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Diese Formelsammlung fasst Formeln und Definitionen der Analysis mit Vektor- und Tensorfeldern zweiter Stufe in der Kontinuumsmechanik zusammen.

• Allgemeines

• Siehe auch

• Nomenklatur

• Kronecker-Delta

• Permutationssymbol

• Basisvektoren

• Ableitung von Skalar-, Vektor- oder Tensorfunktionen

• Gâteaux-Differential

• Fréchet-Ableitung

• Ableitung von Potenzen eines Tensors

• Ableitungen nach dem Ort

• Nabla-Operator

• Gradient

• Divergenz

• Rotation

• Laplace-Operator

• Verknüpfungen

• Grassmann-Entwicklung

• Sätze über Gradient, Divergenz und Rotation

• Helmholtz-Theorem

• Satz über rotationsfreie Felder

• Gaußscher Integralsatz

• Klassischer Integralsatz von Stokes

• Reynoldscher Transportsatz

• Transportsatz für Flächenintegrale

• Transportsatz für Kurvenintegrale

• Kontinuumsmechanik

• Kleine Deformationen

• Starrkörperbewegung

• Ableitungen der Invarianten

• Konvektive Koordinaten

• Geschwindigkeitsgradient

• Objektive Zeitableitungen

• Materielle Zeitableitung

• Fußnoten

• Literatur

• 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 ρ , φ φ , z {\displaystyle \rho ,\varphi ,z}

• #Kugelkoordinaten: r , ϑ ϑ , φ 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:

• Formelsammlung Tensoralgebra#Spur: Sp

• Formelsammlung Tensoralgebra#Transposition: T⊤

• Formelsammlung Tensoralgebra#Inverse: T -1

• Transponierte Inverse: T ⊤-1

• Formelsammlung Tensoralgebra#Skalarprodukt von Tensoren :, von Vektoren ·

• Formelsammlung Tensoralgebra#Kreuzprodukt eines Vektors mit einem Tensor × oder von Vektoren untereinander

• Formelsammlung Tensoralgebra#Dyadisches Produkt ⊗

• Äußeres Tensorprodukt ( a → → ⊗ ⊗ g → → ) # # ( b → → ⊗ ⊗ h → → ) := ( a → → × × b → → ) ⊗ ⊗ ( g → → × × h → → ) {\displaystyle ({\vec {a}}\otimes {\vec {g}})\#({\vec {b}}\otimes {\vec {h}}):=({\vec {a}}\times {\vec {b}})\otimes ({\vec {g}}\times {\vec {h}})}

• Vektorinvariante i → → ( a → → ⊗ ⊗ b → → ) = a → → × × b → → {\displaystyle {\vec {\mathrm {i} }}({\vec {a}}\otimes {\vec {b}})={\vec {a}}\times {\vec {b}}}

• 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 ϑ = ∂ ∂ f r ∂ ∂ ϑ 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:

• Verschiebung u → → = u i e ^ ^ i {\displaystyle {\vec {u}}=u_{i}{\hat {e}}_{i}}

• Geschwindigkeit v → → = v i e ^ ^ i {\displaystyle {\vec {v}}=v_{i}{\hat {e}}_{i}}

• Deformationsgradient F {\displaystyle \mathbf {F} }

• Räumlicher Geschwindigkeitsgradient l {\displaystyle \mathbf {l} }

• der Differentialoperator D/Dt und der Überpunkt steht für die substantielle Zeitableitung

Siehe auch

:

Kronecker-Delta

δ δ i j = δ δ i j = δ δ i j = δ 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.}

Siehe auch

:

Permutationssymbol

ϵ 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 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 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}})}

Siehe auch

:

Kartesische Koordinaten

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.

Siehe auch

:

Zylinderkoordinaten

e ^ ^ ρ ρ = ( cos ⁡ ⁡ ( φ φ ) sin ⁡ ⁡ ( φ φ ) 0 ) , e ^ ^ φ φ = ( − − sin ⁡ ⁡ ( φ φ ) cos ⁡ ⁡ ( φ 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 ρ , φ φ = e ^ ^ φ φ , e ^ ^ φ φ , φ φ = − − e ^ ^ ρ ρ e ^ ^ z , φ 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 ω → → = φ φ ˙ ˙ e ^ ^ z → → e ^ ^ ˙ ˙ ρ ρ / φ φ / z = ω ω → → × × e ^ ^ ρ ρ / φ 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}}

Siehe auch

:

Kugelkoordinaten

e ^ ^ r = ( sin ⁡ ⁡ ( ϑ ϑ ) cos ⁡ ⁡ ( φ φ ) sin ⁡ ⁡ ( ϑ ϑ ) sin ⁡ ⁡ ( φ φ ) cos ⁡ ⁡ ( ϑ ϑ ) ) , e ^ ^ ϑ ϑ = ( cos ⁡ ⁡ ( ϑ ϑ ) cos ⁡ ⁡ ( φ φ ) cos ⁡ ⁡ ( ϑ ϑ ) sin ⁡ ⁡ ( φ φ ) − − sin ⁡ ⁡ ( ϑ ϑ ) ) , e ^ ^ φ φ = ( − − sin ⁡ ⁡ ( φ φ ) cos ⁡ ⁡ ( φ 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 ω → → = ( − − ϑ ϑ ˙ ˙ sin ⁡ ⁡ ( φ φ ) ϑ ϑ ˙ ˙ cos ⁡ ⁡ ( φ φ ) φ φ ˙ ˙ ) = φ φ ˙ ˙ cos ⁡ ⁡ ( ϑ ϑ ) e ^ ^ r − − φ φ ˙ ˙ sin ⁡ ⁡ ( ϑ ϑ ) e ^ ^ ϑ ϑ + ϑ ϑ ˙ ˙ e ^ ^ φ φ → → e ^ ^ ˙ ˙ r / ϑ ϑ / φ φ = ω ω → → × × e ^ ^ r / ϑ ϑ / φ 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}}}

Siehe auch

:

Krummlinige Koordinaten

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 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}}

Siehe auch

:

Gâteaux-Differential

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]]}

Siehe auch

:

Fréchet-Ableitung

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}}} .

( 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#Tensoren vierter Stufe.

