1
=
−
(
T
−
1
⊗
T
⊤
−
1
)
⊤
23
:
T
˙
d
T
−
1
d
T
=
−
(
T
−
1
⊗
T
⊤
−
1
)
⊤
23
(
T
⊤
−
1
)
˙
=
−
T
⊤
−
1
⋅
T
˙
⊤
⋅
T
⊤
−
1
=
−
(
T
⊤
−
1
⊗
T
⊤
−
1
)
⊤
24
:
T
˙
d
T
⊤
−
1
d
T
=
−
(
T
⊤
−
1
⊗
T
⊤
−
1
)
⊤
24
{\displaystyle {\begin{aligned}{\big (}\mathbf {T} ^{-1}{\dot {{\big )}\;}}=&-\mathbf {T} ^{-1}\cdot {\dot {\mathbf {T} }}\cdot {\mathbf {T} }^{-1}=-\left(\mathbf {T} ^{-1}\otimes \mathbf {T} ^{\top -1}\right)^{\stackrel {23}{\top }}:{\dot {\mathbf {T} }}\\{\frac {\mathrm {d} \mathbf {T} ^{-1}}{\mathrm {d} \mathbf {T} }}=&-\left(\mathbf {T} ^{-1}\otimes \mathbf {T} ^{\top -1}\right)^{\stackrel {23}{\top }}\\{\big (}\mathbf {T} ^{\top -1}{\dot {{\big )}\;}}=&-\mathbf {T} ^{\top -1}\cdot {\dot {\mathbf {T} }}^{\top }\cdot {\mathbf {T} }^{\top -1}=-\left(\mathbf {T} ^{\top -1}\otimes \mathbf {T} ^{\top -1}\right)^{\stackrel {24}{\top }}:{\dot {\mathbf {T} }}\\{\frac {\mathrm {d} \mathbf {T} ^{\top -1}}{\mathrm {d} \mathbf {T} }}=&-\left(\mathbf {T} ^{\top -1}\otimes \mathbf {T} ^{\top -1}\right)^{\stackrel {24}{\top }}\end{aligned}}}
siehe Formelsammlung Tensoralgebra#Spezielle Tensoren vierter Stufe .
Allgemein mit n ∈ ℕ, >0, T 0 := 1 :
D
T
n
(
T
)
[
H
]
=
∑
m
=
0
n
−
1
T
m
⋅
H
⋅
T
n
−
m
−
1
d
T
n
d
T
=
(
∑
m
=
0
n
−
1
T
m
⊗
(
T
n
−
m
−
1
)
⊤
)
⊤
23
{\displaystyle {\begin{aligned}\mathrm {D} \mathbf {T} ^{n}(\mathbf {T} )[\mathbf {H} ]=&\sum _{m=0}^{n-1}\mathbf {T} ^{m}\cdot \mathbf {H\cdot T} ^{n-m-1}\\{\frac {\mathrm {d} \mathbf {T} ^{n}}{\mathrm {d} \mathbf {T} }}=&\left(\sum _{m=0}^{n-1}\mathbf {T} ^{m}\otimes \left(\mathbf {T} ^{n-m-1}\right)^{\top }\right)^{\stackrel {23}{\top }}\end{aligned}}}
#Gâteaux-Differential der Inversen:
T
⋅
T
−
1
=
1
→
D
T
(
T
)
[
H
]
⏞
H
⋅
T
−
1
+
T
⋅
D
T
−
1
(
T
)
[
H
]
=
0
→
D
T
−
1
(
T
)
[
H
]
=
−
T
−
1
⋅
H
⋅
T
−
1
=
−
(
T
−
1
⊗
T
⊤
−
1
)
⊤
23
:
H
D
T
⊤
−
1
(
T
)
[
H
]
=
−
T
⊤
−
1
⋅
H
⊤
⋅
T
⊤
−
1
=
−
(
T
⊤
−
1
⊗
T
⊤
−
1
)
⊤
24
:
H
{\displaystyle {\begin{aligned}\mathbf {T\cdot T} ^{-1}=&\mathbf {1} \;\rightarrow \quad \overbrace {\mathrm {D} \mathbf {T} (\mathbf {T} )[\mathbf {H} ]} ^{\mathbf {H} }\cdot \mathbf {T} ^{-1}+\mathbf {T} \cdot \mathrm {D} \mathbf {T} ^{-1}(\mathbf {T} )[\mathbf {H} ]=\mathbf {0} \\\rightarrow \quad \mathrm {D} \mathbf {T} ^{-1}(\mathbf {T} )[\mathbf {H} ]=&-\mathbf {T} ^{-1}\cdot \mathbf {H} \cdot \mathbf {T} ^{-1}=-\left(\mathbf {T} ^{-1}\otimes \mathbf {T} ^{\top -1}\right)^{\stackrel {23}{\top }}:\mathbf {H} \\\mathrm {D} \mathbf {T} ^{\top -1}(\mathbf {T} )[\mathbf {H} ]=&-\mathbf {T} ^{\top -1}\cdot \mathbf {H} ^{\top }\cdot \mathbf {T} ^{\top -1}=-\left(\mathbf {T} ^{\top -1}\otimes \mathbf {T} ^{\top -1}\right)^{\stackrel {24}{\top }}:\mathbf {H} \end{aligned}}}
n ∈ ℕ, >0:
D
T
−
n
(
T
)
[
H
]
=
∑
m
=
1
−
n
0
T
m
⋅
D
T
−
1
(
T
)
[
H
]
⋅
T
1
−
n
−
m
=
−
∑
m
=
1
−
n
0
T
m
−
1
⋅
H
⋅
T
−
n
−
m
d
T
−
n
d
T
=
−
(
∑
m
=
1
−
n
0
T
m
−
1
⊗
(
T
−
n
−
m
)
⊤
)
⊤
23
{\displaystyle {\begin{aligned}\mathrm {D} \mathbf {T} ^{-n}(\mathbf {T} )[\mathbf {H} ]=&\sum _{m=1-n}^{0}\mathbf {T} ^{m}\cdot \mathrm {D} \mathbf {T} ^{-1}(\mathbf {T} )[\mathbf {H} ]\cdot \mathbf {T} ^{1-n-m}\\=&-\sum _{m=1-n}^{0}\mathbf {T} ^{m-1}\cdot \mathbf {H\cdot T} ^{-n-m}\\{\frac {\mathrm {d} \mathbf {T} ^{-n}}{\mathrm {d} \mathbf {T} }}=&-\left(\sum _{m=1-n}^{0}\mathbf {T} ^{m-1}\otimes \left(\mathbf {T} ^{-n-m}\right)^{\top }\right)^{\stackrel {23}{\top }}\end{aligned}}}
D
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⊤
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=
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{\displaystyle {\begin{aligned}\mathrm {D} \mathbf {T} ^{\top -n}(\mathbf {T} )[\mathbf {H} ]=&-\sum _{m=1-n}^{0}\left(\mathbf {T} ^{m-1}\right)^{\top }\cdot \mathbf {H^{\top }\cdot {\big (}T} ^{-n-m}{\big )}^{\top }\\{\frac {\mathrm {d} \mathbf {T} ^{\top -n}}{\mathrm {d} \mathbf {T} }}=&-\left(\sum _{m=1-n}^{0}\left(\mathbf {T} ^{m-1}\right)^{\top }\otimes \left(\mathbf {T} ^{-n-m}\right)^{\top }\right)^{\stackrel {24}{\top }}\end{aligned}}}
Orthogonaler Tensor (Q·Q ⊤ =1 ):
Q
˙
⊤
=
−
Q
⊤
⋅
Q
˙
⋅
Q
⊤
{\displaystyle {\dot {\mathbf {Q} }}^{\top }=-\mathbf {Q} ^{\top }\cdot {\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }}
Ableitungen nach dem Ort
Bearbeiten
#Kartesische Koordinaten
x
→
{\displaystyle {\vec {x}}}
:
∇
=
e
^
i
∂
∂
x
i
{\displaystyle \nabla ={\hat {e}}_{i}{\frac {\partial }{\partial x_{i}}}}
#Zylinderkoordinaten :
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{\displaystyle \nabla ={\vec {e}}_{\rho }{\frac {\partial }{\partial \rho }}+{\frac {1}{\rho }}{\vec {e}}_{\varphi }{\frac {\partial }{\partial \varphi }}+{\vec {e}}_{z}{\frac {\partial }{\partial z}}}
#Kugelkoordinaten :
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{\displaystyle \nabla ={\vec {e}}_{r}{\frac {\partial }{\partial r}}+{\frac {1}{r}}{\vec {e}}_{\vartheta }{\frac {\partial }{\partial \vartheta }}+{\frac {1}{r\sin(\vartheta )}}{\vec {e}}_{\varphi }{\frac {\partial }{\partial \varphi }}}
#Krummlinige Koordinaten
y
→
{\displaystyle {\vec {y}}}
:
∇
=
b
→
j
∂
∂
y
j
{\displaystyle \nabla ={\vec {b}}^{j}{\frac {\partial }{\partial y_{j}}}}
mit
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j
=
∂
y
j
∂
x
i
e
^
i
{\displaystyle {\vec {b}}^{j}={\frac {\partial y_{j}}{\partial x_{i}}}{\hat {e}}_{i}}
.
Definition des Gradienten/Allgemeines
Bearbeiten
Definierende Eigenschaft bei skalar- oder vektorwertiger Funktion f :[1]
f
(
y
→
)
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f
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grad
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{\displaystyle f({\vec {y}})-f({\vec {x}})=\operatorname {grad} (f)\cdot ({\vec {y}}-{\vec {x}})+{\mathcal {O}}(|{\vec {y}}-{\vec {x}}|)}
wenn
y
→
→
x
→
{\displaystyle {\vec {y}}\to {\vec {x}}}
Wenn der Gradient existiert, ist er eindeutig. Berechnung bei skalar- oder vektorwertiger Funktion f :
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{\displaystyle \operatorname {grad} (f)\cdot {\vec {h}}=\left.{\frac {\mathrm {d} }{\mathrm {d} s}}f({\vec {x}}+s{\vec {h}})\right|_{s=0}=\lim _{s\to 0}{\frac {f({\vec {x}}+s{\vec {h}})-f({\vec {x}})}{s}}\quad \forall \;{\vec {h}}\in \mathbb {V} }
Integrabilitätsbedingung : Jedes rotationsfreie Vektorfeld ist das Gradientenfeld eines Skalarpotentials :
rot
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=
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→
∃
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{\displaystyle \operatorname {rot} ({\vec {f}})={\vec {0}}\quad \rightarrow \quad \exists g\colon {\vec {f}}=\operatorname {grad} (g)}
.
