Formelsammlung Tensoranalysis

Kartesische Koordinaten

${\displaystyle x_{1},x_{2},x_{3}\in \mathbb {R} }$

mit Basisvektoren

${\displaystyle {\hat {e}}_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},\quad {\hat {e}}_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},\quad {\hat {e}}_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}$

die Standardbasis oder allgemeiner eine beliebige Orthonormalbasis ist.

Zylinderkoordinaten

${\displaystyle {\hat {e}}_{\rho }={\begin{pmatrix}\cos(\varphi )\\\sin(\varphi )\\0\end{pmatrix}},\quad {\hat {e}}_{\varphi }={\begin{pmatrix}-\sin(\varphi )\\\cos(\varphi )\\0\end{pmatrix}},\quad {\hat {e}}_{z}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}$

${\displaystyle {\hat {e}}_{\rho ,\varphi }={\hat {e}}_{\varphi },\quad {\hat {e}}_{\varphi ,\varphi }=-{\hat {e}}_{\rho }\quad {\hat {e}}_{z,\varphi }={\vec {0}}}$

Winkelgeschwindigkeit#Zylinderkoordinaten:

${\displaystyle {\vec {\omega }}={\dot {\varphi }}{\hat {e}}_{z}\;\rightarrow \;{\dot {\hat {e}}}_{\rho /\varphi /z}={\vec {\omega }}\times {\hat {e}}_{\rho /\varphi /z}}$

Kugelkoordinaten

${\displaystyle {\hat {e}}_{r}={\begin{pmatrix}\sin(\vartheta )\cos(\varphi )\\\sin(\vartheta )\sin(\varphi )\\\cos(\vartheta )\end{pmatrix}},\quad {\hat {e}}_{\vartheta }={\begin{pmatrix}\cos(\vartheta )\cos(\varphi )\\\cos(\vartheta )\sin(\varphi )\\-\sin(\vartheta )\end{pmatrix}},\quad {\hat {e}}_{\varphi }={\begin{pmatrix}-\sin(\varphi )\\\cos(\varphi )\\0\end{pmatrix}}}$

Winkelgeschwindigkeit#Kugelkoordinaten:

${\displaystyle {\begin{aligned}&{\vec {\omega }}={\begin{pmatrix}-{\dot {\vartheta }}\sin(\varphi )\\{\dot {\vartheta }}\cos(\varphi )\\{\dot {\varphi }}\end{pmatrix}}={\dot {\varphi }}\cos(\vartheta ){\hat {e}}_{r}-{\dot {\varphi }}\sin(\vartheta ){\hat {e}}_{\vartheta }+{\dot {\vartheta }}{\hat {e}}_{\varphi }\\&\rightarrow \;{\dot {\hat {e}}}_{r/\vartheta /\varphi }={\vec {\omega }}\times {\hat {e}}_{r/\vartheta /\varphi }\end{aligned}}}$

Krummlinige Koordinaten

${\displaystyle y_{1},y_{2},y_{3}\in \mathbb {R} }$

${\displaystyle {\vec {b}}_{i}={\frac {\partial {\vec {x}}}{\partial y_{i}}},\quad {\vec {b}}^{i}=\operatorname {grad} (y_{i})={\frac {\partial y_{i}}{\partial {\vec {x}}}}\quad \rightarrow \quad {\vec {b}}_{i}\cdot {\vec {b}}^{j}=\delta _{i}^{j}}$

Definition des Gradienten/Allgemeines

Definierende Eigenschaft bei skalar-, vektor- oder tensorwertiger Funktion φ ist die lineare Approximation 𝓛 in einem Punkt:^[1.1]

${\displaystyle \varphi ({\vec {y}})-\varphi ({\vec {x}})={\mathcal {L}}({\vec {y}}-{\vec {x}})+{\mathcal {O}}(|{\vec {y}}-{\vec {x}}|)}$ wenn ${\displaystyle {\vec {y}}\to {\vec {x}}}$

Wenn 𝓛 existiert, ist es eindeutig. Bei einer skalaren Funktion f ist grad(f) ein Vektor:

${\displaystyle {\mathcal {L}}({\vec {y}}-{\vec {x}})=\operatorname {grad} (f)\cdot ({\vec {y}}-{\vec {x}})}$

${\displaystyle \operatorname {grad} (f)=\nabla f=:{\frac {\partial f}{\partial {\vec {x}}}}}$

Bei Vektoren oder Tensoren φ gibt es Rechtsgradienten

${\displaystyle {\mathcal {L}}({\vec {y}}-{\vec {x}})=\operatorname {grad} (\varphi )\cdot ({\vec {y}}-{\vec {x}})}$