Allgemein mit n ∈ ℕ, >0, T0 := 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 T ⊤ ⊤ − − n ( T ) [ H ] = − − ∑ ∑ 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 ) ⊤ ⊤ ) ⊤ ⊤ 24 {\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 }}

Siehe auch

:

Nabla-Operator

#Kartesische Koordinaten x → → {\displaystyle {\vec {x}}} : ∇ ∇ = e ^ ^ i ∂ ∂ ∂ ∂ x i {\displaystyle \nabla ={\hat {e}}_{i}{\frac {\partial }{\partial x_{i}}}}

#Zylinderkoordinaten: ∇ ∇ = e → → ρ ρ ∂ ∂ ∂ ∂ ρ ρ + 1 ρ ρ e → → φ φ ∂ ∂ ∂ ∂ φ φ + e → → z ∂ ∂ ∂ ∂ z {\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: ∇ ∇ = e → → r ∂ ∂ ∂ ∂ r + 1 r e → → ϑ ϑ ∂ ∂ ∂ ∂ ϑ ϑ + 1 r sin ⁡ ⁡ ( ϑ ϑ ) e → → φ φ ∂ ∂ ∂ ∂ φ φ {\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 b → → j = ∂ ∂ y j ∂ ∂ x i e ^ ^ i {\displaystyle {\vec {b}}^{j}={\frac {\partial y_{j}}{\partial x_{i}}}{\hat {e}}_{i}} .

Siehe auch

:

Gradient (Mathematik)

Definierende Eigenschaft bei skalar- oder vektorwertiger Funktion f:cite-ref-hbphys-1-0[1]

f ( y → → ) − − f ( x → → ) = grad ⁡ ⁡ ( f ) ⋅ ⋅ ( y → → − − x → → ) + O ( | y → → − − x → → | ) {\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:

grad ⁡ ⁡ ( f ) ⋅ ⋅ h → → = d d s f ( x → → + s h → → ) | s = 0 = lim s → → 0 f ( x → → + s h → → ) − − f ( x → → ) s ∀ ∀ h → → ∈ ∈ V {\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 ⁡ ⁡ ( f → → ) = 0 → → → → ∃ ∃ g : : f → → = grad ⁡ ⁡ ( g ) {\displaystyle \operatorname {rot} ({\vec {f}})={\vec {0}}\quad \rightarrow \quad \exists g\colon {\vec {f}}=\operatorname {grad} (g)} .

Koordinatenfreie Darstellung als Volumenableitung:

• Volumen v {\displaystyle v} mit

• Oberfläche a {\displaystyle a} mit äußerem vektoriellem Oberflächenelement d a → → {\displaystyle \mathrm {d} {\vec {a}}}

grad ⁡ ⁡ ( f ) = lim v → → 0 ( 1 v ∫ ∫ a f d a → → ) {\displaystyle \operatorname {grad} (f)=\lim _{v\to 0}\left({\frac {1}{v}}\int _{a}f\,\mathrm {d} {\vec {a}}\right)}

Skalarfeld f:

grad ⁡ ⁡ ( f ) = ∇ ∇ f =: ∂ ∂ f ∂ ∂ x → → {\displaystyle \operatorname {grad} (f)=\nabla f=:{\frac {\partial f}{\partial {\vec {x}}}}}

Vektorfeld f → → = f i e ^ ^ i {\displaystyle {\vec {f}}=f_{i}{\hat {e}}_{i}} :cite-ref-altenbach-2-0[2]

g r a d ( f → → ) = ( ∇ ∇ ⊗ ⊗ f → → ) ⊤ ⊤ =: ∂ ∂ f → → ∂ ∂ x → → {\displaystyle \mathrm {grad} ({\vec {f}})=(\nabla \otimes {\vec {f}})^{\top }=:{\frac {\partial {\vec {f}}}{\partial {\vec {x}}}}}

g r a d ( x → → ) = 1 {\displaystyle \mathrm {grad} ({\vec {x}})=\mathbf {1} }

Zusammenhang mit den anderen Differentialoperatoren:

g r a d ( f ) = d i v ( f 1 ) = ∇ ∇ ⋅ ⋅ ( f 1 ) {\displaystyle \mathrm {grad} (f)=\mathrm {div} (f\mathbf {1} )=\nabla \cdot (f\mathbf {1} )}

g r a d ( f ) × × c → → = r o t ( f c → → ) {\displaystyle \mathrm {grad} (f)\times {\vec {c}}=\mathrm {rot} (f{\vec {c}})}

#Kartesische Koordinaten:

g r a d ( f ) = f , i e ^ ^ i {\displaystyle \mathrm {grad} (f)=f_{,i}{\hat {e}}_{i}}

g r a d ( f → → ) = f → → , i ⊗ ⊗ e ^ ^ i = e ^ ^ i ⊗ ⊗ g r a d ( f i ) = f i , j e ^ ^ i ⊗ ⊗ e ^ ^ j {\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 a d ( f ) = f , ρ ρ e ^ ^ ρ ρ + f , φ φ ρ ρ e ^ ^ φ φ + f , z e ^ ^ z {\displaystyle \mathrm {grad} (f)=f_{,\rho }{\hat {e}}_{\rho }+{\frac {f_{,\varphi }}{\rho }}{\hat {e}}_{\varphi }+f_{,z}{\hat {e}}_{z}}

g r a d ( f → → ) = e ^ ^ ρ ρ ⊗ ⊗ g r a d ( f ρ ρ ) + e ^ ^ φ φ ⊗ ⊗ g r a d ( f φ φ ) + e ^ ^ z ⊗ ⊗ g r a d ( f z ) + 1 ρ ρ ( f ρ ρ e ^ ^ φ φ − − f φ φ e ^ ^ ρ ρ ) ⊗ ⊗ e ^ ^ φ φ {\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 a d ( f ) = f , r e ^ ^ r + f , ϑ ϑ r e ^ ^ ϑ ϑ + f , φ φ r sin ⁡ ⁡ ( ϑ ϑ ) e ^ ^ φ 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 d ( f → → ) = e ^ ^ r ⊗ ⊗ g r a d ( f r ) + e ^ ^ ϑ ϑ ⊗ ⊗ g r a d ( f ϑ ϑ ) + e ^ ^ φ φ ⊗ ⊗ g r a d ( f φ φ ) + f r r ( 1 − − e ^ ^ r ⊗ ⊗ e ^ ^ r ) − − e ^ ^ r ⊗ ⊗ f ϑ ϑ e ^ ^ ϑ ϑ + f φ φ e ^ ^ φ φ r + f ϑ ϑ e ^ ^ φ φ − − f φ φ e ^ ^ ϑ ϑ r tan ⁡ ⁡ ( ϑ ϑ ) ⊗ ⊗ e ^ ^ φ φ {\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:

Siehe auch

:

Gradient eines Vektorfeldes

Christoffelsymbole: Γ Γ i j k = g → → i , j ⋅ ⋅ g → → k {\displaystyle \Gamma _{ij}^{k}={\vec {g}}_{i,j}\cdot {\vec {g}}^{k}}