Koordinatenfreie Darstellung als Volumenableitung:
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∫
a
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{\displaystyle \operatorname {grad} (f)=\lim _{v\to 0}\left({\frac {1}{v}}\int _{a}f\,\mathrm {d} {\vec {a}}\right)}
Skalarfeld f :
grad
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=
∇
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=:
∂
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∂
x
→
{\displaystyle \operatorname {grad} (f)=\nabla f=:{\frac {\partial f}{\partial {\vec {x}}}}}
Vektorfeld
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^
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{\displaystyle {\vec {f}}=f_{i}{\hat {e}}_{i}}
:[2]
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{\displaystyle \mathrm {grad} ({\vec {f}})=(\nabla \otimes {\vec {f}})^{\top }=:{\frac {\partial {\vec {f}}}{\partial {\vec {x}}}}}
g
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d
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{\displaystyle \mathrm {grad} ({\vec {x}})=\mathbf {1} }
Zusammenhang mit den anderen Differentialoperatoren:
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{\displaystyle \mathrm {grad} (f)=\mathrm {div} (f\mathbf {1} )=\nabla \cdot (f\mathbf {1} )}
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t
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{\displaystyle \mathrm {grad} (f)\times {\vec {c}}=\mathrm {rot} (f{\vec {c}})}
Gradient in verschiedenen Koordinatensystemen
Bearbeiten
#Kartesische Koordinaten :
g
r
a
d
(
f
)
=
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,
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{\displaystyle \mathrm {grad} (f)=f_{,i}{\hat {e}}_{i}}
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{\displaystyle \mathrm {grad} ({\vec {f}})={\vec {f}}_{,i}\otimes {\hat {e}}_{i}={\hat {e}}_{i}\otimes \mathrm {grad} (f_{i})=f_{i,j}{\hat {e}}_{i}\otimes {\hat {e}}_{j}}
#Zylinderkoordinaten :
g
r
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d
(
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=
f
,
ρ
e
^
ρ
+
f
,
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^
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{\displaystyle \mathrm {grad} (f)=f_{,\rho }{\hat {e}}_{\rho }+{\frac {f_{,\varphi }}{\rho }}{\hat {e}}_{\varphi }+f_{,z}{\hat {e}}_{z}}
g
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{\displaystyle {\begin{aligned}\mathrm {grad} ({\vec {f}})=&{\hat {e}}_{\rho }\otimes \mathrm {grad} (f_{\rho })+{\hat {e}}_{\varphi }\otimes \mathrm {grad} (f_{\varphi })+{\hat {e}}_{z}\otimes \mathrm {grad} (f_{z})\\&+{\frac {1}{\rho }}(f_{\rho }{\hat {e}}_{\varphi }-f_{\varphi }{\hat {e}}_{\rho })\otimes {\hat {e}}_{\varphi }\end{aligned}}}
#Kugelkoordinaten :
g
r
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d
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f
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=
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,
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e
^
r
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,
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r
sin
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ϑ
)
e
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φ
{\displaystyle \mathrm {grad} (f)=f_{,r}{\hat {e}}_{r}+{\frac {f_{,\vartheta }}{r}}{\hat {e}}_{\vartheta }+{\frac {f_{,\varphi }}{r\sin(\vartheta )}}{\hat {e}}_{\varphi }}
g
r
a
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→
)
=
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^
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⊗
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ϑ
⊗
g
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+
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^
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r
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1
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tan
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ϑ
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⊗
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{\displaystyle {\begin{aligned}\mathrm {grad} ({\vec {f}})=&{\hat {e}}_{r}\otimes \mathrm {grad} (f_{r})+{\hat {e}}_{\vartheta }\otimes \mathrm {grad} (f_{\vartheta })+{\hat {e}}_{\varphi }\otimes \mathrm {grad} (f_{\varphi })\\&+{\frac {f_{r}}{r}}(\mathbf {1} -{\hat {e}}_{r}\otimes {\hat {e}}_{r})-{\hat {e}}_{r}\otimes {\frac {f_{\vartheta }{\hat {e}}_{\vartheta }+f_{\varphi }{\hat {e}}_{\varphi }}{r}}+{\frac {f_{\vartheta }{\hat {e}}_{\varphi }-f_{\varphi }{\hat {e}}_{\vartheta }}{r\tan(\vartheta )}}\otimes {\hat {e}}_{\varphi }\end{aligned}}}
#Krummlinige Koordinaten :
Christoffelsymbole :
Γ
i
j
k
=
g
→
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,
j
⋅
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→
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{\displaystyle \Gamma _{ij}^{k}={\vec {g}}_{i,j}\cdot {\vec {g}}^{k}}
Vektorfelder:
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d
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=
Γ
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j
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{\displaystyle \mathrm {grad} ({\vec {g}}_{i})=\Gamma _{ij}^{k}{\vec {g}}_{k}\otimes {\vec {g}}^{j}}
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=
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Γ
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⊗
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{\displaystyle \mathrm {grad} ({\vec {g}}^{k})=-\Gamma _{ij}^{k}{\vec {g}}^{i}\otimes {\vec {g}}^{j}}
g
r
a
d
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g
→
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=
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|
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→
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⊗
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→
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{\displaystyle \mathrm {grad} (f^{i}{\vec {g}}_{i})=\left.f^{i}\right|_{j}{\vec {g}}_{i}\otimes {\vec {g}}^{j}}
g
r
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→
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=
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⊗
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{\displaystyle \mathrm {grad} (f_{i}{\vec {g}}^{i})=\left.f_{i}\right|_{j}{\vec {g}}^{i}\otimes {\vec {g}}^{j}}
Mit den kovarianten Ableitungen
f
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|
j
=
f
,
j
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Γ
k
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f
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{\displaystyle \left.f^{i}\right|_{j}=f_{,j}^{i}+\Gamma _{kj}^{i}f^{k}}
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=
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i
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Γ
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f
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{\displaystyle \left.f_{i}\right|_{j}=f_{i,j}-\Gamma _{ij}^{k}f_{k}}
Tensorfelder:
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r
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d
(
T
)
[
h
→
]
=
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→
⋅
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→
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g
→
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⊗
T
,
k
)
=
(
T
,
k
⊗
g
→
k
)
⋅
h
→
{\displaystyle \mathrm {grad} (\mathbf {T} )[{\vec {h}}]=({\vec {h}}\cdot {\vec {g}}^{k})\mathbf {T} _{,k}={\vec {h}}\cdot ({\vec {g}}^{k}\otimes \mathbf {T} _{,k})=(\mathbf {T} _{,k}\otimes {\vec {g}}^{k})\cdot {\vec {h}}}
Soll das Argument wie beim Vektorgradient rechts vom Operator stehen, dann lautet der Tensorgradient
g
r
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d
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=
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,
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⊗
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→
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{\displaystyle \mathrm {grad} (\mathbf {T} )=\mathbf {T} _{,k}\otimes {\vec {g}}^{k}}
Für ein Tensorfeld zweiter Stufe:
g
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d
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T
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j
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→
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⊗
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)
=
T
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⊗
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⊗
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,
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,
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Γ
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⊗
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⊗
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,
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⊗
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{\displaystyle {\begin{aligned}\mathrm {grad} (T^{ij}{\vec {g}}_{i}\otimes {\vec {g}}_{j})=&\left.T_{ij}\right|_{k}{\vec {g}}^{i}\otimes {\vec {g}}^{j}\otimes {\vec {g}}^{k},\quad \left.T_{ij}\right|_{k}\!\!\!\!\!\!\!\!\!\!\!\!&=T_{ij,k}-\Gamma _{ik}^{l}T_{lj}-\Gamma _{jk}^{l}T_{il}\\\mathrm {grad} (T^{ij}{\vec {g}}_{i}\otimes {\vec {g}}_{j})=&\left.T^{ij}\right|_{k}{\vec {g}}_{i}\otimes {\vec {g}}_{j}\otimes {\vec {g}}^{k},\quad \left.T^{ij}\right|_{k}\!\!\!\!\!\!\!\!\!\!\!\!&=T_{,k}^{ij}+\Gamma _{lk}^{i}T^{lj}+\Gamma _{lk}^{j}T^{il}\\\mathrm {grad} (T_{i}^{.j}{\vec {g}}^{i}\otimes {\vec {g}}_{j})=&\left.T_{i}^{.j}\right|_{k}{\vec {g}}^{i}\otimes {\vec {g}}_{j}\otimes {\vec {g}}^{k},\quad \left.T_{i}^{.j}\right|_{k}\!\!\!\!\!\!\!\!\!\!\!\!&=T_{i,k}^{.j}-\Gamma _{ik}^{l}T_{l}^{.j}+\Gamma _{lk}^{j}T_{i}^{.l}\\\mathrm {grad} (T_{.j}^{i}){\vec {g}}_{i}\otimes {\vec {g}}^{j}=&\left.T_{.j}^{i}\right|_{k}{\vec {g}}_{i}\otimes {\vec {g}}^{j}\otimes {\vec {g}}^{k},\quad \left.T_{.j}^{i}\right|_{k}\!\!\!\!\!\!\!\!\!\!\!\!&=T_{.j,k}^{i}+\Gamma _{lk}^{i}T_{.j}^{l}-\Gamma _{jk}^{l}T_{.l}^{i}\end{aligned}}}
Produktregel für Gradienten
Bearbeiten
g
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=
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{\displaystyle {\begin{array}{rclcl}\mathrm {grad} (fg)&=&(f_{,i}g+fg_{,i}){\hat {e}}_{i}&=&\mathrm {grad} (f)g+f\mathrm {grad} (g)\\\mathrm {grad} (f{\vec {g}})&=&(f_{,i}{\vec {g}}+f{\vec {g}}_{,i})\otimes {\hat {e}}_{i}&=&{\vec {g}}\otimes \mathrm {grad} (f)+f\mathrm {grad} ({\vec {g}})\\\mathrm {grad} ({\vec {f}}\cdot {\vec {g}})&=&\left({\vec {f}}_{,i}\cdot {\vec {g}}+{\vec {f}}\cdot {\vec {g}}_{,i}\right){\hat {e}}_{i}&=&{\vec {g}}\cdot \mathrm {grad} ({\vec {f}})+{\vec {f}}\cdot \mathrm {grad} ({\vec {g}})\\\mathrm {grad} ({\vec {f}}\times {\vec {g}})&=&\left({\vec {f}}_{,i}\times {\vec {g}}+{\vec {f}}\times {\vec {g}}_{,i}\right)\otimes {\hat {e}}_{i}&=&{\vec {f}}\times \mathrm {grad} ({\vec {g}})-{\vec {g}}\times \mathrm {grad} ({\vec {f}})\end{array}}}
In drei Dimensionen ist speziell[3]
g
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{\displaystyle \mathrm {grad} ({\vec {f}}\cdot {\vec {g}})=\mathrm {grad} ({\vec {f}})\cdot {\vec {g}}+\mathrm {grad} ({\vec {g}})\cdot {\vec {f}}+{\vec {f}}\times \mathrm {rot} ({\vec {g}})+{\vec {g}}\times \mathrm {rot} ({\vec {f}})}
Beliebige Basis:
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+
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{\displaystyle \mathrm {grad} (f_{i}{\vec {b}}_{i})={\vec {b}}_{i}\otimes \mathrm {grad} (f_{i})+f_{i}\,\mathrm {grad} ({\vec {b}}_{i})}
Definition der Divergenz/Allgemeines
Bearbeiten
Vektorfeld
f
→
{\displaystyle {\vec {f}}}
:
d
i
v
(
f
→
)
=
∇
⋅
f
→
=
S
p
(
g
r
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→
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)
{\displaystyle \mathrm {div} ({\vec {f}})=\nabla \cdot {\vec {f}}=\mathrm {Sp} {\big (}\mathrm {grad} ({\vec {f}}){\big )}}
d
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→
)
=
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→
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=
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{\displaystyle \mathrm {div} ({\vec {x}})=\mathrm {Sp} {\big (}\mathrm {grad} ({\vec {x}}){\big )}=\mathrm {Sp} (\mathbf {1} )=3}
Klassische Definition für ein Tensorfeld T :[1]
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v
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⋅
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→
=
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⊤
⋅
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→
)
∀
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∈
V
{\displaystyle \mathrm {div} (\mathbf {T} )\cdot {\vec {c}}=\mathrm {div} \left(\mathbf {T} ^{\top }\cdot {\vec {c}}\right)\quad \forall {\vec {c}}\in \mathbb {V} }
→
d
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v
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T
)
=
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⋅
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{\displaystyle \mathrm {div} (\mathbf {T} )=\nabla \cdot \left(\mathbf {T} ^{\top }\right)}
Koordinatenfreie Darstellung:
d
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v
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f
→
)
=
lim
v
→
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1
v
∫
a
f
→
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→
)
{\displaystyle \mathrm {div} ({\vec {f}})=\lim _{v\to 0}\left({\frac {1}{v}}\int _{a}{\vec {f}}\;\cdot \mathrm {d} {\vec {a}}\right)}
Zusammenhang mit den anderen Differentialoperatoren:
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{\displaystyle {\begin{array}{lcccl}\mathrm {div} ({\vec {f}})&=&\nabla \cdot {\vec {f}}&=&\mathrm {Sp(grad} ({\vec {f}}))\\\mathrm {div} (f\mathbf {1} )&=&\nabla \cdot (f\mathbf {1} )&=&\mathrm {grad} (f)\end{array}}}
Divergenz in verschiedenen Koordinatensystemen
Bearbeiten
#Kartesische Koordinaten :
d
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)
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{\displaystyle \mathrm {div} ({\vec {f}})={\vec {f}}_{,i}\cdot {\hat {e}}_{i}=f_{i,i}}
d
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)
=
T
,
i
⋅
e
^
i
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T
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j
,
j
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{\displaystyle \mathrm {div} (\mathbf {T} )=\mathbf {T} _{,i}\cdot {\hat {e}}_{i}=T_{ij,j}{\hat {e}}_{i}}
∇
⋅
T
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e
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i
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T
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i
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,
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{\displaystyle \nabla \cdot \mathbf {T} ={\hat {e}}_{i}\cdot \mathbf {T} _{,i}=T_{ij,i}{\hat {e}}_{j}=T_{ji,j}{\hat {e}}_{i}}
#Zylinderkoordinaten :
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1
ρ
∂
∂
ρ
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ρ
f
ρ
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1
ρ
f
φ
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φ
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,
z
{\displaystyle \mathrm {div} ({\vec {f}})={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}(\rho f_{\rho })+{\frac {1}{\rho }}f_{\varphi ,\varphi }+f_{z,z}}
d
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v
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T
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ρ
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e
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ρ
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φ
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)
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ρ
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1
ρ
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T
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{\displaystyle {\begin{aligned}\mathrm {div} (\mathbf {T} )=&\left(T_{\rho \rho ,\rho }+{\frac {1}{\rho }}(T_{\rho \varphi ,\varphi }+T_{\rho \rho }-T_{\varphi \varphi })+T_{\rho z,z}\right){\hat {e}}_{\rho }\\&+\left(T_{\varphi \rho ,\rho }+{\frac {1}{\rho }}(T_{\varphi \varphi ,\varphi }+T_{\rho \varphi }+T_{\varphi \rho })+T_{\varphi z,z}\right){\hat {e}}_{\varphi }\\&+\left(T_{z\rho ,\rho }+{\frac {1}{\rho }}(T_{z\varphi ,\varphi }+T_{z\rho })+T_{zz,z}\right){\hat {e}}_{z}\end{aligned}}}
∇
⋅
T
=
d
i
v
(
T
⊤
)
{\displaystyle \nabla \cdot \mathbf {T} =\mathrm {div} \left(\mathbf {T} ^{\top }\right)}
ergibt sich hieraus durch Vertauschen von Tab durch Tba .