${\displaystyle \mathrm {grad} ({\vec {f}})=(\nabla \otimes {\vec {f}})^{\top }=:{\frac {\partial {\vec {f}}}{\partial {\vec {x}}}}}$

und Linksgradienten

${\displaystyle {\mathcal {L}}({\vec {y}}-{\vec {x}})=({\vec {y}}-{\vec {x}})\cdot {\widetilde {\operatorname {grad} }}(\varphi )}$

${\displaystyle {\widetilde {\mathrm {grad} }}({\vec {f}})=\nabla \otimes {\vec {f}}}$

siehe Krummlinige Koordinaten#Allgemeine krummlinige Koordinatensysteme. Hier wird die Rechtsversion verwendet.

Berechnung für skalar-, vektor- oder allgemein tensorwertige Funktion φ:

${\displaystyle \operatorname {grad} (\varphi )\cdot {\vec {h}}=\left.{\frac {\mathrm {d} }{\mathrm {d} s}}\varphi ({\vec {x}}+s{\vec {h}})\right|_{s=0}=\lim _{s\to 0}{\frac {\varphi ({\vec {x}}+s{\vec {h}})-\varphi ({\vec {x}})}{s}}\quad \forall \;{\vec {h}}\in \mathbb {V} }$

Integrabilitätsbedingung: Jedes rotationsfreie Vektorfeld ist das Gradientenfeld eines Skalarpotentials:

${\displaystyle \operatorname {rot} ({\vec {f}})={\vec {0}}\quad \rightarrow \quad \exists g\colon {\vec {f}}=\operatorname {grad} (g)}$.

Koordinatenfreie Darstellung als Volumenableitung:

• Volumen ${\displaystyle v}$ mit
• Oberfläche ${\displaystyle a}$ mit äußerem vektoriellem Oberflächenelement ${\displaystyle \mathrm {d} {\vec {a}}}$

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

${\displaystyle \mathrm {grad} ({\vec {x}})=\mathbf {1} }$

Zusammenhang mit den anderen Differentialoperatoren:

${\displaystyle \mathrm {grad} (f)=\mathrm {div} (f\mathbf {1} )=\nabla \cdot (f\mathbf {1} )}$

${\displaystyle \mathrm {grad} (f)\times {\vec {c}}=\mathrm {rot} (f{\vec {c}})}$

${\displaystyle \mathrm {rot} ({\vec {f}})\times {\vec {c}}={\big (}\mathrm {grad} ({\vec {f}})-\mathrm {grad} ({\vec {f}})^{\top }{\big )}\cdot {\vec {c}}}$

Nützliche Formeln für Gradienten

${\displaystyle {\begin{aligned}\mathrm {grad} (r)=&{\frac {\vec {x}}{r}}\\\mathrm {grad} {\big (}f(r){\big )}=&{\frac {\mathrm {d} f(r)}{\mathrm {d} r}}{\frac {\vec {x}}{r}}\\\mathrm {grad} (r^{-1})=&-{\frac {\vec {x}}{r^{3}}}\end{aligned}}}$

mit ${\displaystyle r=|{\vec {x}}|}$^[2.1]

Gradient in verschiedenen Koordinatensystemen

#Kartesische Koordinaten:

${\displaystyle \mathrm {grad} (f)=f_{,i}{\hat {e}}_{i}}$

${\displaystyle \mathrm {grad} ({\vec {f}})={\vec {f}}_{,i}\otimes {\hat {e}}_{i}={\hat {e}}_{i}\otimes \mathrm {grad} (f_{i})=f_{i,j}{\hat {e}}_{i}\otimes {\hat {e}}_{j}}$

#Zylinderkoordinaten:

${\displaystyle \mathrm {grad} (f)=f_{,\rho }{\hat {e}}_{\rho }+{\frac {f_{,\varphi }}{\rho }}{\hat {e}}_{\varphi }+f_{,z}{\hat {e}}_{z}}$

${\displaystyle {\begin{aligned}\mathrm {grad} ({\vec {f}})=&{\hat {e}}_{\rho }\otimes \mathrm {grad} (f_{\rho })+{\hat {e}}_{\varphi }\otimes \mathrm {grad} (f_{\varphi })+{\hat {e}}_{z}\otimes \mathrm {grad} (f_{z})\\&+{\frac {1}{\rho }}(f_{\rho }{\hat {e}}_{\varphi }-f_{\varphi }{\hat {e}}_{\rho })\otimes {\hat {e}}_{\varphi }\end{aligned}}}$

#Kugelkoordinaten:

${\displaystyle \mathrm {grad} (f)=f_{,r}{\hat {e}}_{r}+{\frac {f_{,\vartheta }}{r}}{\hat {e}}_{\vartheta }+{\frac {f_{,\varphi }}{r\sin(\vartheta )}}{\hat {e}}_{\varphi }}$

${\displaystyle {\begin{aligned}\mathrm {grad} ({\vec {f}})=&{\hat {e}}_{r}\otimes \mathrm {grad} (f_{r})+{\hat {e}}_{\vartheta }\otimes \mathrm {grad} (f_{\vartheta })+{\hat {e}}_{\varphi }\otimes \mathrm {grad} (f_{\varphi })\\&+{\frac {f_{r}}{r}}(\mathbf {1} -{\hat {e}}_{r}\otimes {\hat {e}}_{r})-{\hat {e}}_{r}\otimes {\frac {f_{\vartheta }{\hat {e}}_{\vartheta }+f_{\varphi }{\hat {e}}_{\varphi }}{r}}+{\frac {f_{\vartheta }{\hat {e}}_{\varphi }-f_{\varphi }{\hat {e}}_{\vartheta }}{r\tan(\vartheta )}}\otimes {\hat {e}}_{\varphi }\end{aligned}}}$

#Krummlinige Koordinaten:

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

Vektorfelder:

${\displaystyle \mathrm {grad} ({\vec {g}}_{i})=\Gamma _{ij}^{k}{\vec {g}}_{k}\otimes {\vec {g}}^{j}}$

${\displaystyle \mathrm {grad} ({\vec {g}}^{k})=-\Gamma _{ij}^{k}{\vec {g}}^{i}\otimes {\vec {g}}^{j}}$

${\displaystyle \mathrm {grad} (f^{i}{\vec {g}}_{i})=\left.f^{i}\right|_{j}{\vec {g}}_{i}\otimes {\vec {g}}^{j}}$

${\displaystyle \mathrm {grad} (f_{i}{\vec {g}}^{i})=\left.f_{i}\right|_{j}{\vec {g}}^{i}\otimes {\vec {g}}^{j}}$

Mit den kovarianten Ableitungen

${\displaystyle \left.f^{i}\right|_{j}=f_{,j}^{i}+\Gamma _{kj}^{i}f^{k}}$

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

Tensorfelder:

${\displaystyle \mathrm {grad} (\mathbf {T} )=\mathbf {T} _{,k}\otimes {\vec {g}}^{k}}$

Für ein Tensorfeld zweiter Stufe:

${\displaystyle {\begin{aligned}\mathrm {grad} (T^{ij}{\vec {g}}_{i}\otimes {\vec {g}}_{j})=&\left.T_{ij}\right|_{k}{\vec {g}}^{i}\otimes {\vec {g}}^{j}\otimes {\vec {g}}^{k},\quad \left.T_{ij}\right|_{k}\!\!\!\!\!\!\!\!\!\!\!\!&=T_{ij,k}-\Gamma _{ik}^{l}T_{lj}-\Gamma _{jk}^{l}T_{il}\\\mathrm {grad} (T^{ij}{\vec {g}}_{i}\otimes {\vec {g}}_{j})=&\left.T^{ij}\right|_{k}{\vec {g}}_{i}\otimes {\vec {g}}_{j}\otimes {\vec {g}}^{k},\quad \left.T^{ij}\right|_{k}\!\!\!\!\!\!\!\!\!\!\!\!&=T_{,k}^{ij}+\Gamma _{lk}^{i}T^{lj}+\Gamma _{lk}^{j}T^{il}\\\mathrm {grad} (T_{i}^{.j}{\vec {g}}^{i}\otimes {\vec {g}}_{j})=&\left.T_{i}^{.j}\right|_{k}{\vec {g}}^{i}\otimes {\vec {g}}_{j}\otimes {\vec {g}}^{k},\quad \left.T_{i}^{.j}\right|_{k}\!\!\!\!\!\!\!\!\!\!\!\!&=T_{i,k}^{.j}-\Gamma _{ik}^{l}T_{l}^{.j}+\Gamma _{lk}^{j}T_{i}^{.l}\\\mathrm {grad} (T_{.j}^{i}){\vec {g}}_{i}\otimes {\vec {g}}^{j}=&\left.T_{.j}^{i}\right|_{k}{\vec {g}}_{i}\otimes {\vec {g}}^{j}\otimes {\vec {g}}^{k},\quad \left.T_{.j}^{i}\right|_{k}\!\!\!\!\!\!\!\!\!\!\!\!&=T_{.j,k}^{i}+\Gamma _{lk}^{i}T_{.j}^{l}-\Gamma _{jk}^{l}T_{.l}^{i}\end{aligned}}}$