Vektorfelder:

g r a d ( g → → i ) = Γ Γ i j k g → → k ⊗ ⊗ g → → j {\displaystyle \mathrm {grad} ({\vec {g}}_{i})=\Gamma _{ij}^{k}{\vec {g}}_{k}\otimes {\vec {g}}^{j}}

g r a d ( g → → k ) = − − Γ Γ i j k g → → i ⊗ ⊗ g → → j {\displaystyle \mathrm {grad} ({\vec {g}}^{k})=-\Gamma _{ij}^{k}{\vec {g}}^{i}\otimes {\vec {g}}^{j}}

g r a d ( f i g → → i ) = f i | j g → → i ⊗ ⊗ g → → j {\displaystyle \mathrm {grad} (f^{i}{\vec {g}}_{i})=\left.f^{i}\right|_{j}{\vec {g}}_{i}\otimes {\vec {g}}^{j}}

g r a d ( f i g → → i ) = f i | j g → → i ⊗ ⊗ g → → j {\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 i | j = f , j i + Γ Γ k j i f k {\displaystyle \left.f^{i}\right|_{j}=f_{,j}^{i}+\Gamma _{kj}^{i}f^{k}}

f i | j = f i , j − − Γ Γ i j k f k {\displaystyle \left.f_{i}\right|_{j}=f_{i,j}-\Gamma _{ij}^{k}f_{k}}

Tensorfelder:

g r a d ( T ) [ h → → ] = ( h → → ⋅ ⋅ g → → k ) T , k = h → → ⋅ ⋅ ( g → → k ⊗ ⊗ 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 a d ( T ) = T , k ⊗ ⊗ g → → k {\displaystyle \mathrm {grad} (\mathbf {T} )=\mathbf {T} _{,k}\otimes {\vec {g}}^{k}}

Für ein Tensorfeld zweiter Stufe:

g r a d ( T i j g → → i ⊗ ⊗ g → → j ) = T i j | k g → → i ⊗ ⊗ g → → j ⊗ ⊗ g → → k , T i j | k = T i j , k − − Γ Γ i k l T l j − − Γ Γ j k l T i l g r a d ( T i j g → → i ⊗ ⊗ g → → j ) = T i j | k g → → i ⊗ ⊗ g → → j ⊗ ⊗ g → → k , T i j | k = T , k i j + Γ Γ l k i T l j + Γ Γ l k j T i l g r a d ( T i . j g → → i ⊗ ⊗ g → → j ) = T i . j | k g → → i ⊗ ⊗ g → → j ⊗ ⊗ g → → k , T i . j | k = T i , k . j − − Γ Γ i k l T l . j + Γ Γ l k j T i . l g r a d ( T . j i ) g → → i ⊗ ⊗ g → → j = T . j i | k g → → i ⊗ ⊗ g → → j ⊗ ⊗ g → → k , T . j i | k = T . j , k i + Γ Γ l k i T . j l − − Γ Γ j k l T . l i {\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}}}

g r a d ( f g ) = ( f , i g + f g , i ) e ^ ^ i = g r a d ( f ) g + f g r a d ( g ) g r a d ( f g → → ) = ( f , i g → → + f g → → , i ) ⊗ ⊗ e ^ ^ i = g → → ⊗ ⊗ g r a d ( f ) + f g r a d ( g → → ) g r a d ( f → → ⋅ ⋅ g → → ) = ( f → → , i ⋅ ⋅ g → → + f → → ⋅ ⋅ g → → , i ) e ^ ^ i = g → → ⋅ ⋅ g r a d ( f → → ) + f → → ⋅ ⋅ g r a d ( g → → ) g r a d ( f → → × × g → → ) = ( f → → , i × × g → → + f → → × × g → → , i ) ⊗ ⊗ e ^ ^ i = f → → × × g r a d ( g → → ) − − g → → × × g r a d ( f → → ) {\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 speziellcite-ref-3[3]

g r a d ( f → → ⋅ ⋅ g → → ) = g r a d ( f → → ) ⋅ ⋅ g → → + g r a d ( g → → ) ⋅ ⋅ f → → + f → → × × r o t ( g → → ) + g → → × × r o t ( f → → ) {\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:

g r a d ( f i b → → i ) = b → → i ⊗ ⊗ g r a d ( f i ) + f i g r a d ( b → → i ) {\displaystyle \mathrm {grad} (f_{i}{\vec {b}}_{i})={\vec {b}}_{i}\otimes \mathrm {grad} (f_{i})+f_{i}\,\mathrm {grad} ({\vec {b}}_{i})}

Siehe auch

:

Divergenz eines Vektorfeldes

Vektorfeld f → → {\displaystyle {\vec {f}}} :

d i v ( f → → ) = ∇ ∇ ⋅ ⋅ f → → = S p ( g r a d ( f → → ) ) {\displaystyle \mathrm {div} ({\vec {f}})=\nabla \cdot {\vec {f}}=\mathrm {Sp} {\big (}\mathrm {grad} ({\vec {f}}){\big )}}

d i v ( x → → ) = S p ( g r a d ( x → → ) ) = S p ( 1 ) = 3 {\displaystyle \mathrm {div} ({\vec {x}})=\mathrm {Sp} {\big (}\mathrm {grad} ({\vec {x}}){\big )}=\mathrm {Sp} (\mathbf {1} )=3}

Klassische Definition für ein Tensorfeld T:cite-ref-hbphys-1-1[1]

d i v ( T ) ⋅ ⋅ c → → = d i v ( T ⊤ ⊤ ⋅ ⋅ c → → ) ∀ ∀ c → → ∈ ∈ 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 i v ( T ) = ∇ ∇ ⋅ ⋅ ( T ⊤ ⊤ ) {\displaystyle \mathrm {div} (\mathbf {T} )=\nabla \cdot \left(\mathbf {T} ^{\top }\right)}

Koordinatenfreie Darstellung:

• Volumen v {\displaystyle v} mit

• Oberfläche a {\displaystyle a} mit äußerem vektoriellem Oberflächenelement d a → → {\displaystyle \mathrm {d} {\vec {a}}}

d i v ( f → → ) = lim v → → 0 ( 1 v ∫ ∫ a f → → ⋅ ⋅ d a → → ) {\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:

d i v ( f → → ) = ∇ ∇ ⋅ ⋅ f → → = S p ( g r a d ( f → → ) ) d i v ( f 1 ) = ∇ ∇ ⋅ ⋅ ( f 1 ) = g r a d ( f ) {\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}}}

#Kartesische Koordinaten:

d i v ( f → → ) = f → → , i ⋅ ⋅ e ^ ^ i = f i , i {\displaystyle \mathrm {div} ({\vec {f}})={\vec {f}}_{,i}\cdot {\hat {e}}_{i}=f_{i,i}}

d i v ( T ) = T , i ⋅ ⋅ e ^ ^ i = T i j , j e ^ ^ i {\displaystyle \mathrm {div} (\mathbf {T} )=\mathbf {T} _{,i}\cdot {\hat {e}}_{i}=T_{ij,j}{\hat {e}}_{i}}