#Kugelkoordinaten :
d
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v
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f
→
)
=
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r
,
r
+
2
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r
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ϑ
,
ϑ
r
+
f
ϑ
cos
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ϑ
)
+
f
φ
,
φ
r
sin
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ϑ
)
d
i
v
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T
)
=
(
T
r
r
,
r
+
2
T
r
r
−
T
ϑ
ϑ
−
T
φ
φ
+
T
r
ϑ
,
ϑ
r
+
T
r
φ
,
φ
+
T
r
ϑ
cos
(
ϑ
)
r
sin
(
ϑ
)
)
e
^
r
(
T
ϑ
r
,
r
+
2
T
ϑ
r
+
T
r
ϑ
+
T
ϑ
ϑ
,
ϑ
r
+
(
T
ϑ
ϑ
−
T
φ
φ
)
cos
(
ϑ
)
+
T
ϑ
φ
,
φ
r
sin
(
ϑ
)
)
e
^
ϑ
(
T
φ
r
,
r
+
2
T
φ
r
+
T
r
φ
+
T
φ
ϑ
,
ϑ
r
+
(
T
ϑ
φ
+
T
φ
ϑ
)
cos
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ϑ
)
+
T
φ
φ
,
φ
r
sin
(
ϑ
)
)
e
^
φ
{\displaystyle {\begin{aligned}\mathrm {div} ({\vec {f}})=&f_{r,r}+{\frac {2f_{r}+f_{\vartheta ,\vartheta }}{r}}+{\frac {f_{\vartheta }\cos(\vartheta )+f_{\varphi ,\varphi }}{r\sin(\vartheta )}}\\\mathrm {div} (\mathbf {T} )=&\left(T_{rr,r}+{\frac {2T_{rr}-T_{\vartheta \vartheta }-T_{\varphi \varphi }+T_{r\vartheta ,\vartheta }}{r}}+{\frac {T_{r\varphi ,\varphi }+T_{r\vartheta }\cos(\vartheta )}{r\sin(\vartheta )}}\right){\hat {e}}_{r}\\&\left(T_{\vartheta r,r}+{\frac {2T_{\vartheta r}+T_{r\vartheta }+T_{\vartheta \vartheta ,\vartheta }}{r}}+{\frac {(T_{\vartheta \vartheta }-T_{\varphi \varphi })\cos(\vartheta )+T_{\vartheta \varphi ,\varphi }}{r\sin(\vartheta )}}\right){\hat {e}}_{\vartheta }\\&\left(T_{\varphi r,r}+{\frac {2T_{\varphi r}+T_{r\varphi }+T_{\varphi \vartheta ,\vartheta }}{r}}+{\frac {(T_{\vartheta \varphi }+T_{\varphi \vartheta })\cos(\vartheta )+T_{\varphi \varphi ,\varphi }}{r\sin(\vartheta )}}\right){\hat {e}}_{\varphi }\end{aligned}}}
∇
⋅
T
=
d
i
v
(
T
⊤
)
{\displaystyle \nabla \cdot \mathbf {T} =\mathrm {div} \left(\mathbf {T} ^{\top }\right)}
ergibt sich hieraus durch Vertauschen von Tab durch Tba .
Produktregel für Divergenzen
Bearbeiten
d
i
v
(
f
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→
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∇
⋅
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⋅
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→
+
f
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i
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g
→
)
{\displaystyle \mathrm {div} (f{\vec {g}})=\nabla \cdot (f{\vec {g}})=\left(f_{,i}{\vec {g}}+f{\vec {g}}_{,i}\right)\cdot {\hat {e}}_{i}=\mathrm {grad} (f)\cdot {\vec {g}}+f\mathrm {div} ({\vec {g}})}
d
i
v
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g
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∇
⋅
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g
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i
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−
f
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⋅
r
o
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(
g
→
)
{\displaystyle \mathrm {div} ({\vec {f}}\times {\vec {g}})=\nabla \cdot ({\vec {f}}\times {\vec {g}})=\left({\vec {f}}_{,i}\times {\vec {g}}+{\vec {f}}\times {\vec {g}}_{,i}\right)\cdot {\hat {e}}_{i}={\vec {g}}\cdot \mathrm {rot} ({\vec {f}})-{\vec {f}}\cdot \mathrm {rot} ({\vec {g}})}
d
i
v
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f
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⊗
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=
(
f
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f
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i
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f
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⋅
g
→
+
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i
v
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i
v
(
f
T
)
=
(
f
,
i
T
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f
T
,
i
)
⋅
e
^
i
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⋅
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r
a
d
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f
)
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f
d
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)
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⋅
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⋅
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⋅
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,
i
)
⋅
e
^
i
=
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v
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⊤
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⋅
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+
T
⊤
:
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r
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f
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)
d
i
v
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f
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×
T
)
=
(
f
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i
×
T
+
f
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×
T
,
i
)
⋅
e
^
i
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(
g
r
a
d
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f
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)
⋅
T
⊤
)
+
f
→
×
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i
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T
)
{\displaystyle {\begin{aligned}\mathrm {div} ({\vec {f}}\otimes {\vec {g}})=&\left({\vec {f}}_{,i}\otimes {\vec {g}}+{\vec {f}}\otimes {\vec {g}}_{,i}\right)\cdot {\hat {e}}_{i}\!\!\!\!\!\!\!\!\!\!&=&\mathrm {grad} ({\vec {f}})\cdot {\vec {g}}+\mathrm {div} ({\vec {g}}){\vec {f}}\\\mathrm {div} (f\mathbf {T} )=&(f_{,i}\mathbf {T} +f\mathbf {T} _{,i})\cdot {\hat {e}}_{i}\!\!\!\!\!\!\!\!\!\!&=&\mathbf {T} \cdot \mathrm {grad} (f)+f\mathrm {div} (\mathbf {T} )\\\mathrm {div} (\mathbf {T} \cdot {\vec {f}})=&\left(\mathbf {T} _{,i}\cdot {\vec {f}}+\mathbf {T} \cdot {\vec {f}}_{,i}\right)\cdot {\hat {e}}_{i}\!\!\!\!\!\!\!\!\!\!&=&\mathrm {div} (\mathbf {T} ^{\top })\cdot {\vec {f}}+\mathbf {T} ^{\top }:\mathrm {grad} ({\vec {f}})\\\mathrm {div} ({\vec {f}}\times \mathbf {T} )=&({\vec {f}}_{,i}\times \mathbf {T} +{\vec {f}}\times \mathbf {T} _{,i})\cdot {\hat {e}}_{i}\!\!\!\!\!\!\!\!\!\!&=&{\vec {\mathrm {i} }}\left(\mathrm {grad} ({\vec {f}})\cdot \mathbf {T} ^{\top }\right)+{\vec {f}}\times \mathrm {div} (\mathbf {T} )\end{aligned}}}
∇
⋅
(
f
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⊗
g
→
)
=
e
^
i
⋅
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f
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,
i
⊗
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(
∇
⋅
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g
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+
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∇
⊗
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⊤
⋅
f
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∇
⋅
(
f
T
)
=
e
^
i
⋅
(
f
,
i
T
+
f
T
,
i
)
=
(
∇
f
)
⋅
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+
f
∇
⋅
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∇
⋅
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T
⋅
f
→
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=
e
^
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⋅
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T
,
i
⋅
f
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+
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⋅
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,
i
)
=
(
∇
⋅
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)
⋅
f
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+
T
:
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∇
⊗
f
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×
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=
e
^
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T
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i
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⋅
T
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×
f
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i
→
(
(
∇
⊗
f
→
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⊤
⋅
T
)
{\displaystyle {\begin{aligned}\nabla \cdot ({\vec {f}}\otimes {\vec {g}})=&{\hat {e}}_{i}\cdot \left({\vec {f}}_{,i}\otimes {\vec {g}}+{\vec {f}}\otimes {\vec {g}}_{,i}\right)\!\!\!\!\!\!\!\!\!\!&=&(\nabla \cdot {\vec {f}}){\vec {g}}+(\nabla \otimes {\vec {g}})^{\top }\cdot {\vec {f}}\\\nabla \cdot (f\mathbf {T} )=&{\hat {e}}_{i}\cdot (f_{,i}\mathbf {T} +f\mathbf {T} _{,i})\!\!\!\!\!\!\!\!\!\!&=&(\nabla f)\cdot \mathbf {T} +f\nabla \cdot \mathbf {T} \\\nabla \cdot (\mathbf {T} \cdot {\vec {f}})=&{\hat {e}}_{i}\cdot \left(\mathbf {T} _{,i}\cdot {\vec {f}}+\mathbf {T} \cdot {\vec {f}}_{,i}\right)\!\!\!\!\!\!\!\!\!\!&=&(\nabla \cdot \mathbf {T} )\cdot {\vec {f}}+\mathbf {T} :(\nabla \otimes {\vec {f}})\\\nabla \cdot (\mathbf {T} \times {\vec {f}})=&{\hat {e}}_{i}\cdot (\mathbf {T} _{,i}\times {\vec {f}}+\mathbf {T} \times {\vec {f}}_{,i})\!\!\!\!\!\!\!\!\!\!&=&(\nabla \cdot \mathbf {T} )\times {\vec {f}}-{\vec {\mathrm {i} }}\left((\nabla \otimes {\vec {f}})^{\top }\cdot \mathbf {T} \right)\end{aligned}}}
Beliebige Basis:
d
i
v
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f
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b
→
i
)
=
∇
⋅
(
f
i
b
→
i
)
=
g
r
a
d
(
f
i
)
⋅
b
→
i
+
f
i
d
i
v
(
b
→
i
)
{\displaystyle \mathrm {div} (f_{i}{\vec {b}}_{i})=\nabla \cdot (f_{i}{\vec {b}}_{i})=\mathrm {grad} (f_{i})\cdot {\vec {b}}_{i}+f_{i}\,\mathrm {div} ({\vec {b}}_{i})}
d
i
v
(
T
i
j
b
→
i
⊗
b
→
j
)
=
(
g
r
a
d
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T
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)
⋅
b
→
j
)
b
→
i
+
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r
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d
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b
→
i
)
⋅
b
→
j
+
d
i
v
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b
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j
)
b
→
i
)
{\displaystyle \mathrm {div} (T^{ij}{\vec {b}}_{i}\otimes {\vec {b}}_{j})=(\mathrm {grad} (T^{ij})\cdot {\vec {b}}_{j}){\vec {b}}_{i}+T^{ij}\,{\big (}\mathrm {grad} ({\vec {b}}_{i})\cdot {\vec {b}}_{j}+\mathrm {div} ({\vec {b}}_{j}){\vec {b}}_{i}{\big )}}
∇
⋅
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T
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⊗
b
→
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=
(
(
∇
T
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)
⋅
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i
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∇
⋅
b
→
i
)
b
→
j
+
(
∇
b
→
j
)
⋅
b
→
i
)
{\displaystyle \nabla \cdot (T^{ij}{\vec {b}}_{i}\otimes {\vec {b}}_{j})={\big (}(\nabla T^{ij})\cdot {\vec {b}}_{i}{\big )}{\vec {b}}_{j}+T^{ij}\,{\big (}(\nabla \cdot {\vec {b}}_{i}){\vec {b}}_{j}+(\nabla {\vec {b}}_{j})\cdot {\vec {b}}_{i}{\big )}}
Produkt mit Konstanten:
d
i
v
(
f
C
)
=
C
⋅
g
r
a
d
(
f
)
→
d
i
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f
1
)
=
g
r
a
d
(
f
)
{\displaystyle \mathrm {div} (f\mathbf {C} )=\mathbf {C} \cdot \mathrm {grad} (f)\quad \rightarrow \quad \mathrm {div} (f\mathbf {1} )=\mathrm {grad} (f)}
∇
⋅
(
f
C
)
=
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r
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d
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f
)
⋅
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→
∇
⋅
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f
1
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=
∇
f
{\displaystyle \nabla \cdot (f\mathbf {C} )=\mathrm {grad} (f)\cdot \mathbf {C} \quad \rightarrow \quad \nabla \cdot (f\mathbf {1} )=\nabla f}
d
i
v
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C
⋅
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→
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=
C
⊤
:
g
r
a
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{\displaystyle {\begin{aligned}\mathrm {div} (\mathbf {C} \cdot {\vec {f}})=\mathbf {C} ^{\top }:\mathrm {grad} ({\vec {f}})\quad \rightarrow \quad \mathrm {div} ({\vec {f}})=&\mathrm {div} (\mathbf {1} \cdot {\vec {f}})=\mathbf {1} :\mathrm {grad} ({\vec {f}})\\=&\mathrm {Sp} (\mathrm {grad} ({\vec {f}}))\end{aligned}}}
Definition der Rotation/Allgemeines
Bearbeiten
Vektorfeld
f
→
{\displaystyle {\vec {f}}}
:
r
o
t
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f
→
)
=
∇
×
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→
{\displaystyle \mathrm {rot} ({\vec {f}})=\nabla \times {\vec {f}}}
Klassische Definition für ein Tensorfeld T :[1]
r
o
t
(
T
)
⋅
c
→
=
r
o
t
(
T
⊤
⋅
c
→
)
∀
c
→
∈
V
{\displaystyle \mathrm {rot} (\mathbf {T} )\cdot {\vec {c}}=\mathrm {rot} \left(\mathbf {T} ^{\top }\cdot {\vec {c}}\right)\quad \forall {\vec {c}}\in \mathbb {V} }
→
r
o
t
(
T
)
=
∇
×
(
T
⊤
)
{\displaystyle \mathrm {rot} (\mathbf {T} )=\nabla \times \left(\mathbf {T} ^{\top }\right)}
Allgemeine Identitäten:
T
=
T
⊤
→
S
p
(
r
o
t
(
T
)
)
=
S
p
(
∇
×
T
)
=
0
{\displaystyle \mathbf {T=T} ^{\top }\quad \rightarrow \quad \mathrm {Sp{\big (}rot} (\mathbf {T} ){\big )}=\mathrm {Sp} (\nabla \times \mathbf {T} )=0}
r
o
t
(
x
→
)
=
0
→
{\displaystyle \mathrm {rot} ({\vec {x}})={\vec {0}}}
Integrabilitätsbedingung[4] : Jedes divergenzfreie Vektorfeld ist die Rotation eines Vektorfeldes:
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→
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∃
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:
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r
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→
)
{\displaystyle \mathrm {div} ({\vec {f}})=0\quad \rightarrow \quad \exists {\vec {g}}\colon {\vec {f}}=\mathrm {rot} ({\vec {g}})}
.