Produktregel für Gradienten

${\displaystyle {\begin{array}{rclcl}\mathrm {grad} (fg)&=&(f_{,i}g+fg_{,i}){\hat {e}}_{i}&=&\mathrm {grad} (f)g+f\mathrm {grad} (g)\\\mathrm {grad} (f{\vec {g}})&=&(f_{,i}{\vec {g}}+f{\vec {g}}_{,i})\otimes {\hat {e}}_{i}&=&{\vec {g}}\otimes \mathrm {grad} (f)+f\mathrm {grad} ({\vec {g}})\\\mathrm {grad} ({\vec {f}}\cdot {\vec {g}})&=&\left({\vec {f}}_{,i}\cdot {\vec {g}}+{\vec {f}}\cdot {\vec {g}}_{,i}\right){\hat {e}}_{i}&=&{\vec {g}}\cdot \mathrm {grad} ({\vec {f}})+{\vec {f}}\cdot \mathrm {grad} ({\vec {g}})\\&&&=&\mathrm {grad} ({\vec {f}})^{\top }\cdot {\vec {g}}+\mathrm {grad} ({\vec {g}})^{\top }\cdot {\vec {f}}\\\mathrm {grad} ({\vec {f}}\times {\vec {g}})&=&\left({\vec {f}}_{,i}\times {\vec {g}}+{\vec {f}}\times {\vec {g}}_{,i}\right)\otimes {\hat {e}}_{i}&=&{\vec {f}}\times \mathrm {grad} ({\vec {g}})-{\vec {g}}\times \mathrm {grad} ({\vec {f}})\end{array}}}$

In drei Dimensionen ist speziell^[2]

${\displaystyle \mathrm {grad} ({\vec {f}}\cdot {\vec {g}})=\mathrm {grad} ({\vec {f}})\cdot {\vec {g}}+\mathrm {grad} ({\vec {g}})\cdot {\vec {f}}+{\vec {f}}\times \mathrm {rot} ({\vec {g}})+{\vec {g}}\times \mathrm {rot} ({\vec {f}})}$

Beliebige Basis:

${\displaystyle \mathrm {grad} (f_{i}{\vec {b}}_{i})={\vec {b}}_{i}\otimes \mathrm {grad} (f_{i})+f_{i}\,\mathrm {grad} ({\vec {b}}_{i})}$

Definition der Divergenz/Allgemeines

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

${\displaystyle \mathrm {div} ({\vec {f}})=\nabla \cdot {\vec {f}}=\mathrm {Sp} {\big (}\mathrm {grad} ({\vec {f}}){\big )}}$

${\displaystyle \mathrm {div} ({\vec {x}})=\mathrm {Sp} {\big (}\mathrm {grad} ({\vec {x}}){\big )}=\mathrm {Sp} (\mathbf {1} )=3}$

Klassische Definition für ein Tensorfeld T:^[1]

${\displaystyle \mathrm {div} (\mathbf {T} )\cdot {\vec {c}}=\mathrm {div} \left(\mathbf {T} ^{\top }\cdot {\vec {c}}\right)\quad \forall {\vec {c}}\in \mathbb {V} }$

→ ${\displaystyle \mathrm {div} (\mathbf {T} )=\nabla \cdot \left(\mathbf {T} ^{\top }\right)}$

Koordinatenfreie Darstellung:

• Volumen ${\displaystyle v}$ mit
• Oberfläche ${\displaystyle a}$ mit äußerem vektoriellem Oberflächenelement ${\displaystyle \mathrm {d} {\vec {a}}}$

${\displaystyle \mathrm {div} ({\vec {f}})=\lim _{v\to 0}\left({\frac {1}{v}}\int _{a}{\vec {f}}\;\cdot \mathrm {d} {\vec {a}}\right)}$

Zusammenhang mit den anderen Differentialoperatoren:

${\displaystyle {\begin{array}{lcccl}\mathrm {div} ({\vec {f}})&=&\nabla \cdot {\vec {f}}&=&\mathrm {Sp(grad} ({\vec {f}}))\\\mathrm {div} (f\mathbf {1} )&=&\nabla \cdot (f\mathbf {1} )&=&\mathrm {grad} (f)\end{array}}}$

Nützliche Formeln für Divergenzen

${\displaystyle {\begin{aligned}\mathrm {div} {\big (}f(r){\vec {x}}{\big )}=&3f(r)+r{\frac {\mathrm {d} f(r)}{\mathrm {d} r}}\end{aligned}}}$

mit ${\displaystyle r=|{\vec {x}}|}$^[2.1]