∇ ∇ ⋅ ⋅ T = e ^ ^ i ⋅ ⋅ T , i = T i j , i e ^ ^ j = T j i , j e ^ ^ i {\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:

d i v ( f → → ) = 1 ρ ρ ∂ ∂ ∂ ∂ ρ ρ ( ρ ρ f ρ ρ ) + 1 ρ ρ f φ φ , φ φ + f z , 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 i v ( T ) = ( T ρ ρ ρ ρ , ρ ρ + 1 ρ ρ ( T ρ ρ φ φ , φ φ + T ρ ρ ρ ρ − − T φ φ φ φ ) + T ρ ρ z , z ) e ^ ^ ρ ρ + ( T φ φ ρ ρ , ρ ρ + 1 ρ ρ ( T φ φ φ φ , φ φ + T ρ ρ φ φ + T φ φ ρ ρ ) + T φ φ z , z ) e ^ ^ φ φ + ( T z ρ ρ , ρ ρ + 1 ρ ρ ( T z φ φ , φ φ + T z ρ ρ ) + T z z , z ) e ^ ^ z {\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 i v ( f → → ) = f r , r + 2 f r + f ϑ ϑ , ϑ ϑ r + f ϑ ϑ cos ⁡ ⁡ ( ϑ ϑ ) + f φ φ , φ φ r sin ⁡ ⁡ ( ϑ ϑ ) d i v ( T ) = ( T r r , r + 2 T r r − − T ϑ ϑ ϑ ϑ − − T φ φ φ φ + T r ϑ ϑ , ϑ ϑ r + T r φ φ , φ φ + T r ϑ ϑ cos ⁡ ⁡ ( ϑ ϑ ) r sin ⁡ ⁡ ( ϑ ϑ ) ) e ^ ^ r ( T ϑ ϑ r , r + 2 T ϑ ϑ r + T r ϑ ϑ + T ϑ ϑ ϑ ϑ , ϑ ϑ r + ( T ϑ ϑ ϑ ϑ − − T φ φ φ φ ) cos ⁡ ⁡ ( ϑ ϑ ) + T ϑ ϑ φ φ , φ φ r sin ⁡ ⁡ ( ϑ ϑ ) ) e ^ ^ ϑ ϑ ( T φ φ r , r + 2 T φ φ r + T r φ φ + T φ φ ϑ ϑ , ϑ ϑ r + ( T ϑ ϑ φ φ + T φ φ ϑ ϑ ) cos ⁡ ⁡ ( ϑ ϑ ) + T φ φ φ φ , φ φ r sin ⁡ ⁡ ( ϑ ϑ ) ) e ^ ^ φ 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.

d i v ( f g → → ) = ∇ ∇ ⋅ ⋅ ( f g → → ) = ( f , i g → → + f g → → , i ) ⋅ ⋅ e ^ ^ i = g r a d ( f ) ⋅ ⋅ g → → + f d i v ( 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 ( f → → × × g → → ) = ∇ ∇ ⋅ ⋅ ( f → → × × g → → ) = ( f → → , i × × g → → + f → → × × g → → , i ) ⋅ ⋅ e ^ ^ i = g → → ⋅ ⋅ r o t ( f → → ) − − f → → ⋅ ⋅ r o t ( 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 ( f → → ⊗ ⊗ g → → ) = ( f → → , i ⊗ ⊗ g → → + f → → ⊗ ⊗ g → → , i ) ⋅ ⋅ e ^ ^ i = g r a d ( f → → ) ⋅ ⋅ g → → + d i v ( g → → ) f → → d i v ( f T ) = ( f , i T + f T , i ) ⋅ ⋅ e ^ ^ i = T ⋅ ⋅ g r a d ( f ) + f d i v ( T ) d i v ( T ⋅ ⋅ f → → ) = ( T , i ⋅ ⋅ f → → + T ⋅ ⋅ f → → , i ) ⋅ ⋅ e ^ ^ i = d i v ( T ⊤ ⊤ ) ⋅ ⋅ f → → + T ⊤ ⊤ : g r a d ( f → → ) d i v ( f → → × × T ) = ( f → → , i × × T + f → → × × T , i ) ⋅ ⋅ e ^ ^ i = i → → ( g r a d ( f → → ) ⋅ ⋅ T ⊤ ⊤ ) + f → → × × d i v ( 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 → → ⊗ ⊗ g → → ) = e ^ ^ i ⋅ ⋅ ( f → → , i ⊗ ⊗ g → → + f → → ⊗ ⊗ g → → , i ) = ( ∇ ∇ ⋅ ⋅ f → → ) g → → + ( ∇ ∇ ⊗ ⊗ g → → ) ⊤ ⊤ ⋅ ⋅ f → → ∇ ∇ ⋅ ⋅ ( f T ) = e ^ ^ i ⋅ ⋅ ( f , i T + f T , i ) = ( ∇ ∇ f ) ⋅ ⋅ T + f ∇ ∇ ⋅ ⋅ T ∇ ∇ ⋅ ⋅ ( T ⋅ ⋅ f → → ) = e ^ ^ i ⋅ ⋅ ( T , i ⋅ ⋅ f → → + T ⋅ ⋅ f → → , i ) = ( ∇ ∇ ⋅ ⋅ T ) ⋅ ⋅ f → → + T : ( ∇ ∇ ⊗ ⊗ f → → ) ∇ ∇ ⋅ ⋅ ( T × × f → → ) = e ^ ^ i ⋅ ⋅ ( T , i × × f → → + T × × f → → , i ) = ( ∇ ∇ ⋅ ⋅ T ) × × f → → − − i → → ( ( ∇ ∇ ⊗ ⊗ f → → ) ⊤ ⊤ ⋅ ⋅ 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 ( f i 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 ( T i j ) ⋅ ⋅ b → → j ) b → → i + T i j ( g r a d ( b → → i ) ⋅ ⋅ b → → j + d i v ( b → → 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 )}}

∇ ∇ ⋅ ⋅ ( T i j b → → i ⊗ ⊗ b → → j ) = ( ( ∇ ∇ T i j ) ⋅ ⋅ b → → i ) b → → j + T i j ( ( ∇ ∇ ⋅ ⋅ 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 v ( 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 ) = g r a d ( f ) ⋅ ⋅ C → → ∇ ∇ ⋅ ⋅ ( f 1 ) = ∇ ∇ 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 ( C ⋅ ⋅ f → → ) = C ⊤ ⊤ : g r a d ( f → → ) → → d i v ( f → → ) = d i v ( 1 ⋅ ⋅ f → → ) = 1 : g r a d ( f → → ) = S p ( g r a d ( f → → ) ) {\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}}}