Siehe auch #Satz über rotationsfreie Felder .
Koordinatenfreie Darstellung:
r
o
t
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f
→
)
=
−
lim
v
→
0
(
1
v
∫
a
f
→
×
d
a
→
)
{\displaystyle \mathrm {rot} ({\vec {f}})=-\lim _{v\rightarrow 0}\left({\frac {1}{v}}\int _{a}{\vec {f}}\times \mathrm {d} {\vec {a}}\right)}
Zusammenhang mit den anderen Differentialoperatoren:
r
o
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f
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→
)
=
g
r
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d
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f
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×
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→
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⊗
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{\displaystyle {\begin{aligned}\mathrm {rot} (f{\vec {c}})=&\mathrm {grad} (f)\times {\vec {c}}\\\mathrm {rot} ({\vec {f}})=&-{\vec {\mathrm {i} }}{\big (}\mathrm {grad} ({\vec {f}}){\big )}={\vec {\mathrm {i} }}(\nabla \otimes {\vec {f}})=\nabla \times {\vec {f}}\end{aligned}}}
Rotation in verschiedenen Koordinatensystemen
Bearbeiten
#Kartesische Koordinaten :
r
o
t
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f
→
)
=
f
j
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i
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^
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×
e
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ϵ
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k
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=
(
f
3
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3
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+
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1
)
e
^
2
+
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,
2
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3
{\displaystyle {\begin{aligned}\mathrm {rot} ({\vec {f}})=&f_{j,i}{\hat {e}}_{i}\times {\hat {e}}_{j}=\epsilon _{ijk}f_{j,i}{\hat {e}}_{k}\\=&(f_{3,2}-f_{2,3}){\hat {e}}_{1}+(f_{1,3}-f_{3,1}){\hat {e}}_{2}+(f_{2,1}-f_{1,2}){\hat {e}}_{3}\end{aligned}}}
r
o
t
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T
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{\displaystyle \mathrm {rot} (\mathbf {T} )={\hat {e}}_{i}\times \mathbf {T} _{,i}^{\top }={\hat {e}}_{i}\times T_{lj,i}{\hat {e}}_{j}\otimes {\hat {e}}_{l}=\epsilon _{ijk}T_{lj,i}{\hat {e}}_{k}\otimes {\hat {e}}_{l}}
#Zylinderkoordinaten :
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{\displaystyle \mathrm {rot} ({\vec {f}})={\frac {f_{z,\varphi }-\rho f_{\varphi ,z}}{\rho }}{\hat {e}}_{\rho }+(f_{\rho ,z}-f_{z,\rho }){\hat {e}}_{\varphi }+{\frac {f_{\varphi }+\rho f_{\varphi ,\rho }-f_{\rho ,\varphi }}{\rho }}{\hat {e}}_{z}}
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{\displaystyle \mathrm {rot} (\mathbf {T} )={\hat {e}}_{\rho }\times (\mathbf {T} _{,\rho }^{\top })+{\frac {1}{\rho }}{\hat {e}}_{\varphi }\times (\mathbf {T} _{,\varphi }^{\top })+{\hat {e}}_{z}\times (\mathbf {T} _{,z}^{\top })}
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{\displaystyle \nabla \times \mathbf {T} ={\hat {e}}_{\rho }\times \mathbf {T} _{,\rho }+{\frac {1}{\rho }}{\hat {e}}_{\varphi }\times \mathbf {T} _{,\varphi }+{\hat {e}}_{z}\times \mathbf {T} _{,z}}
#Kugelkoordinaten :
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{\displaystyle {\begin{aligned}\mathrm {rot} ({\vec {f}})=&{\frac {f_{\varphi ,\vartheta }\sin(\vartheta )+f_{\varphi }\cos(\vartheta )-f_{\vartheta ,\varphi }}{r\sin(\vartheta )}}{\hat {e}}_{r}+\left({\frac {f_{r,\varphi }}{r\sin(\vartheta )}}-{\frac {f_{\varphi }+rf_{\varphi ,r}}{r}}\right){\hat {e}}_{\vartheta }\\&+{\frac {f_{\vartheta }+rf_{\vartheta ,r}-f_{r,\vartheta }}{r}}{\hat {e}}_{\varphi }\end{aligned}}}
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{\displaystyle \mathrm {rot} (\mathbf {T} )={\hat {e}}_{r}\times (\mathbf {T} _{,r}^{\top })+{\frac {1}{r}}{\hat {e}}_{\vartheta }\times (\mathbf {T} _{,\vartheta }^{\top })+{\frac {1}{r\sin(\vartheta )}}{\hat {e}}_{\varphi }\times (\mathbf {T} _{,\varphi }^{\top })}
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{\displaystyle \nabla \times \mathbf {T} ={\hat {e}}_{r}\times \mathbf {T} _{,r}+{\frac {1}{r}}{\hat {e}}_{\vartheta }\times \mathbf {T} _{,\vartheta }+{\frac {1}{r\sin(\vartheta )}}{\hat {e}}_{\varphi }\times \mathbf {T} _{,\varphi }}
Produktregel für Rotationen
Bearbeiten
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{\displaystyle {\begin{aligned}\mathrm {rot} (f{\vec {g}})=&{\hat {e}}_{i}\times (f_{,i}{\vec {g}}+f{\vec {g}}_{,i})=\mathrm {grad} (f)\times {\vec {g}}+f\mathrm {rot} ({\vec {g}})\\\mathrm {rot} ({\vec {f}}\times {\vec {g}})=&{\hat {e}}_{i}\times \left({\vec {f}}_{,i}\times {\vec {g}}+{\vec {f}}\times {\vec {g}}_{,i}\right)\\=&({\hat {e}}_{i}\cdot {\vec {g}}){\vec {f}}_{,i}-\left({\hat {e}}_{i}\cdot {\vec {f}}_{,i}\right){\vec {g}}+\left({\hat {e}}_{i}\cdot {\vec {g}}_{,i}\right){\vec {f}}-({\hat {e}}_{i}\cdot {\vec {f}}){\vec {g}}_{,i}\\=&\mathrm {grad} ({\vec {f}})\cdot {\vec {g}}-\mathrm {div} ({\vec {f}}){\vec {g}}+\mathrm {div} ({\vec {g}}){\vec {f}}-\mathrm {grad} ({\vec {g}})\cdot {\vec {f}}\\=&\mathrm {div} ({\vec {f}}\otimes {\vec {g}})-\mathrm {div} ({\vec {g}}\otimes {\vec {f}})=\nabla \cdot ({\vec {g}}\otimes {\vec {f}})-\nabla \cdot ({\vec {f}}\otimes {\vec {g}})\end{aligned}}}
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{\displaystyle {\begin{aligned}\mathrm {rot} ({\vec {f}}\otimes {\vec {g}})=&{\hat {e}}_{i}\times \left({\vec {g}}_{,i}\otimes {\vec {f}}+{\vec {g}}\otimes {\vec {f}}_{,i}\right)\!\!\!\!\!\!\!\!\!\!&=&\mathrm {rot} ({\vec {g}})\otimes {\vec {f}}-{\vec {g}}\times \mathrm {grad} ({\vec {f}})^{\top }\\\mathrm {rot} (f\mathbf {T} )=&{\hat {e}}_{k}\times (f_{,k}\mathbf {T} ^{\top }+f\mathbf {T} _{,k}^{\top })\!\!\!\!\!\!\!\!\!\!&=&\mathrm {grad} (f)\times (\mathbf {T} ^{\top })+f\mathrm {rot} (\mathbf {T} )\end{aligned}}}
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{\displaystyle {\begin{aligned}\mathrm {rot} (\mathbf {T} \cdot {\vec {f}})=&{\hat {e}}_{k}\times {\big (}\mathbf {T} _{,k}\cdot {\vec {f}}+\mathbf {T} \cdot {\vec {f}}_{,k}{\big )}\\=&\mathrm {rot} (\mathbf {T} ^{\top })\cdot {\vec {f}}+{\vec {\mathrm {i} }}\left({\hat {e}}_{k}\otimes \mathbf {T} \cdot {\vec {f}}_{,k}\right)\\=&\mathrm {rot} (\mathbf {T} ^{\top })\cdot {\vec {f}}-{\vec {\mathrm {i} }}\left(\mathbf {T} \cdot \mathrm {grad} ({\vec {f}})\right)\\\mathrm {rot} ({\vec {f}}\times \mathbf {T} )=&-\mathrm {rot} \left((\mathbf {T} ^{\top }\times {\vec {f}})^{\top }\right)\\=&-\nabla \times \left(\mathbf {T} ^{\top }\times {\vec {f}}\right)\\=&-(\nabla \times \mathbf {T} ^{\top })\times {\vec {f}}+\mathbf {T} ^{\top }\#(\nabla \otimes {\vec {f}})\\=&-\mathrm {rot} (\mathbf {T} )\times {\vec {f}}+\left(\mathbf {T} \#\mathrm {grad} ({\vec {f}})\right)^{\top }\end{aligned}}}
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{\displaystyle {\begin{aligned}\nabla \times ({\vec {f}}\otimes {\vec {g}})=&{\hat {e}}_{i}\times \left({\vec {f}}_{,i}\otimes {\vec {g}}+{\vec {f}}\otimes {\vec {g}}_{,i}\right)\!\!\!\!\!\!\!\!\!\!&=&(\nabla \times {\vec {f}})\otimes {\vec {g}}-{\vec {f}}\times (\nabla \otimes ({\vec {g}})\\\nabla \times (f\mathbf {T} )=&{\hat {e}}_{k}\times (f_{,k}\mathbf {T} +f\mathbf {T} _{,k})\!\!\!\!\!\!\!\!\!\!&=&(\nabla f)\times \mathbf {T} +f\nabla \times \mathbf {T} \end{aligned}}}
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{\displaystyle {\begin{aligned}\nabla \times (\mathbf {T} \cdot {\vec {f}})=&{\hat {e}}_{k}\times (\mathbf {T} _{,k}\cdot {\vec {f}}+\mathbf {T} \cdot {\vec {f}}_{,k})\\=&(\nabla \times \mathbf {T} )\cdot {\vec {f}}+{\vec {\mathrm {i} }}\left({\hat {e}}_{k}\otimes \mathbf {T} \cdot {\vec {f}}_{,k}\right)\\=&(\nabla \times \mathbf {T} )\cdot {\vec {f}}-{\vec {\mathrm {i} }}{\big (}\mathbf {T} \cdot (\nabla \otimes {\vec {f}})^{\top }{\big )}\\\nabla \times (\mathbf {T} \times {\vec {f}})=&{\hat {e}}_{k}\times (\mathbf {T} _{,k}\times {\vec {f}}+(\mathbf {T} \cdot {\hat {e}}_{i})\otimes {\hat {e}}_{i}\times {\vec {f}}_{,k})\\=&(\nabla \times \mathbf {T} )\times {\vec {f}}-(\mathbf {T} \cdot {\hat {e}}_{i})\times {\hat {e}}_{k}\otimes {\hat {e}}_{i}\times {\vec {f}}_{,k}\\=&(\nabla \times \mathbf {T} )\times {\vec {f}}-\mathbf {T} \#(\nabla \otimes {\vec {f}})\end{aligned}}}
Beliebige Basis:
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{\displaystyle \mathrm {rot} (f^{i}{\vec {b}}_{i})=\mathrm {grad} (f^{i})\times {\vec {b}}_{i}+f^{i}\,\mathrm {rot} ({\vec {b}}_{i})}
Produkt mit Konstanten:
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{\displaystyle {\begin{array}{rcl}\mathrm {rot} (\mathbf {C} \cdot {\vec {f}})&=&-{\vec {\mathrm {i} }}\left(\mathbf {C} \cdot \mathrm {grad} ({\vec {f}})\right)\\&&\rightarrow \quad \mathrm {rot} ({\vec {f}})=\mathrm {rot} (\mathbf {1} \cdot {\vec {f}})=-{\vec {\mathrm {i} }}\left(\mathrm {grad} ({\vec {f}})\right)\\\mathrm {rot} ({\vec {f}}\times \mathbf {1} )&=&\mathbf {1} \#\mathrm {grad} ({\vec {f}})^{\top }=\mathrm {grad} ({\vec {f}})-\mathrm {div} ({\vec {f}})\mathbf {1} \end{array}}}
In divergenzfreien Feldern ist also:
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{\displaystyle \mathrm {rot} ({\vec {f}}\times \mathbf {1} )=\mathrm {grad} ({\vec {f}})}
Definition/Allgemeines
Bearbeiten
Δ
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{\displaystyle \Delta :=\nabla \cdot \nabla =\nabla ^{2}}
Zusammenhang mit anderen Differentialoperatoren:
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{\displaystyle {\begin{array}{rclcl}\Delta f&=&\mathrm {div{\big (}grad} (f){\big )}&=&\nabla \cdot (\nabla f)\\\Delta {\vec {f}}&=&\mathrm {div{\big (}grad} ({\vec {f}}){\big )}&=&\nabla \cdot (\nabla \otimes {\vec {f}})\end{array}}}
„Vektorieller Laplace-Operator“:
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{\displaystyle \Delta {\vec {f}}=\mathrm {grad{\big (}div} ({\vec {f}}){\big )}-\mathrm {rot{\big (}rot} ({\vec {f}}){\big )}}
Laplace-Operator in verschiedenen Koordinatensystemen
Bearbeiten
#Kartesische Koordinaten :
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{\displaystyle {\begin{aligned}\Delta f=&f_{,kk}\\\Delta {\vec {f}}=&\Delta f_{i}{\hat {e}}_{i}=f_{i,kk}{\hat {e}}_{i}\\\Delta \mathbf {T} =&\Delta T_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=T_{ij,kk}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\end{aligned}}}
#Zylinderkoordinaten :
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{\displaystyle {\begin{aligned}\Delta f=&{\frac {f_{,\rho }}{\rho }}+f_{,\rho \rho }+{\frac {f_{,\varphi \varphi }}{\rho ^{2}}}+f_{,zz}\\\Delta {\vec {f}}=&\left(\Delta f_{\rho }-{\frac {2f_{\varphi ,\varphi }+f_{\rho }}{\rho ^{2}}}\right){\hat {e}}_{\rho }+\left(\Delta f_{\varphi }+{\frac {2f_{\rho ,\varphi }-f_{\varphi }}{\rho ^{2}}}\right){\hat {e}}_{\varphi }+\Delta f_{z}{\hat {e}}_{z}\end{aligned}}}
#Kugelkoordinaten :
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{\displaystyle {\begin{aligned}\Delta f=&{\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin(\vartheta )}}{\frac {\partial }{\partial \vartheta }}\left(\sin(\vartheta )\,{\frac {\partial f}{\partial \vartheta }}\right)+{\frac {1}{r^{2}\sin ^{2}(\vartheta )}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}\\=&{\frac {2f_{,r}}{r}}+f_{,rr}+{\frac {f_{,\vartheta }\cos(\vartheta )+f_{,\vartheta \vartheta }\sin(\vartheta )}{r^{2}\sin(\vartheta )}}+{\frac {f_{,\varphi \varphi }}{r^{2}\sin ^{2}(\vartheta )}}\\\Delta {\vec {f}}=&\left(\Delta f_{r}-{\frac {2}{r^{2}}}(f_{r}+f_{\vartheta ,\vartheta })-2{\frac {f_{\varphi ,\varphi }+f_{\vartheta }\cos(\vartheta )}{r^{2}\sin(\vartheta )}}\right){\hat {e}}_{r}\\&+\left(\Delta f_{\vartheta }+{\frac {2f_{r,\vartheta }}{r^{2}}}-{\frac {f_{\vartheta }+2f_{\varphi ,\varphi }\cos(\vartheta )}{r^{2}\sin ^{2}(\vartheta )}}\right){\hat {e}}_{\vartheta }\\&+\left(\Delta f_{\varphi }-{\frac {f_{\varphi }-2f_{\vartheta ,\varphi }\cos(\vartheta )-2f_{r,\varphi }\sin(\vartheta )}{r^{2}\sin ^{2}(\vartheta )}}\right){\hat {e}}_{\varphi }\end{aligned}}}
Wegen der in der Literatur teilweise abweichenden Definitionen der Differentialoperatoren kann es in der Literatur zu abweichenden Formeln kommen. Wenn die Definitionen der Literatur hier eingesetzt werden, gehen die hiesigen Formeln in die der Literatur über.
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{\displaystyle {\begin{array}{rclcl}\mathrm {div(rot} ({\vec {f}}))&=&\nabla \cdot (\nabla \times {\vec {f}})&=&0\\\mathrm {rot(grad} (f))&=&\nabla \times \nabla f&=&{\vec {0}}\\\mathrm {div(grad} (f)\times \mathrm {grad} (g))&=&\nabla \cdot (\nabla f\times \nabla g)=\nabla g\cdot (\nabla \times \nabla f)&=&0\\\mathrm {rot{\big (}grad} ({\vec {f}}){\big )}&=&\nabla \times (\nabla \otimes {\vec {f}})&=&\mathbf {0} \\\mathrm {div{\big (}rot} (\mathbf {T} )^{\top }{\big )}&=&\nabla \cdot (\nabla \times \mathbf {T} )&=&{\vec {0}}\end{array}}}
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{\displaystyle {\begin{array}{rclcl}\mathrm {div{\big (}grad} (f){\big )}&=&\nabla \cdot (\nabla f)=(\nabla \cdot \nabla )f&=&\Delta f\\\mathrm {div{\big (}grad} ({\vec {f}}){\big )}&=&\nabla \cdot (\nabla \otimes {\vec {f}})=(\nabla \cdot \nabla ){\vec {f}}&=&\Delta {\vec {f}}\end{array}}}
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{\displaystyle {\begin{array}{rclcl}\mathrm {div{\big (}grad} ({\vec {f}})^{\top }{\big )}&=&\nabla \cdot (\nabla \otimes {\vec {f}}^{\top })=f_{i,ij}{\hat {e}}_{j}&=&\mathrm {grad{\big (}div} ({\vec {f}}){\big )}\\\mathrm {rot{\big (}grad} ({\vec {f}})^{\top }{\big )}&=&\nabla \times {\big (}(\nabla \otimes {\vec {f}})^{\top }{\big )}=\nabla \times {\big (}{\vec {f}}_{,i}\otimes {\hat {e}}_{i}{\big )}&=&\mathrm {grad{\big (}rot} ({\vec {f}}){\big )}\end{array}}}
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{\displaystyle {\begin{array}{rclcl}\mathrm {rot{\big (}rot} ({\vec {f}}){\big )}&=&\nabla \times (\nabla \times {\vec {f}})=\nabla (\nabla \cdot {\vec {f}})-\Delta {\vec {f}}&=&\mathrm {grad(div} ({\vec {f}}))-\Delta {\vec {f}}\\\mathrm {rot{\big (}rot} (\mathbf {T} )^{\top }{\big )}^{\top }&=&{\big (}\nabla \times (\nabla \times (\mathbf {T} ^{\top })){\big )}^{\top }\\&=&{\big (}\nabla \otimes \nabla \cdot \mathbf {T} ^{\top }{\big )}^{\top }-(\nabla \cdot \nabla )\mathbf {T} &=&\mathrm {grad(div} (\mathbf {T} ))-\Delta \mathbf {T} \end{array}}}
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{\displaystyle {\begin{array}{rcl}\mathrm {rot{\big (}rot} (\mathbf {T} ^{\top }){\big )}&=&-\Delta \mathbf {T} -\mathrm {grad{\big (}grad(Sp} (\mathbf {T} )){\big )}+\mathrm {grad{\big (}div} (\mathbf {T} ){\big )}+\mathrm {grad{\big (}div} (\mathbf {T} ^{\top }){\big )}^{\top }\\&&+\left[\Delta \mathrm {Sp} (\mathbf {T} )-\mathrm {div{\big (}div} (\mathbf {T} ){\big )}\right]\mathbf {1} \end{array}}}
Bei symmetrischem T = T ⊤ gilt:
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{\displaystyle {\begin{array}{rcl}\mathrm {rot{\big (}rot} (\mathbf {T} ){\big )}&=&-\Delta \mathbf {T} -\mathrm {grad{\big (}grad(Sp} (\mathbf {T} )){\big )}+\mathrm {grad{\big (}div} (\mathbf {T} ){\big )}+\mathrm {grad{\big (}div} (\mathbf {T} ){\big )}^{\top }\\&&+\left[\Delta \mathrm {Sp} (\mathbf {T} )-\mathrm {div{\big (}div} (\mathbf {T} ){\big )}\right]\mathbf {1} \end{array}}}
Wenn zusätzlich
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{\displaystyle \mathbf {T} =\mathbf {T} ^{\top }=\mathbf {G} -\mathrm {Sp} (\mathbf {G} )\mathbf {1} }
dann ist:
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{\displaystyle \mathrm {rot{\big (}rot} (\mathbf {T} ){\big )}=-\Delta \mathbf {G} +\mathrm {grad{\big (}div} (\mathbf {G} ){\big )}+\mathrm {grad{\big (}div} (\mathbf {G} ){\big )}^{\top }-\mathrm {div{\big (}div} (\mathbf {G} ){\big )}\mathbf {1} }
Der Laplace-Operator kann zwischen den anderen Operatoren wie ein Skalar behandelt werden, also an beliebiger Stelle in die Formeln eingesetzt werden, z. B.:
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{\displaystyle {\begin{array}{l}\Delta \mathrm {rot(rot} ({\vec {f}}))=\mathrm {rot(\Delta rot} ({\vec {f}}))=\mathrm {rot(rot} (\Delta {\vec {f}}))=\ldots \\\ldots =\Delta \mathrm {grad(div} ({\vec {f}}))-\Delta \Delta {\vec {f}}=\mathrm {grad} (\Delta \mathrm {div} ({\vec {f}}))-\Delta \Delta {\vec {f}}=\mathrm {grad(div} (\Delta {\vec {f}}))-\Delta \Delta {\vec {f}}\end{array}}}
Grassmann-Entwicklung
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{\displaystyle {\begin{aligned}{\vec {f}}\times \mathrm {rot} ({\vec {f}})=&{\frac {1}{2}}\mathrm {grad} ({\vec {f}}\cdot {\vec {f}})-\mathrm {grad} ({\vec {f}})\cdot {\vec {f}}\\=&{\big (}\mathrm {grad} ({\vec {f}})^{\top }-\mathrm {grad} ({\vec {f}}){\big )}\cdot {\vec {f}}={\vec {\mathrm {i} }}{\big (}\mathrm {grad} ({\vec {f}}){\big )}\times {\vec {f}}\end{aligned}}}
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{\displaystyle \mathrm {grad} ({\vec {f}})\cdot {\vec {f}}={\frac {1}{2}}\mathrm {grad} ({\vec {f}}\cdot {\vec {f}})-{\vec {f}}\times \mathrm {rot} ({\vec {f}})}
Sätze über Gradient, Divergenz und Rotation
Bearbeiten
Ein Vektorfeld, dessen Divergenz und Rotation verschwindet, ist harmonisch :
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{\displaystyle \operatorname {div} ({\vec {f}})=0\;{\text{und}}\;\mathrm {rot} ({\vec {f}})={\vec {0}}\quad \rightarrow \quad \Delta {\vec {f}}={\vec {0}}}
Jedes Vektorfeld lässt sich eindeutig in einen divergenzfreien und einen rotationsfreien Anteil zerlegen. Den Integrabilitätsbedingungen für Rotationen und Gradienten zufolge ist der erste Anteil ein Rotationsfeld und der zweite ein Gradientenfeld.