Divergenz in verschiedenen Koordinatensystemen

#Kartesische Koordinaten:

${\displaystyle \mathrm {div} ({\vec {f}})={\vec {f}}_{,i}\cdot {\hat {e}}_{i}=f_{i,i}}$

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

${\displaystyle \nabla \cdot \mathbf {T} ={\hat {e}}_{i}\cdot \mathbf {T} _{,i}=T_{ij,i}{\hat {e}}_{j}=T_{ji,j}{\hat {e}}_{i}}$

#Zylinderkoordinaten:

${\displaystyle \mathrm {div} ({\vec {f}})={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}(\rho f_{\rho })+{\frac {1}{\rho }}f_{\varphi ,\varphi }+f_{z,z}}$

${\displaystyle {\begin{aligned}\mathrm {div} (\mathbf {T} )=&\left(T_{\rho \rho ,\rho }+{\frac {1}{\rho }}(T_{\rho \varphi ,\varphi }+T_{\rho \rho }-T_{\varphi \varphi })+T_{\rho z,z}\right){\hat {e}}_{\rho }\\&+\left(T_{\varphi \rho ,\rho }+{\frac {1}{\rho }}(T_{\varphi \varphi ,\varphi }+T_{\rho \varphi }+T_{\varphi \rho })+T_{\varphi z,z}\right){\hat {e}}_{\varphi }\\&+\left(T_{z\rho ,\rho }+{\frac {1}{\rho }}(T_{z\varphi ,\varphi }+T_{z\rho })+T_{zz,z}\right){\hat {e}}_{z}\end{aligned}}}$

${\displaystyle \nabla \cdot \mathbf {T} =\mathrm {div} \left(\mathbf {T} ^{\top }\right)}$ ergibt sich hieraus durch Vertauschen von T_ab durch T_ba.

#Kugelkoordinaten:

${\displaystyle {\begin{aligned}\mathrm {div} ({\vec {f}})=&f_{r,r}+{\frac {2f_{r}+f_{\vartheta ,\vartheta }}{r}}+{\frac {f_{\vartheta }\cos(\vartheta )+f_{\varphi ,\varphi }}{r\sin(\vartheta )}}\\\mathrm {div} (\mathbf {T} )=&\left(T_{rr,r}+{\frac {2T_{rr}-T_{\vartheta \vartheta }-T_{\varphi \varphi }+T_{r\vartheta ,\vartheta }}{r}}+{\frac {T_{r\varphi ,\varphi }+T_{r\vartheta }\cos(\vartheta )}{r\sin(\vartheta )}}\right){\hat {e}}_{r}\\&\left(T_{\vartheta r,r}+{\frac {2T_{\vartheta r}+T_{r\vartheta }+T_{\vartheta \vartheta ,\vartheta }}{r}}+{\frac {(T_{\vartheta \vartheta }-T_{\varphi \varphi })\cos(\vartheta )+T_{\vartheta \varphi ,\varphi }}{r\sin(\vartheta )}}\right){\hat {e}}_{\vartheta }\\&\left(T_{\varphi r,r}+{\frac {2T_{\varphi r}+T_{r\varphi }+T_{\varphi \vartheta ,\vartheta }}{r}}+{\frac {(T_{\vartheta \varphi }+T_{\varphi \vartheta })\cos(\vartheta )+T_{\varphi \varphi ,\varphi }}{r\sin(\vartheta )}}\right){\hat {e}}_{\varphi }\end{aligned}}}$

${\displaystyle \nabla \cdot \mathbf {T} =\mathrm {div} \left(\mathbf {T} ^{\top }\right)}$ ergibt sich hieraus durch Vertauschen von T_ab durch T_ba.

Produktregel für Divergenzen

${\displaystyle \mathrm {div} (f{\vec {g}})=\nabla \cdot (f{\vec {g}})=\left(f_{,i}{\vec {g}}+f{\vec {g}}_{,i}\right)\cdot {\hat {e}}_{i}=\mathrm {grad} (f)\cdot {\vec {g}}+f\mathrm {div} ({\vec {g}})}$

${\displaystyle \mathrm {div} ({\vec {f}}\times {\vec {g}})=\nabla \cdot ({\vec {f}}\times {\vec {g}})=\left({\vec {f}}_{,i}\times {\vec {g}}+{\vec {f}}\times {\vec {g}}_{,i}\right)\cdot {\hat {e}}_{i}={\vec {g}}\cdot \mathrm {rot} ({\vec {f}})-{\vec {f}}\cdot \mathrm {rot} ({\vec {g}})}$