Siehe auch

:

Rotation eines Vektorfeldes

Vektorfeld f → → {\displaystyle {\vec {f}}} :

r o t ( f → → ) = ∇ ∇ × × f → → {\displaystyle \mathrm {rot} ({\vec {f}})=\nabla \times {\vec {f}}}

Klassische Definition für ein Tensorfeld T:cite-ref-hbphys-1-2[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}}}

Integrabilitätsbedingungcite-ref-4[4]: Jedes divergenzfreie Vektorfeld ist die Rotation eines Vektorfeldes:

d i v ( f → → ) = 0 → → ∃ ∃ g → → : : f → → = r o t ( g → → ) {\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:

• Volumen v {\displaystyle v} mit

• Oberfläche a {\displaystyle a} mit äußerem vektoriellem Oberflächenelement d a → → {\displaystyle \mathrm {d} {\vec {a}}}

r o t ( 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 t ( f c → → ) = g r a d ( f ) × × c → → r o t ( f → → ) = − − i → → ( g r a d ( f → → ) ) = i → → ( ∇ ∇ ⊗ ⊗ f → → ) = ∇ ∇ × × f → → {\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}}}

#Kartesische Koordinaten:

r o t ( f → → ) = e ^ ^ i × × f → → , i = f j , i e ^ ^ i × × e ^ ^ j = ϵ ϵ i j k f j , i e ^ ^ k = ( f 3 , 2 − − f 2 , 3 ) e ^ ^ 1 + ( f 1 , 3 − − f 3 , 1 ) e ^ ^ 2 + ( f 2 , 1 − − f 1 , 2 ) e ^ ^ 3 {\displaystyle {\begin{aligned}\mathrm {rot} ({\vec {f}})=&{\hat {e}}_{i}\times {\vec {f}}_{,i}=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 ( T ) = e ^ ^ i × × T , i ⊤ ⊤ = e ^ ^ i × × T l j , i e ^ ^ j ⊗ ⊗ e ^ ^ l = ϵ ϵ i j k T l j , i e ^ ^ k ⊗ ⊗ e ^ ^ l {\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:

r o t ( f → → ) = f z , φ φ − − ρ ρ f φ φ , z ρ ρ e ^ ^ ρ ρ + ( f ρ ρ , z − − f z , ρ ρ ) e ^ ^ φ φ + f φ φ + ρ ρ f φ φ , ρ ρ − − f ρ ρ , φ φ ρ ρ e ^ ^ z {\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}}

r o t ( T ) = e ^ ^ ρ ρ × × ( T , ρ ρ ⊤ ⊤ ) + 1 ρ ρ e ^ ^ φ φ × × ( T , φ φ ⊤ ⊤ ) + e ^ ^ z × × ( T , z ⊤ ⊤ ) {\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 })}

∇ ∇ × × T = e ^ ^ ρ ρ × × T , ρ ρ + 1 ρ ρ e ^ ^ φ φ × × T , φ φ + e ^ ^ z × × T , z {\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:

r o t ( f → → ) = f φ φ , ϑ ϑ sin ⁡ ⁡ ( ϑ ϑ ) + f φ φ cos ⁡ ⁡ ( ϑ ϑ ) − − f ϑ ϑ , φ φ r sin ⁡ ⁡ ( ϑ ϑ ) e ^ ^ r + ( f r , φ φ r sin ⁡ ⁡ ( ϑ ϑ ) − − f φ φ + r f φ φ , r r ) e ^ ^ ϑ ϑ + f ϑ ϑ + r f ϑ ϑ , r − − f r , ϑ ϑ r e ^ ^ φ φ {\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}}}

r o t ( T ) = e ^ ^ r × × ( T , r ⊤ ⊤ ) + 1 r e ^ ^ ϑ ϑ × × ( T , ϑ ϑ ⊤ ⊤ ) + 1 r sin ⁡ ⁡ ( ϑ ϑ ) e ^ ^ φ φ × × ( T , φ φ ⊤ ⊤ ) {\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 })}

∇ ∇ × × T = e ^ ^ r × × T , r + 1 r e ^ ^ ϑ ϑ × × T , ϑ ϑ + 1 r sin ⁡ ⁡ ( ϑ ϑ ) e ^ ^ φ φ × × T , φ φ {\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 }}

r o t ( f g → → ) = e ^ ^ i × × ( f , i g → → + f g → → , i ) = g r a d ( f ) × × g → → + f r o t ( g → → ) r o t ( f → → × × g → → ) = e ^ ^ i × × ( f → → , i × × g → → + f → → × × g → → , i ) = ( e ^ ^ i ⋅ ⋅ g → → ) f → → , i − − ( e ^ ^ i ⋅ ⋅ f → → , i ) g → → + ( e ^ ^ i ⋅ ⋅ g → → , i ) f → → − − ( e ^ ^ i ⋅ ⋅ f → → ) g → → , i = g r a d ( f → → ) ⋅ ⋅ g → → − − d i v ( f → → ) g → → + d i v ( g → → ) f → → − − g r a d ( g → → ) ⋅ ⋅ f → → = d i v ( f → → ⊗ ⊗ g → → ) − − d i v ( g → → ⊗ ⊗ f → → ) = ∇ ∇ ⋅ ⋅ ( g → → ⊗ ⊗ f → → ) − − ∇ ∇ ⋅ ⋅ ( f → → ⊗ ⊗ g → → ) {\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}}}

r o t ( f → → ⊗ ⊗ g → → ) = e ^ ^ i × × ( g → → , i ⊗ ⊗ f → → + g → → ⊗ ⊗ f → → , i ) = r o t ( g → → ) ⊗ ⊗ f → → − − g → → × × g r a d ( f → → ) ⊤ ⊤ r o t ( f T ) = e ^ ^ k × × ( f , k T ⊤ ⊤ + f T , k ⊤ ⊤ ) = g r a d ( f ) × × ( T ⊤ ⊤ ) + f r o t ( T ) {\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}}}

r o t ( T ⋅ ⋅ f → → ) = e ^ ^ k × × ( T , k ⋅ ⋅ f → → + T ⋅ ⋅ f → → , k ) = r o t ( T ⊤ ⊤ ) ⋅ ⋅ f → → + i → → ( e ^ ^ k ⊗ ⊗ T ⋅ ⋅ f → → , k ) = r o t ( T ⊤ ⊤ ) ⋅ ⋅ f → → − − i → → ( T ⋅ ⋅ g r a d ( f → → ) ) r o t ( f → → × × T ) = − − r o t ( ( T ⊤ ⊤ × × f → → ) ⊤ ⊤ ) = − − ∇ ∇ × × ( T ⊤ ⊤ × × f → → ) = − − ( ∇ ∇ × × T ⊤ ⊤ ) × × f → → + T ⊤ ⊤ # # ( ∇ ∇ ⊗ ⊗ f → → ) = − − r o t ( T ) × × f → → + ( T # # g r a d ( f → → ) ) ⊤ ⊤ {\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}}}