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{\displaystyle {\begin{array}{rclccl}{\vec {f}}={\vec {f}}_{1}+{\vec {f}}_{2}:&&&\mathrm {div} ({\vec {f}}_{1})=0&{\text{und}}&\operatorname {rot} ({\vec {f}}_{2})={\vec {0}}\\\leftrightarrow \exists g,{\vec {g}}:&&{\vec {f}}=&\operatorname {rot} ({\vec {g}})&+&\mathrm {grad} (g)\end{array}}}
Satz über rotationsfreie Felder
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{\displaystyle {\begin{array}{rrcll}{\textsf {I}}:&\mathrm {rot} ({\vec {u}}):={\hat {e}}_{k}\times {\vec {u}}_{,k}={\vec {0}}&\rightarrow &\exists f\colon &{\vec {u}}=\mathrm {grad} (f)\\{\textsf {II}}:&\mathrm {rot} (\mathbf {T} )=\mathbf {0} &\rightarrow &\exists {\vec {u}}\colon &\mathbf {T} =\mathrm {grad} ({\vec {u}})\\{\textsf {III}}:&\mathrm {rot} (\mathbf {T} )=\mathbf {0} \;{\text{und}}\;\mathrm {Sp} (\mathbf {T} )=0&\rightarrow &\exists \mathbf {W} \colon &\mathbf {T} =\mathrm {rot} (\mathbf {W} )\;{\text{und}}\;\mathbf {W} =-\mathbf {W} ^{\top }\end{array}}}
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{\displaystyle {\begin{array}{rrcll}{\textsf {II}}:&\nabla \times (\mathbf {T} ^{\top })=\mathbf {0} &\rightarrow &\exists {\vec {u}}\colon &\mathbf {T} =\mathrm {grad} ({\vec {u}})\\{\textsf {III}}:&\nabla \times (\mathbf {T} ^{\top })=\mathbf {0} \;{\text{und}}\;\mathrm {Sp} (\mathbf {T} )=0&\rightarrow &\exists \mathbf {W} \colon &\mathbf {T} =\mathrm {rot} (\mathbf {W} )\;{\text{und}}\;\mathbf {W} =-\mathbf {W} ^{\top }\end{array}}}
Gaußscher Integralsatz
Bearbeiten
Volumen
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{\displaystyle v}
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{\displaystyle \mathrm {d} v}
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{\displaystyle a}
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Skalar-, vektor- oder tensorwertige Funktion
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{\displaystyle f,{\vec {f}},\mathbf {T} }
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{\displaystyle {\begin{array}{rcl}\int _{v}\mathrm {grad} (f)\,\mathrm {d} v&=&\int _{a}f\,\mathrm {d} {\vec {a}}\\\int _{v}\mathrm {grad} ({\vec {f}})\,\mathrm {d} v&=&\int _{a}{\vec {f}}\otimes \mathrm {d} {\vec {a}}\\\int _{v}\mathrm {div} ({\vec {f}})\,\mathrm {d} v&=&\int _{a}{\vec {f}}\cdot \mathrm {d} {\vec {a}}\\\int _{v}\mathrm {rot} ({\vec {f}})\,\mathrm {d} v&=&-\int _{a}{\vec {f}}\times \mathrm {d} {\vec {a}}\\\int _{v}\mathrm {div} (\mathbf {T} )\,\mathrm {d} v&=&\int _{a}\mathbf {T} \cdot \mathrm {d} {\vec {a}}\\\int _{v}\nabla \cdot \mathbf {T} \,\mathrm {d} v&=&\int _{a}\mathbf {T} ^{\top }\cdot \mathrm {d} {\vec {a}}\end{array}}}
Mit der #Produktregel für Gradienten , #Produktregel für Divergenzen und #Produktregel für Rotationen können Formeln für die partielle Integration im Mehrdimensionalen abgeleitet werden, beispielsweise:
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{\displaystyle {\begin{aligned}\mathrm {grad} (fg)=&\mathrm {grad} (f)g+f\mathrm {grad} (g)\\\rightarrow \int _{v}\mathrm {grad} (f)g\,\mathrm {d} v=&\int _{a}fg\,\mathrm {d} {\vec {a}}-\int _{v}f\mathrm {grad} (g)\,\mathrm {d} v\end{aligned}}}
Klassischer Integralsatz von Stokes
Bearbeiten
Gegeben:
Fläche
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{\displaystyle \int _{a}\mathrm {rot} ({\vec {f}})\cdot \mathrm {d} {\vec {a}}=\oint _{b}{\vec {f}}\cdot \mathrm {d} {\vec {b}}}
Mit der #Produktregel für Rotationen können Formeln für die partielle Integration im Mehrdimensionalen abgeleitet werden, beispielsweise:
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{\displaystyle {\begin{aligned}\mathrm {rot} (f{\vec {g}})=&\mathrm {grad} (f)\times {\vec {g}}+f\mathrm {rot} ({\vec {g}})\\\rightarrow \int _{a}{\big (}\mathrm {grad} (f)\times {\vec {g}}{\big )}\cdot \mathrm {d} {\vec {a}}=&\oint _{b}f{\vec {g}}\cdot \mathrm {d} {\vec {b}}-\int _{a}f\mathrm {rot} ({\vec {g}})\cdot \mathrm {d} {\vec {a}}\end{aligned}}}
Reynoldscher Transportsatz
Bearbeiten
Gegeben:
Zeit
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{\displaystyle t}
Zeitabhängiges Volumen
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{\displaystyle v}
mit Volumenform
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{\displaystyle \mathrm {d} v}
mit
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)
{\displaystyle {\vec {v}}({\vec {x}},t)}
Eine skalare oder vektorwertige Dichtefunktion pro Volumeneinheit
f
(
x
→
,
t
)
{\displaystyle f({\vec {x}},t)}
, die mit den sich bewegenden Partikeln transportiert wird.
Die Integrale Größe für das Volumen:
∫
v
f
→
(
x
→
,
t
)
d
v
{\displaystyle \int _{v}{\vec {f}}({\vec {x}},t)\,\mathrm {d} v}
Skalare Funktion
f
(
x
→
,
t
)
{\displaystyle f({\vec {x}},t)}
:
d
d
t
∫
v
f
d
v
=
∫
v
∂
f
∂
t
d
v
+
∫
a
f
(
v
→
⋅
d
a
→
)
=
∫
v
(
∂
f
∂
t
+
d
i
v
(
f
v
→
)
)
d
v
=
∫
v
(
∂
f
∂
t
+
g
r
a
d
(
f
)
⋅
v
→
+
d
i
v
(
v
→
)
f
)
d
v
=
∫
v
(
f
˙
+
d
i
v
(
v
→
)
f
)
d
v
{\displaystyle {\begin{array}{rcl}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}f\,\mathrm {d} v&=&\int _{v}{\frac {\partial f}{\partial t}}\,\mathrm {d} v+\int _{a}f({\vec {v}}\cdot \mathrm {d} {\vec {a}})=\int _{v}\left({\frac {\partial f}{\partial t}}+\mathrm {div} (f{\vec {v}})\right)\,\mathrm {d} v\\&=&\int _{v}\left({\frac {\partial f}{\partial t}}+\mathrm {grad} (f)\cdot {\vec {v}}+\mathrm {div} ({\vec {v}})\,f\right)\,\mathrm {d} v=\int _{v}\left({\dot {f}}+\mathrm {div} ({\vec {v}})\,f\right)\,\mathrm {d} v\end{array}}}
Vektorwertige Funktion
f
→
(
x
→
,
t
)
{\displaystyle {\vec {f}}({\vec {x}},t)}
:
d
d
t
∫
v
f
→
d
v
=
∫
v
∂
f
→
∂
t
d
v
+
∫
a
f
→
(
v
→
⋅
d
a
→
)
=
∫
v
(
∂
f
→
∂
t
+
d
i
v
(
v
→
⊗
f
→
)
)
d
v
=
∫
v
(
∂
f
→
∂
t
+
g
r
a
d
(
f
→
)
⋅
v
→
+
d
i
v
(
v
→
)
f
→
)
d
v
=
∫
v
(
f
→
˙
+
d
i
v
(
v
→
)
f
→
)
d
v
{\displaystyle {\begin{array}{rcl}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{v}{\vec {f}}\,\mathrm {d} v&=&\int _{v}{\frac {\partial {\vec {f}}}{\partial t}}\,\mathrm {d} v+\int _{a}{\vec {f}}({\vec {v}}\cdot \mathrm {d} {\vec {a}})=\int _{v}\left({\frac {\partial {\vec {f}}}{\partial t}}+\mathrm {div} ({\vec {v}}\otimes {\vec {f}})\right)\,\mathrm {d} v\\&=&\int _{v}\left({\frac {\partial {\vec {f}}}{\partial t}}+\mathrm {grad} ({\vec {f}})\cdot {\vec {v}}+\mathrm {div} ({\vec {v}}){\vec {f}}\right)\,\mathrm {d} v=\int _{v}({\dot {\vec {f}}}+\mathrm {div} ({\vec {v}}){\vec {f}})\,\mathrm {d} v\end{array}}}
Transportsatz für Flächenintegrale
Bearbeiten
Gegeben:
Zeit
t
{\displaystyle t}
Ortsvektoren
x
→
∈
v
{\displaystyle {\vec {x}}\in v}
Geschwindigkeitsfeld:
v
→
(
x
→
,
t
)
{\displaystyle {\vec {v}}({\vec {x}},t)}
Zeitabhängige Fläche
a
:
[
0
,
1
]
2
↦
v
{\displaystyle a\colon [0,1]^{2}\mapsto v}
, die mit dem Geschwindigkeitsfeld transportiert wird und auf der mit räumlichem, vektoriellem Oberflächenelement
d
a
→
{\displaystyle \mathrm {d} {\vec {a}}}
im Volumen v integriert wird
Eine skalare oder vektorwertige Feldgröße
f
(
x
→
,
t
)
{\displaystyle f({\vec {x}},t)}
, die mit den sich bewegenden Partikeln transportiert wird.
Die Integrale Größe auf der Fläche:
∫
a
f
(
x
→
,
t
)
⋅
d
a
→
{\displaystyle \int _{a}f({\vec {x}},t)\cdot \mathrm {d} {\vec {a}}}
Skalare Funktion
f
(
x
→
,
t
)
{\displaystyle f({\vec {x}},t)}
:
d
d
t
∫
a
f
d
a
→
=
∫
a
[
f
˙
1
+
f
div
(
v
→
)
1
−
f
grad
(
v
→
)
⊤
]
⋅
d
a
→
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{a}f\,\mathrm {d} {\vec {a}}=\int _{a}[{\dot {f}}\mathbf {1} +f\operatorname {div} ({\vec {v}})\mathbf {1} -f\operatorname {grad} ({\vec {v}})^{\top }]\cdot \,\mathrm {d} {\vec {a}}}
Vektorwertige Funktion
f
→
(
x
→
,
t
)
{\displaystyle {\vec {f}}({\vec {x}},t)}
:
d
d
t
∫
a
f
→
⋅
d
a
→
=
∫
a
[
f
→
˙
+
f
→
div
(
v
→
)
−
grad
(
v
→
)
⋅
f
→
]
⋅
d
a
→
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{a}{\vec {f}}\cdot \,\mathrm {d} {\vec {a}}=\int _{a}[{\dot {\vec {f}}}+{\vec {f}}\operatorname {div} ({\vec {v}})-\operatorname {grad} ({\vec {v}})\cdot {\vec {f}}]\cdot \,\mathrm {d} {\vec {a}}}
Transportsatz für Kurvenintegrale
Bearbeiten
Gegeben:
Zeit
t
{\displaystyle t}
Ortsvektoren
x
→
∈
v
{\displaystyle {\vec {x}}\in v}
Geschwindigkeitsfeld:
v
→
(
x
→
,
t
)
{\displaystyle {\vec {v}}({\vec {x}},t)}
Zeitabhängige Kurve
b
:
[
0
,
1
)
↦
v
{\displaystyle b\colon [0,1)\mapsto v}
, die mit dem Geschwindigkeitsfeld transportiert wird und entlang derer mit räumlichem, vektoriellem Linienelement
d
b
→
{\displaystyle \mathrm {d} {\vec {b}}}
im Volumen v integriert wird
Eine skalare oder vektorwertige Feldgröße
f
(
x
→
,
t
)
{\displaystyle f({\vec {x}},t)}
, die mit den sich bewegenden Partikeln transportiert wird.
Die Integrale Größe entlang des Weges:
∫
b
f
(
x
→
,
t
)
⋅
d
b
→
{\displaystyle \int _{b}f({\vec {x}},t)\cdot \mathrm {d} {\vec {b}}}
Skalare Funktion
f
(
x
→
,
t
)
{\displaystyle f({\vec {x}},t)}
:
d
d
t
∮
b
f
d
b
→
=
∮
b
(
f
˙
1
+
f
g
r
a
d
v
→
)
⋅
d
b
→
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\oint _{b}f\,\mathrm {d} {\vec {b}}=\oint _{b}({\dot {f}}\mathbf {1} +f\,\mathrm {grad} {\vec {v}})\cdot \mathrm {d} {\vec {b}}}
Vektorwertige Funktion
f
→
(
x
→
,
t
)
{\displaystyle {\vec {f}}({\vec {x}},t)}
:
d
d
t
∮
b
f
→
⋅
d
b
→
=
∮
b
(
f
→
˙
+
f
→
⋅
g
r
a
d
v
→
)
⋅
d
b
→
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\oint _{b}{\vec {f}}\cdot \mathrm {d} {\vec {b}}=\oint _{b}({\dot {\vec {f}}}+{\vec {f}}\cdot \mathrm {grad} {\vec {v}})\cdot \mathrm {d} {\vec {b}}}
Kleine Deformationen
Bearbeiten
Ingenieursdehnungen:
ε
=
ε
i
j
e
^
i
⊗
e
^
j
=
1
2
(
u
i
,
j
+
u
j
,
i
)
e
^
i
⊗
e
^
j
{\displaystyle {\boldsymbol {\varepsilon }}=\varepsilon _{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}={\frac {1}{2}}(u_{i,j}+u_{j,i}){\hat {e}}_{i}\otimes {\hat {e}}_{j}}
Kompatibilitätsbedingungen :
r
o
t
(
r
o
t
(
ε
)
)
=
∇
×
(
∇
×
ε
)
⊤
=
0
↓
2
ε
12
,
12
−
ε
22
,
11
−
ε
11
,
22
=
0
2
ε
13
,
13
−
ε
33
,
11
−
ε
11
,
33
=
0
2
ε
23
,
23
−
ε
33
,
22
−
ε
22
,
33
=
0
ε
11
,
23
+
ε
23
,
11
−
ε
12
,
13
−
ε
13
,
12
=
0
ε
22
,
13
+
ε
13
,
22
−
ε
12
,
23
−
ε
23
,
12
=
0
ε
12
,
33
+
ε
33
,
12
−
ε
13
,
23
−
ε
23
,
13
=
0
{\displaystyle {\begin{array}{rcl}\mathrm {rot{\big (}rot} ({\boldsymbol {\varepsilon }}){\big )}=\nabla \times (\nabla \times {\boldsymbol {\varepsilon }})^{\top }&=&\mathbf {0} \\&\downarrow &\\2\varepsilon _{12,12}-\varepsilon _{22,11}-\varepsilon _{11,22}&=&0\\2\varepsilon _{13,13}-\varepsilon _{33,11}-\varepsilon _{11,33}&=&0\\2\varepsilon _{23,23}-\varepsilon _{33,22}-\varepsilon _{22,33}&=&0\\\varepsilon _{11,23}+\varepsilon _{23,11}-\varepsilon _{12,13}-\varepsilon _{13,12}&=&0\\\varepsilon _{22,13}+\varepsilon _{13,22}-\varepsilon _{12,23}-\varepsilon _{23,12}&=&0\\\varepsilon _{12,33}+\varepsilon _{33,12}-\varepsilon _{13,23}-\varepsilon _{23,13}&=&0\end{array}}}
Orthogonaler Tensor
Q
{\displaystyle \mathbf {Q} }
beschreibt die Drehung.