∇ ∇ × × ( f → → ⊗ ⊗ g → → ) = e ^ ^ i × × ( f → → , i ⊗ ⊗ g → → + f → → ⊗ ⊗ g → → , i ) = ( ∇ ∇ × × f → → ) ⊗ ⊗ g → → − − f → → × × ( ∇ ∇ ⊗ ⊗ ( g → → ) ∇ ∇ × × ( f T ) = e ^ ^ k × × ( f , k T + f T , k ) = ( ∇ ∇ f ) × × T + f ∇ ∇ × × T {\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}}}

∇ ∇ × × ( T ⋅ ⋅ f → → ) = e ^ ^ k × × ( T , k ⋅ ⋅ f → → + T ⋅ ⋅ f → → , k ) = ( ∇ ∇ × × T ) ⋅ ⋅ f → → + i → → ( e ^ ^ k ⊗ ⊗ T ⋅ ⋅ f → → , k ) = ( ∇ ∇ × × T ) ⋅ ⋅ f → → − − i → → ( T ⋅ ⋅ ( ∇ ∇ ⊗ ⊗ f → → ) ⊤ ⊤ ) ∇ ∇ × × ( T × × f → → ) = e ^ ^ k × × ( T , k × × f → → + ( T ⋅ ⋅ e ^ ^ i ) ⊗ ⊗ e ^ ^ i × × f → → , k ) = ( ∇ ∇ × × T ) × × f → → − − ( T ⋅ ⋅ e ^ ^ i ) × × e ^ ^ k ⊗ ⊗ e ^ ^ i × × f → → , k = ( ∇ ∇ × × T ) × × f → → − − T # # ( ∇ ∇ ⊗ ⊗ f → → ) {\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:

r o t ( f i b → → i ) = g r a d ( f i ) × × b → → i + f i r o t ( b → → i ) {\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:

r o t ( C ⋅ ⋅ f → → ) = − − i → → ( C ⋅ ⋅ g r a d ( f → → ) ) → → r o t ( f → → ) = r o t ( 1 ⋅ ⋅ f → → ) = − − i → → ( g r a d ( f → → ) ) r o t ( f → → × × 1 ) = 1 # # g r a d ( f → → ) ⊤ ⊤ = g r a d ( f → → ) − − d i v ( f → → ) 1 {\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: r o t ( f → → × × 1 ) = g r a d ( f → → ) {\displaystyle \mathrm {rot} ({\vec {f}}\times \mathbf {1} )=\mathrm {grad} ({\vec {f}})}

Siehe auch

:

Laplace-Operator

Δ Δ := ∇ ∇ ⋅ ⋅ ∇ ∇ = ∇ ∇ 2 {\displaystyle \Delta :=\nabla \cdot \nabla =\nabla ^{2}}

Zusammenhang mit anderen Differentialoperatoren:

Δ Δ f = d i v ( g r a d ( f ) ) = ∇ ∇ ⋅ ⋅ ( ∇ ∇ f ) Δ Δ f → → = d i v ( g r a d ( f → → ) ) = ∇ ∇ ⋅ ⋅ ( ∇ ∇ ⊗ ⊗ f → → ) {\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“:

Δ Δ f → → = g r a d ( d i v ( f → → ) ) − − r o t ( r o t ( f → → ) ) {\displaystyle \Delta {\vec {f}}=\mathrm {grad{\big (}div} ({\vec {f}}){\big )}-\mathrm {rot{\big (}rot} ({\vec {f}}){\big )}}

#Kartesische Koordinaten:

Δ Δ f = f , k k Δ Δ f → → = Δ Δ f i e ^ ^ i = f i , k k e ^ ^ i Δ Δ T = Δ Δ T i j e ^ ^ i ⊗ ⊗ e ^ ^ j = T i j , k k e ^ ^ i ⊗ ⊗ e ^ ^ j {\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:

Δ Δ f = f , ρ ρ ρ ρ + f , ρ ρ ρ ρ + f , φ φ φ φ ρ ρ 2 + f , z z Δ Δ f → → = ( Δ Δ f ρ ρ − − 2 f φ φ , φ φ + f ρ ρ ρ ρ 2 ) e ^ ^ ρ ρ + ( Δ Δ f φ φ + 2 f ρ ρ , φ φ − − f φ φ ρ ρ 2 ) e ^ ^ φ φ + Δ Δ f z e ^ ^ z {\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:

Δ Δ f = 1 r 2 ∂ ∂ ∂ ∂ r ( r 2 ∂ ∂ f ∂ ∂ r ) + 1 r 2 sin ⁡ ⁡ ( ϑ ϑ ) ∂ ∂ ∂ ∂ ϑ ϑ ( sin ⁡ ⁡ ( ϑ ϑ ) ∂ ∂ f ∂ ∂ ϑ ϑ ) + 1 r 2 sin 2 ⁡ ⁡ ( ϑ ϑ ) ∂ ∂ 2 f ∂ ∂ φ φ 2 = 2 f , r r + f , r r + f , ϑ ϑ cos ⁡ ⁡ ( ϑ ϑ ) + f , ϑ ϑ ϑ ϑ sin ⁡ ⁡ ( ϑ ϑ ) r 2 sin ⁡ ⁡ ( ϑ ϑ ) + f , φ φ φ φ r 2 sin 2 ⁡ ⁡ ( ϑ ϑ ) Δ Δ f → → = ( Δ Δ f r − − 2 r 2 ( f r + f ϑ ϑ , ϑ ϑ ) − − 2 f φ φ , φ φ + f ϑ ϑ cos ⁡ ⁡ ( ϑ ϑ ) r 2 sin ⁡ ⁡ ( ϑ ϑ ) ) e ^ ^ r + ( Δ Δ f ϑ ϑ + 2 f r , ϑ ϑ r 2 − − f ϑ ϑ + 2 f φ φ , φ φ cos ⁡ ⁡ ( ϑ ϑ ) r 2 sin 2 ⁡ ⁡ ( ϑ ϑ ) ) e ^ ^ ϑ ϑ + ( Δ Δ f φ φ − − f φ φ − − 2 f ϑ ϑ , φ φ cos ⁡ ⁡ ( ϑ ϑ ) − − 2 f r , φ φ sin ⁡ ⁡ ( ϑ ϑ ) r 2 sin 2 ⁡ ⁡ ( ϑ ϑ ) ) e ^ ^ φ φ {\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.

d i v ( r o t ( f → → ) ) = ∇ ∇ ⋅ ⋅ ( ∇ ∇ × × f → → ) = 0 r o t ( g r a d ( f ) ) = ∇ ∇ × × ∇ ∇ f = 0 → → d i v ( g r a d ( f ) × × g r a d ( g ) ) = ∇ ∇ ⋅ ⋅ ( ∇ ∇ f × × ∇ ∇ g ) = ∇ ∇ g ⋅ ⋅ ( ∇ ∇ × × ∇ ∇ f ) = 0 r o t ( g r a d ( f → → ) ) = ∇ ∇ × × ( ∇ ∇ ⊗ ⊗ f → → ) = 0 d i v ( r o t ( T ) ⊤ ⊤ ) = ∇ ∇ ⋅ ⋅ ( ∇ ∇ × × T ) = 0 → → {\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}}}

d i v ( g r a d ( f ) ) = ∇ ∇ ⋅ ⋅ ( ∇ ∇ f ) = ( ∇ ∇ ⋅ ⋅ ∇ ∇ ) f = Δ Δ f d i v ( g r a d ( f → → ) ) = ∇ ∇ ⋅ ⋅ ( ∇ ∇ ⊗ ⊗ f → → ) = ( ∇ ∇ ⋅ ⋅ ∇ ∇ ) f → → = Δ Δ f → → {\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}}}

d i v ( g r a d ( f → → ) ⊤ ⊤ ) = ∇ ∇ ⋅ ⋅ ( ∇ ∇ ⊗ ⊗ f → → ⊤ ⊤ ) = f i , i j e ^ ^ j = g r a d ( d i v ( f → → ) ) r o t ( g r a d ( f → → ) ⊤ ⊤ ) = ∇ ∇ × × ( ( ∇ ∇ ⊗ ⊗ f → → ) ⊤ ⊤ ) = ∇ ∇ × × ( f → → , i ⊗ ⊗ e ^ ^ i ) = g r a d ( r o t ( f → → ) ) {\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}}}

r o t ( r o t ( f → → ) ) = ∇ ∇ × × ( ∇ ∇ × × f → → ) = ∇ ∇ ( ∇ ∇ ⋅ ⋅ f → → ) − − Δ Δ f → → = g r a d ( d i v ( f → → ) ) − − Δ Δ f → → r o t ( r o t ( T ) ⊤ ⊤ ) ⊤ ⊤ = ( ∇ ∇ × × ( ∇ ∇ × × ( T ⊤ ⊤ ) ) ) ⊤ ⊤ = ( ∇ ∇ ⊗ ⊗ ∇ ∇ ⋅ ⋅ T ⊤ ⊤ ) ⊤ ⊤ − − ( ∇ ∇ ⋅ ⋅ ∇ ∇ ) T = g r a d ( d i v ( T ) ) − − Δ Δ T {\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}}}

r o t ( r o t ( T ⊤ ⊤ ) ) = − − Δ Δ T − − g r a d ( g r a d ( S p ( T ) ) ) + g r a d ( d i v ( T ) ) + g r a d ( d i v ( T ⊤ ⊤ ) ) ⊤ ⊤ + [ Δ Δ S p ( T ) − − d i v ( d i v ( T ) ) ] 1 {\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:

r o t ( r o t ( T ) ) = − − Δ Δ T − − g r a d ( g r a d ( S p ( T ) ) ) + g r a d ( d i v ( T ) ) + g r a d ( d i v ( T ) ) ⊤ ⊤ + [ Δ Δ S p ( T ) − − d i v ( d i v ( T ) ) ] 1 {\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 T = T ⊤ ⊤ = G − − S p ( G ) 1 {\displaystyle \mathbf {T} =\mathbf {T} ^{\top }=\mathbf {G} -\mathrm {Sp} (\mathbf {G} )\mathbf {1} } dann ist:

r o t ( r o t ( T ) ) = − − Δ Δ G + g r a d ( d i v ( G ) ) + g r a d ( d i v ( G ) ) ⊤ ⊤ − − d i v ( d i v ( G ) ) 1 {\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.:

Δ Δ r o t ( r o t ( f → → ) ) = r o t ( Δ Δ r o t ( f → → ) ) = r o t ( r o t ( Δ Δ f → → ) ) = … … … … = Δ Δ g r a d ( d i v ( f → → ) ) − − Δ Δ Δ Δ f → → = g r a d ( Δ Δ d i v ( f → → ) ) − − Δ Δ Δ Δ f → → = g r a d ( d i v ( Δ Δ f → → ) ) − − Δ Δ Δ Δ f → → {\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}}}

f → → × × r o t ( f → → ) = 1 2 g r a d ( f → → ⋅ ⋅ f → → ) − − g r a d ( f → → ) ⋅ ⋅ f → → = ( g r a d ( f → → ) ⊤ ⊤ − − g r a d ( f → → ) ) ⋅ ⋅ f → → = i → → ( g r a d ( f → → ) ) × × f → → {\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}}}

g r a d ( f → → ) ⋅ ⋅ f → → = 1 2 g r a d ( f → → ⋅ ⋅ f → → ) − − f → → × × r o t ( f → → ) {\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}})}

Ein Vektorfeld, dessen Divergenz und Rotation verschwindet, ist harmonisch:

div ⁡ ⁡ ( f → → ) = 0 und r o t ( f → → ) = 0 → → → → Δ Δ f → → = 0 → → {\displaystyle \operatorname {div} ({\vec {f}})=0\;{\text{und}}\;\mathrm {rot} ({\vec {f}})={\vec {0}}\quad \rightarrow \quad \Delta {\vec {f}}={\vec {0}}}

Siehe auch

:

Helmholtz-Theorem

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.

f → → = f → → 1 + f → → 2 : d i v ( f → → 1 ) = 0 und rot ⁡ ⁡ ( f → → 2 ) = 0 → → ↔ ↔ ∃ ∃ g , g → → : f → → = rot ⁡ ⁡ ( g → → ) + g r a d ( g ) {\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}}}

Siehe auch

:

Poincaré-Lemma

I : r o t ( u → → ) := e ^ ^ k × × u → → , k = 0 → → → → ∃ ∃ f : : u → → = g r a d ( f ) II : r o t ( T ) = 0 → → ∃ ∃ u → → : : T = g r a d ( u → → ) III : r o t ( T ) = 0 und S p ( T ) = 0 → → ∃ ∃ W : : T = r o t ( W ) und W = − − W ⊤ ⊤ {\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}}}

oder

II : ∇ ∇ × × ( T ⊤ ⊤ ) = 0 → → ∃ ∃ u → → : : T = g r a d ( u → → ) III : ∇ ∇ × × ( T ⊤ ⊤ ) = 0 und S p ( T ) = 0 → → ∃ ∃ W : : T = r o t ( W ) und W = − − W ⊤ ⊤ {\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}}}