Ω
:=
Q
˙
⋅
Q
⊤
=
(
Q
⋅
Q
˙
⊤
)
⊤
=
−
Q
⋅
Q
˙
⊤
{\displaystyle \mathbf {\Omega } :={\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }={(\mathbf {Q} \cdot {\dot {\mathbf {Q} }}^{\top })}^{\top }=-\mathbf {Q} \cdot {\dot {\mathbf {Q} }}^{\top }}
Vektorinvariante oder dualer axialer Vektor
ω
→
{\displaystyle {\vec {\omega }}}
des schiefsymmetrischen Tensors
Ω
{\displaystyle \mathbf {\Omega } }
ist die Winkelgeschwindigkeit :
Ω
⋅
r
→
=
ω
→
×
r
→
∀
r
→
{\displaystyle \mathbf {\Omega } \cdot {\vec {r}}={\vec {\omega }}\times {\vec {r}}{\quad \forall \;}{\vec {r}}}
Starrkörperbewegung mit
r
→
=
c
o
n
s
t
.
{\displaystyle {\vec {r}}=\mathrm {const.} }
:
x
→
=
f
→
+
Q
⋅
r
→
→
r
→
=
Q
⊤
⋅
(
x
→
−
f
→
)
{\displaystyle {\vec {x}}={\vec {f}}+\mathbf {Q} \cdot {\vec {r}}\quad \rightarrow \quad {\vec {r}}=\mathbf {Q} ^{\top }\cdot ({\vec {x}}-{\vec {f}})}
v
→
=
f
→
˙
+
Q
˙
⋅
r
→
=
f
→
˙
+
Q
˙
⋅
Q
⊤
⋅
(
x
→
−
f
→
)
=
f
→
˙
+
Ω
⋅
(
x
→
−
f
→
)
=
f
→
˙
+
ω
→
×
(
x
→
−
f
→
)
{\displaystyle {\vec {v}}={\dot {\vec {f}}}+{\dot {\mathbf {Q} }}\cdot {\vec {r}}={\dot {\vec {f}}}+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }\cdot ({\vec {x}}-{\vec {f}})={\dot {\vec {f}}}+\mathbf {\Omega } \cdot ({\vec {x}}-{\vec {f}})={\dot {\vec {f}}}+{\vec {\omega }}\times ({\vec {x}}-{\vec {f}})}
Ableitungen der Invarianten
Bearbeiten
∂
I
1
(
T
)
∂
T
=
∂
S
p
(
T
)
∂
T
=
1
{\displaystyle {\frac {\partial \mathrm {I} _{1}(\mathbf {T} )}{\partial \mathbf {T} }}={\frac {\partial \mathrm {Sp} (\mathbf {T} )}{\partial \mathbf {T} }}=\mathbf {1} }
∂
I
2
(
T
)
∂
T
=
I
1
(
T
)
1
−
T
⊤
{\displaystyle {\frac {\partial \mathrm {I} _{2}(\mathbf {T} )}{\partial \mathbf {T} }}=\mathrm {I} _{1}(\mathbf {T} )\mathbf {1} -\mathbf {T} ^{\top }}
∂
I
3
(
T
)
∂
T
=
∂
d
e
t
(
T
)
∂
T
=
d
e
t
(
T
)
T
⊤
−
1
=
c
o
f
(
T
)
=
T
⊤
⋅
T
⊤
−
I
1
(
T
)
T
⊤
+
I
2
(
T
)
1
{\displaystyle {\frac {\partial \mathrm {I} _{3}(\mathbf {T} )}{\partial \mathbf {T} }}={\frac {\partial \mathrm {det} (\mathbf {T} )}{\partial \mathbf {T} }}=\mathrm {det} (\mathbf {T} )\mathbf {T} ^{\top -1}=\mathrm {cof} (\mathbf {T} )=\mathbf {T^{\top }\cdot T^{\top }} -\mathrm {I} _{1}(\mathbf {T} )\mathbf {T} ^{\top }+\mathrm {I} _{2}(\mathbf {T} )\mathbf {1} }
mit der transponiert inversen T ⊤-1 und dem Kofaktor cof(T ) des Tensors T .
Funktion
f
{\displaystyle f}
der Invarianten:
∂
f
∂
T
(
I
1
(
T
)
,
I
2
(
T
)
,
I
3
(
T
)
)
=
(
∂
f
∂
I
1
+
I
1
∂
f
∂
I
2
+
I
2
∂
f
∂
I
3
)
1
−
(
∂
f
∂
I
2
+
I
1
∂
f
∂
I
3
)
T
⊤
+
∂
f
∂
I
3
T
⊤
⋅
T
⊤
{\displaystyle {\begin{aligned}{\frac {\partial f}{\partial \mathbf {T} }}(\mathrm {I} _{1}(\mathbf {T} ),\,\mathrm {I} _{2}(\mathbf {T} ),\,\mathrm {I} _{3}(\mathbf {T} ))=&\left({\frac {\partial f}{\partial \mathrm {I} _{1}}}+\mathrm {I} _{1}{\frac {\partial f}{\partial \mathrm {I} _{2}}}+\mathrm {I} _{2}{\frac {\partial f}{\partial \mathrm {I} _{3}}}\right)\mathbf {1} -\left({\frac {\partial f}{\partial \mathrm {I} _{2}}}+\mathrm {I} _{1}{\frac {\partial f}{\partial \mathrm {I} _{3}}}\right)\mathbf {T} ^{\top }\\&+{\frac {\partial f}{\partial \mathrm {I} _{3}}}\mathbf {T} ^{\top }\cdot \mathbf {T} ^{\top }\end{aligned}}}
Ableitung der Frobenius-Norm :
∂
∥
T
∥
∂
T
=
T
∥
T
∥
{\displaystyle {\frac {\partial \parallel \mathbf {T} \parallel }{\partial \mathbf {T} }}={\frac {\mathbf {T} }{\parallel \mathbf {T} \parallel }}}
Eigenwerte (aus der impliziten Ableitung des charakteristischen Polynoms ):
T
⋅
v
→
=
λ
v
→
→
d
e
t
(
T
−
λ
1
)
=
−
λ
3
+
I
1
λ
2
−
I
2
λ
+
I
3
=
0
{\displaystyle \mathbf {T} \cdot {\vec {v}}=\lambda {\vec {v}}\quad \rightarrow \quad \mathrm {det} (\mathbf {T} -\lambda \mathbf {1} )=-\lambda ^{3}+\mathrm {I} _{1}\lambda ^{2}-\mathrm {I} _{2}\lambda +\mathrm {I} _{3}=0}
→
d
λ
d
T
=
(
λ
2
−
λ
I
1
+
I
2
)
1
+
(
λ
−
I
1
)
T
⊤
+
T
⊤
⋅
T
⊤
3
λ
2
−
2
I
1
λ
+
I
2
{\displaystyle {\dfrac {\mathrm {d} \lambda }{\mathrm {d} \mathbf {T} }}={\dfrac {(\lambda ^{2}-\lambda \mathrm {I} _{1}+\mathrm {I} _{2})\mathbf {1} +(\lambda -\mathrm {I} _{1})\mathbf {T} ^{\top }+\mathbf {T^{\top }\cdot T^{\top }} }{3\lambda ^{2}-2\mathrm {I} _{1}\lambda +\mathrm {I} _{2}}}}
Eigenwerte symmetrischer Tensoren:
T
⋅
v
→
=
λ
v
→
→
∂
λ
∂
T
=
v
→
⊗
v
→
{\displaystyle \mathbf {T} \cdot {\vec {v}}=\lambda {\vec {v}}\quad \rightarrow \quad {\frac {\partial \lambda }{\partial \mathbf {T} }}={\vec {v}}\otimes {\vec {v}}}
Eigenwerte von
T
=
∑
i
=
1
3
λ
i
v
→
i
⊗
v
→
i
{\displaystyle \mathbf {T} =\sum _{i=1}^{3}\lambda _{i}\,{\vec {v}}_{i}\otimes {\vec {v}}^{i}}
, wo
v
→
i
{\displaystyle {\vec {v}}^{i}}
dual zu den Eigenvektoren
v
→
i
{\displaystyle {\vec {v}}_{i}}
sind
(
v
→
i
⋅
v
→
j
=
δ
i
j
)
{\displaystyle ({\vec {v}}_{i}\cdot {\vec {v}}^{j}=\delta _{i}^{j})}
:
∂
λ
i
∂
T
=
v
→
i
⊗
v
→
i
{\displaystyle {\frac {\partial \lambda _{i}}{\partial \mathbf {T} }}={\vec {v}}^{i}\otimes {\vec {v}}_{i}}
(keine Summe)
Die Eigenwerte von
T
=
c
v
→
1
⊗
v
→
1
+
a
(
v
→
2
⊗
v
→
2
+
v
→
3
⊗
v
→
3
)
+
b
(
v
→
2
⊗
v
→
3
−
v
→
3
⊗
v
→
2
)
{\displaystyle \mathbf {T} =c\,{\vec {v}}_{1}\otimes {\vec {v}}^{1}+a({\vec {v}}_{2}\otimes {\vec {v}}^{2}+{\vec {v}}_{3}\otimes {\vec {v}}^{3})+b({\vec {v}}_{2}\otimes {\vec {v}}^{3}-{\vec {v}}_{3}\otimes {\vec {v}}^{2})}
sind
λ
1
=
c
,
λ
2
=
a
+
i
b
,
λ
3
=
a
−
i
b
{\displaystyle \lambda _{1}=c,\,\lambda _{2}=a+\mathrm {i} b,\,\lambda _{3}=a-\mathrm {i} b}
mit den Eigenvektoren
v
→
1
,
w
→
2
=
v
→
2
+
i
v
→
3
,
w
→
3
=
v
→
2
−
i
v
→
3
{\displaystyle {\vec {v}}_{1},\,{\vec {w}}_{2}={\vec {v}}_{2}+\mathrm {i} {\vec {v}}_{3},\,{\vec {w}}_{3}={\vec {v}}_{2}-\mathrm {i} {\vec {v}}_{3}}
. Hier ist:
∂
λ
1
∂
T
=
v
→
1
⊗
v
→
1
,
∂
λ
k
∂
T
=
1
2
w
→
k
⊗
w
→
k
¯
,
k
=
2
,
3
{\displaystyle {\frac {\partial \lambda _{1}}{\partial \mathbf {T} }}={\vec {v}}^{1}\otimes {\vec {v}}_{1},\quad {\frac {\partial \lambda _{k}}{\partial \mathbf {T} }}={\frac {1}{2}}{\overline {{\vec {w}}^{k}\otimes {\vec {w}}_{k}}},\quad k=2,3}
(keine Summe)
mit
w
→
2
=
v
→
2
+
i
v
→
3
,
w
→
3
=
v
→
2
−
i
v
→
3
{\displaystyle {\vec {w}}^{2}={\vec {v}}^{2}+\mathrm {i} {\vec {v}}^{3},\,{\vec {w}}^{3}={\vec {v}}^{2}-\mathrm {i} {\vec {v}}^{3}}
und der Überstrich markiert den konjugiert komplexen Wert.