Siehe auch

:

Gaußscher Integralsatz

• Volumen v {\displaystyle v} mit Volumenform d v {\displaystyle \mathrm {d} v} und

• Oberfläche a {\displaystyle a} mit äußerem vektoriellem Oberflächenelement d a → → {\displaystyle \mathrm {d} {\vec {a}}}

• Ortsvektoren x → → ∈ ∈ v {\displaystyle {\vec {x}}\in v}

• Skalar-, vektor- oder tensorwertige Funktion f , f → → , T {\displaystyle f,{\vec {f}},\mathbf {T} } des Ortes x → → {\displaystyle {\vec {x}}} :

∫ ∫ v g r a d ( f ) d v = ∫ ∫ a f d a → → ∫ ∫ v g r a d ( f → → ) d v = ∫ ∫ a f → → ⊗ ⊗ d a → → ∫ ∫ v d i v ( f → → ) d v = ∫ ∫ a f → → ⋅ ⋅ d a → → ∫ ∫ v r o t ( f → → ) d v = − − ∫ ∫ a f → → × × d a → → ∫ ∫ v d i v ( T ) d v = ∫ ∫ a T ⋅ ⋅ d a → → ∫ ∫ v ∇ ∇ ⋅ ⋅ T d v = ∫ ∫ a T ⊤ ⊤ ⋅ ⋅ d a → → {\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:

g r a d ( f g ) = g r a d ( f ) g + f g r a d ( g ) → → ∫ ∫ v g r a d ( f ) g d v = ∫ ∫ a f g d a → → − − ∫ ∫ v f g r a d ( g ) d v {\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}}}

Siehe auch

:

Satz von Stokes

Gegeben:

• Fläche a {\displaystyle a} mit äußerem vektoriellem Oberflächenelement d a → → {\displaystyle \mathrm {d} {\vec {a}}}

• Berandungskurve b {\displaystyle b} der Fläche a {\displaystyle a} mit Linienelement d b → → {\displaystyle \mathrm {d} {\vec {b}}}

• Ortsvektoren x → → ∈ ∈ a {\displaystyle {\vec {x}}\in a}

Vektorwertige Funktion f → → ( x → → , t ) {\displaystyle {\vec {f}}({\vec {x}},t)} :

∫ ∫ a r o t ( f → → ) ⋅ ⋅ d a → → = ∮ ∮ b f → → ⋅ ⋅ d b → → {\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:

r o t ( f g → → ) = g r a d ( f ) × × g → → + f r o t ( g → → ) → → ∫ ∫ a ( g r a d ( f ) × × g → → ) ⋅ ⋅ d a → → = ∮ ∮ b f g → → ⋅ ⋅ d b → → − − ∫ ∫ a f r o t ( g → → ) ⋅ ⋅ d a → → {\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}}}

Siehe auch

:

Reynolds’scher Transportsatz

Gegeben:

• Zeit t {\displaystyle t}

• Zeitabhängiges Volumen v {\displaystyle v} mit Volumenform d v {\displaystyle \mathrm {d} v} mit

• Oberfläche des Volumes a {\displaystyle a} und äußerem vektoriellem Oberflächenelement d a → → {\displaystyle \mathrm {d} {\vec {a}}}

• Ortsvektoren x → → ∈ ∈ v {\displaystyle {\vec {x}}\in v}

• Geschwindigkeitsfeld: v → → ( x → → , t ) {\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}}}

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}}}

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}}}

Ingenieursdehnungen:

ε ε = 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 ) ) = ∇ ∇ × × ( ∇ ∇ × × ε ε ) ⊤ ⊤ = 0 ↓ ↓ 2 ε ε 12 , 12 − − ε ε 22 , 11 − − ε ε 11 , 22 = 0 2 ε ε 13 , 13 − − ε ε 33 , 11 − − ε ε 11 , 33 = 0 2 ε ε 23 , 23 − − ε ε 33 , 22 − − ε ε 22 , 33 = 0 ε ε 11 , 23 + ε ε 23 , 11 − − ε ε 12 , 13 − − ε ε 13 , 12 = 0 ε ε 22 , 13 + ε ε 13 , 22 − − ε ε 12 , 23 − − ε ε 23 , 12 = 0 ε ε 12 , 33 + ε ε 33 , 12 − − ε ε 13 , 23 − − ε 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 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 ω → → {\displaystyle {\vec {\omega }}} des schiefsymmetrischen Tensors Ω Ω {\displaystyle \mathbf {\Omega } } ist die Winkelgeschwindigkeit:

Ω Ω ⋅ ⋅ r → → = ω 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 Ω ⋅ ⋅ ( x → → − − f → → ) = f → → ˙ ˙ + ω 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}})}

∂ ∂ 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 λ v → → → → d e t ( T − − λ λ 1 ) = − − λ λ 3 + I 1 λ λ 2 − − I 2 λ 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 λ d T = ( λ λ 2 − − λ λ I 1 + I 2 ) 1 + ( λ λ − − I 1 ) T ⊤ ⊤ + T ⊤ ⊤ ⋅ ⋅ T ⊤ ⊤ 3 λ λ 2 − − 2 I 1 λ 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 λ v → → → → ∂ ∂ λ 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 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 δ i j ) {\displaystyle ({\vec {v}}_{i}\cdot {\vec {v}}^{j}=\delta _{i}^{j})} :

∂ ∂ λ 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 λ 1 = c , λ λ 2 = a + i b , λ 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 λ 1 ∂ ∂ T = v → → 1 ⊗ ⊗ v → → 1 , ∂ ∂ λ 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.

Siehe auch

:

Konvektive Koordinaten

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 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}}

Siehe auch

:

Geschwindigkeitsgradient

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 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}})}

Siehe auch

:

Euklidische Transformation

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 })}

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 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}}}

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 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}}}

Siehe auch

:

Substantielle Ableitung

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 ρ ∂ ∂ f ∂ ∂ ρ ρ + v φ φ ρ ρ ∂ ∂ f ∂ ∂ φ 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 φ r sin ⁡ ⁡ ( ϑ ϑ ) ∂ ∂ f ∂ ∂ φ φ + v ϑ ϑ r ∂ ∂ f ∂ ∂ ϑ 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.

cite-note-hbphys-11. ↑ 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.

cite-note-altenbach-22. ↑ 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.

cite-note-33. ↑ 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.

cite-note-44. ↑ 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.

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