Konvektive Koordinaten
Bearbeiten
Konvektive Koordinaten
y
1
,
y
2
,
y
3
∈
R
{\displaystyle y_{1},y_{2},y_{3}\in \mathbb {R} }
Kovariante Basisvektoren
B
→
i
=
d
X
→
d
y
i
{\displaystyle {\vec {B}}_{i}={\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} y_{i}}}}
,
b
→
i
=
d
x
→
d
y
i
{\displaystyle {\vec {b}}_{i}={\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} y_{i}}}}
Kontravariante Basisvektoren
B
→
i
=
d
y
i
d
X
→
:=
G
R
A
D
(
y
i
)
{\displaystyle {\vec {B}}^{i}={\frac {\mathrm {d} y_{i}}{\mathrm {d} {\vec {X}}}}:=\mathrm {GRAD} (y_{i})}
,
b
→
i
=
d
y
i
d
x
→
:=
g
r
a
d
(
y
i
)
{\displaystyle {\vec {b}}^{i}={\frac {\mathrm {d} y_{i}}{\mathrm {d} {\vec {x}}}}:=\mathrm {grad} (y_{i})}
B
→
i
⋅
B
→
j
=
b
→
i
⋅
b
→
j
=
δ
i
j
{\displaystyle {\vec {B}}_{i}\cdot {\vec {B}}^{j}={\vec {b}}_{i}\cdot {\vec {b}}^{j}=\delta _{i}^{j}}
Deformationsgradient
F
=
b
→
i
⊗
B
→
i
{\displaystyle \mathbf {F} ={\vec {b}}_{i}\otimes {\vec {B}}^{i}}
Räumlicher Geschwindigkeitsgradient
l
=
b
→
˙
i
⊗
b
→
i
=
−
b
→
i
⊗
b
→
˙
i
{\displaystyle \mathbf {l} ={\dot {\vec {b}}}_{i}\otimes {\vec {b}}^{i}=-{\vec {b}}_{i}\otimes {\dot {\vec {b}}}^{i}}
Kovarianter Tensor
T
=
T
i
j
b
→
i
⊗
b
→
j
{\displaystyle \mathbf {T} =T_{ij}{\vec {b}}^{i}\otimes {\vec {b}}^{j}}
Kontravarianter Tensor
T
=
T
i
j
b
→
i
⊗
b
→
j
{\displaystyle \mathbf {T} =T^{ij}{\vec {b}}_{i}\otimes {\vec {b}}_{j}}
Geschwindigkeitsgradient
Bearbeiten
Räumlicher Geschwindigkeitsgradient:
l
=
g
r
a
d
(
v
→
)
=
F
˙
⋅
F
−
1
{\displaystyle \mathbf {l} =\mathrm {grad} ({\vec {v}})={\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}}
Divergenz der Geschwindigkeit:
d
i
v
(
v
→
)
=
S
p
(
l
)
{\displaystyle \mathrm {div} ({\vec {v}})=\mathrm {Sp} (\mathbf {l} )}
Winkelgeschwindigkeit oder Wirbelstärke ist der duale axiale Vektor
ω
→
=
l
→
A
=
−
1
2
i
→
(
l
)
=
1
2
r
o
t
(
v
→
)
{\displaystyle {\vec {\omega }}={\stackrel {A}{\vec {\mathbf {l} }}}=-{\frac {1}{2}}{\vec {\mathrm {i} }}(\mathbf {l} )={\frac {1}{2}}\mathrm {rot} ({\vec {v}})}
D
D
t
d
e
t
(
F
)
=
d
e
t
(
F
)
F
⊤
−
1
:
F
˙
=
d
e
t
(
F
)
S
p
(
F
˙
⋅
F
−
1
)
=
d
e
t
(
F
)
d
i
v
(
v
→
)
{\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\mathrm {det} (\mathbf {F} )=\mathrm {det} (\mathbf {F} )\mathbf {F} ^{\top -1}:{\dot {\mathbf {F} }}=\mathrm {det} (\mathbf {F} )\mathrm {Sp} ({\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1})=\mathrm {det} (\mathbf {F} )\,\mathrm {div} ({\vec {v}})}
Objektive Zeitableitungen
Bearbeiten
Bezeichnungen wie in #Konvektive Koordinaten .
Räumlicher Geschwindigkeitsgradient
l
=
b
→
˙
i
⊗
b
→
i
=
−
b
→
i
⊗
b
→
˙
i
=
d
+
w
{\displaystyle \mathbf {l} ={\dot {\vec {b}}}_{i}\otimes {\vec {b}}^{i}=-{\vec {b}}_{i}\otimes {\dot {\vec {b}}}\,^{i}=\mathbf {d} +\mathbf {w} }
Räumliche Verzerrungsgeschwindigkeit
d
=
1
2
(
l
+
l
⊤
)
{\displaystyle \mathbf {d} ={\frac {1}{2}}(\mathbf {l} +\mathbf {l} ^{\top })}
Wirbel- oder Spintensor
w
=
1
2
(
l
−
l
⊤
)
{\displaystyle \mathbf {w} ={\frac {1}{2}}(\mathbf {l} -\mathbf {l} ^{\top })}
Objektive Zeitableitungen von Vektoren
Bearbeiten
Gegeben:
v
→
=
v
i
b
→
i
=
v
i
b
→
i
{\displaystyle {\vec {v}}=v_{i}{\vec {b}}^{i}=v^{i}{\vec {b}}_{i}}
:
v
→
Δ
=
v
→
˙
+
l
⊤
⋅
v
→
=
v
˙
i
b
→
i
v
→
∇
=
v
→
˙
−
l
⋅
v
→
=
v
˙
i
b
→
i
v
→
∘
=
v
→
˙
−
w
⋅
v
→
{\displaystyle {\begin{array}{rclcl}{\stackrel {\Delta }{\vec {v}}}&=&{\dot {\vec {v}}}+\mathbf {l} ^{\top }\cdot {\vec {v}}&=&{\dot {v}}_{i}{\vec {b}}^{i}\\{\stackrel {\nabla }{\vec {v}}}&=&{\dot {\vec {v}}}-\mathbf {l} \cdot {\vec {v}}&=&{\dot {v}}^{i}{\vec {b}}_{i}\\{\stackrel {\circ }{\vec {v}}}&=&{\dot {\vec {v}}}-\mathbf {w} \cdot {\vec {v}}\end{array}}}
Objektive Zeitableitungen von Tensoren
Bearbeiten
Gegeben:
T
=
T
i
j
b
→
i
⊗
b
→
j
=
T
i
j
b
→
i
⊗
b
→
j
{\displaystyle \mathbf {T} =T_{ij}{\vec {b}}^{i}\otimes {\vec {b}}^{j}=T^{ij}{\vec {b}}_{i}\otimes {\vec {b}}_{j}}
T
Δ
=
T
˙
+
T
⋅
l
+
l
⊤
⋅
T
=
T
˙
i
j
b
→
i
⊗
b
→
j
T
∇
=
T
˙
−
l
⋅
T
−
T
⋅
l
⊤
=
T
˙
i
j
b
→
i
⊗
b
→
j
T
∘
=
T
˙
+
T
⋅
w
−
w
⋅
T
T
⋄
=
T
˙
+
S
p
(
l
)
T
−
l
⋅
T
−
T
⋅
l
⊤
{\displaystyle {\begin{array}{rclcl}{\stackrel {\Delta }{\mathbf {T} }}&=&{\dot {\mathbf {T} }}+\mathbf {T\cdot l} +\mathbf {l} ^{\top }\cdot \mathbf {T} &=&{\dot {T}}_{ij}{\vec {b}}^{i}\otimes {\vec {b}}^{j}\\{\stackrel {\nabla }{\mathbf {T} }}&=&{\dot {\mathbf {T} }}-\mathbf {l\cdot T} -\mathbf {T\cdot l} ^{\top }&=&{\dot {T}}^{ij}{\vec {b}}_{i}\otimes {\vec {b}}_{j}\\{\stackrel {\circ }{\mathbf {T} }}&=&{\dot {\mathbf {T} }}+\mathbf {T\cdot w} -\mathbf {w\cdot T} \\{\stackrel {\diamond }{\mathbf {T} }}&=&{\dot {\mathbf {T} }}+\mathrm {Sp} (\mathbf {l} )\mathbf {T} -\mathbf {l\cdot T} -\mathbf {T\cdot l} ^{\top }\end{array}}}
Materielle Zeitableitung
Bearbeiten
f
˙
(
x
→
,
t
)
=
D
f
D
t
=
∂
f
∂
t
+
g
r
a
d
(
f
)
⋅
v
→
=
∂
f
∂
t
+
(
v
→
⋅
∇
)
f
{\displaystyle {\dot {f}}({\vec {x}},t)={\frac {\mathrm {D} f}{\mathrm {D} t}}={\frac {\partial f}{\partial t}}+\mathrm {grad} (f)\cdot {\vec {v}}={\frac {\partial f}{\partial t}}+({\vec {v}}\cdot \nabla )f}
f
→
˙
(
x
→
,
t
)
=
D
f
→
D
t
=
∂
f
→
∂
t
+
g
r
a
d
(
f
→
)
⋅
v
→
=
∂
f
→
∂
t
+
(
v
→
⋅
∇
)
f
→
{\displaystyle {\dot {\vec {f}}}({\vec {x}},t)={\frac {\mathrm {D} {\vec {f}}}{\mathrm {D} t}}={\frac {\partial {\vec {f}}}{\partial t}}+\mathrm {grad} ({\vec {f}})\cdot {\vec {v}}={\frac {\partial {\vec {f}}}{\partial t}}+({\vec {v}}\cdot \nabla ){\vec {f}}}
#Kartesische Koordinaten :
D
f
D
t
:=
∂
f
∂
t
+
v
x
∂
f
∂
x
+
v
y
∂
f
∂
y
+
v
z
∂
f
∂
z
{\displaystyle {\frac {\mathrm {D} f}{\mathrm {D} t}}:={\frac {\partial f}{\partial t}}+v_{x}{\frac {\partial f}{\partial x}}+v_{y}{\frac {\partial f}{\partial y}}+v_{z}{\frac {\partial f}{\partial z}}}
#Zylinderkoordinaten :
D
f
D
t
:=
∂
f
∂
t
+
v
ρ
∂
f
∂
ρ
+
v
φ
ρ
∂
f
∂
φ
+
v
z
∂
f
∂
z
{\displaystyle {\frac {\mathrm {D} f}{\mathrm {D} t}}:={\frac {\partial f}{\partial t}}+v_{\rho }{\frac {\partial f}{\partial \rho }}+{\frac {v_{\varphi }}{\rho }}{\frac {\partial f}{\partial \varphi }}+v_{z}{\frac {\partial f}{\partial z}}}
#Kugelkoordinaten :
D
f
D
t
:=
∂
f
∂
t
+
v
r
∂
f
∂
r
+
v
φ
r
sin
(
ϑ
)
∂
f
∂
φ
+
v
ϑ
r
∂
f
∂
ϑ
{\displaystyle {\frac {\mathrm {D} f}{\mathrm {D} t}}:={\frac {\partial f}{\partial t}}+v_{r}{\frac {\partial f}{\partial r}}+{\frac {v_{\varphi }}{r\sin(\vartheta )}}{\frac {\partial f}{\partial \varphi }}+{\frac {v_{\vartheta }}{r}}{\frac {\partial f}{\partial \vartheta }}}
Materielle Zeitableitungen von Vektoren werden mittels
D
f
→
D
t
=
D
f
i
D
t
e
^
i
{\displaystyle {\tfrac {\mathrm {D} {\vec {f}}}{\mathrm {D} t}}={\tfrac {\mathrm {D} f_{i}}{\mathrm {D} t}}{\hat {e}}_{i}}
daraus zusammengesetzt.
↑ a b c Morton E. Gurtin: „The linear theory of elasticity. “ In: S. Flügge (Hrsg.): Handbuch der Physik. Band VIa/2.: Festkörpermechanik II / C. Truesdell (Bandherausgeber). Springer, Berlin 1972, ISBN 3-540-05535-5 , S. 10 ff.
↑ In der Literatur (z. B. Altenbach 2012) wird auch die transponierte Beziehung benutzt:
grad
~
(
f
→
)
=
∇
⊗
f
→
=
e
^
i
⊗
∂
f
→
∂
x
i
=
f
j
∂
x
i
e
^
i
⊗
e
^
j
=
grad
(
f
→
)
⊤
{\displaystyle {\tilde {\operatorname {grad} }}({\vec {f}})=\nabla \otimes {\vec {f}}={\hat {e}}_{i}\otimes {\frac {\partial {\vec {f}}}{\partial x_{i}}}=f_{j}{\partial x_{i}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=\operatorname {grad} ({\vec {f}})^{\top }}
Dann muss, um die Formeln zu vergleichen,
g
r
a
d
~
(
f
→
)
{\displaystyle {\tilde {\mathrm {grad} }}({\vec {f}})}
und
g
r
a
d
(
f
→
)
⊤
{\displaystyle \mathrm {grad} ({\vec {f}})^{\top }}
vertauscht werden.
↑ Wolfgang Werner: Vektoren und Tensoren als universelle Sprache in Physik und Technik . Tensoralgebra und Tensoranalysis. Band 1 . Springer Vieweg Verlag, Wiesbaden 2019, ISBN 978-3-658-25271-7 , S. 367 , doi :10.1007/978-3-658-25272-4 .
↑ R. Greve (2003), S. 111.
H. Altenbach: Kontinuumsmechanik . Springer, 2012, ISBN 978-3-642-24118-5 .
M. Bestehorn: Hydrodynamik und Strukturbildung . Springer, 2006, ISBN 978-3-540-33796-6 .
Adolf J. Schwab : Begriffswelt der Feldtheorie. Praxisnahe, anschauliche Einführung. Elektromagnetische Felder, Maxwellsche Gleichungen, Gradient, Rotation, Divergenz . 6., unveränderte Auflage. Springer, Berlin u. a. 2002, ISBN 3-540-42018-5 .
Konrad Königsberger : Analysis . überarbeitete Auflage. Band 2. 4 . Springer, Berlin u. a. 2000, ISBN 3-540-43580-8 .
Ralf Greve: Kontinuumsmechanik . Springer, 2003, ISBN 3-540-00760-1 .
C. Truesdell: Festkörpermechanik II . In: S. Flügge (Hrsg.): Handbuch der Physik . Band VIa/2. Springer, 1972, ISBN 3-540-05535-5 .