Formelsammlung Tensoranalysis

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

\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}

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

{\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}}

#Krummlinige Koordinaten:

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

{\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}}

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

Vektorfelder:

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

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

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

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

Mit den kovarianten Ableitungen

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

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

Tensorfelder:

\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

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

Für ein Tensorfeld zweiter Stufe:

{\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

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

In drei Dimensionen ist speziell^[3]

\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:

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

Divergenz

Definition der Divergenz/Allgemeines

Vektorfeld ${\vec {f}}$ :

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

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

Klassische Definition für ein Tensorfeld T:^[1]

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

→

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

Koordinatenfreie Darstellung:

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

\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:

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

Divergenz in verschiedenen Koordinatensystemen

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

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

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

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

{\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}}

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

{\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}}

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

Produktregel für Divergenzen

\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}})

\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}})

{\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}}

{\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:

\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})

\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 )}

\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:

\mathrm {div} (f\mathbf {C} )=\mathbf {C} \cdot \mathrm {grad} (f)\quad \rightarrow \quad \mathrm {div} (f\mathbf {1} )=\mathrm {grad} (f)

\nabla \cdot (f\mathbf {C} )=\mathrm {grad} (f)\cdot \mathbf {C} \quad \rightarrow \quad \nabla \cdot (f\mathbf {1} )=\nabla f

{\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}}

Rotation

Definition der Rotation/Allgemeines

Vektorfeld ${\vec {f}}$ :

\mathrm {rot} ({\vec {f}})=\nabla \times {\vec {f}}

Klassische Definition für ein Tensorfeld T:^[1]

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

→

\mathrm {rot} (\mathbf {T} )=\nabla \times \left(\mathbf {T} ^{\top }\right)

Allgemeine Identitäten:

\mathbf {T=T} ^{\top }\quad \rightarrow \quad \mathrm {Sp{\big (}rot} (\mathbf {T} ){\big )}=\mathrm {Sp} (\nabla \times \mathbf {T} )=0

\mathrm {rot} ({\vec {x}})={\vec {0}}

Integrabilitätsbedingung^[4]: Jedes divergenzfreie Vektorfeld ist die Rotation eines Vektorfeldes:

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

\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:

{\begin{aligned}\mathrm {rot} (f{\vec {c}})=&\mathrm {grad} (f)\times {\vec {c}}\\\mathrm {rot} ({\vec {f}})=&-{\vec {\mathrm {i} }}{\big (}\mathrm {grad} ({\vec {f}}){\big )}={\vec {\mathrm {i} }}(\nabla \otimes {\vec {f}})=\nabla \times {\vec {f}}\end{aligned}}

Rotation in verschiedenen Koordinatensystemen

{\begin{aligned}\mathrm {rot} ({\vec {f}})=&f_{j,i}{\hat {e}}_{i}\times {\hat {e}}_{j}=\epsilon _{ijk}f_{j,i}{\hat {e}}_{k}\\=&(f_{3,2}-f_{2,3}){\hat {e}}_{1}+(f_{1,3}-f_{3,1}){\hat {e}}_{2}+(f_{2,1}-f_{1,2}){\hat {e}}_{3}\end{aligned}}

\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}

\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}

\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 })

\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}

{\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}}

\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 })

\nabla \times \mathbf {T} ={\hat {e}}_{r}\times \mathbf {T} _{,r}+{\frac {1}{r}}{\hat {e}}_{\vartheta }\times \mathbf {T} _{,\vartheta }+{\frac {1}{r\sin(\vartheta )}}{\hat {e}}_{\varphi }\times \mathbf {T} _{,\varphi }

Produktregel für Rotationen

{\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}}

{\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}}

${\begin{aligned}\mathrm {rot} (\mathbf {T} \cdot {\vec {f}})=&{\hat {e}}_{k}\times {\big (}\mathbf {T} _{,k}\cdot {\vec {f}}+\mathbf {T} \cdot {\vec {f}}_{,k}{\big )}\\=&\mathrm {rot} (\mathbf {T} ^{\top })\cdot {\vec {f}}+{\vec {\mathrm {i} }}\left({\hat {e}}_{k}\otimes \mathbf {T} \cdot {\vec {f}}_{,k}\right)\\=&\mathrm {rot} (\mathbf {T} ^{\top })\cdot {\vec {f}}-{\vec {\mathrm {i} }}\left(\mathbf {T} \cdot \mathrm {grad} ({\vec {f}})\right)\\\mathrm {rot} ({\vec {f}}\times \mathbf {T} )=&-\mathrm {rot} \left((\mathbf {T} ^{\top }\times {\vec {f}})^{\top }\right)\\=&-\nabla \times \left(\mathbf {T} ^{\top }\times {\vec {f}}\right)\\=&-(\nabla \times \mathbf {T} ^{\top })\times {\vec {f}}+\mathbf {T} ^{\top }\#(\nabla \otimes {\vec {f}})\\=&-\mathrm {rot} (\mathbf {T} )\times {\vec {f}}+\left(\mathbf {T} \#\mathrm {grad} ({\vec {f}})\right)^{\top }\end{aligned}}$

{\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}}

${\begin{aligned}\nabla \times (\mathbf {T} \cdot {\vec {f}})=&{\hat {e}}_{k}\times (\mathbf {T} _{,k}\cdot {\vec {f}}+\mathbf {T} \cdot {\vec {f}}_{,k})\\=&(\nabla \times \mathbf {T} )\cdot {\vec {f}}+{\vec {\mathrm {i} }}\left({\hat {e}}_{k}\otimes \mathbf {T} \cdot {\vec {f}}_{,k}\right)\\=&(\nabla \times \mathbf {T} )\cdot {\vec {f}}-{\vec {\mathrm {i} }}{\big (}\mathbf {T} \cdot (\nabla \otimes {\vec {f}})^{\top }{\big )}\\\nabla \times (\mathbf {T} \times {\vec {f}})=&{\hat {e}}_{k}\times (\mathbf {T} _{,k}\times {\vec {f}}+(\mathbf {T} \cdot {\hat {e}}_{i})\otimes {\hat {e}}_{i}\times {\vec {f}}_{,k})\\=&(\nabla \times \mathbf {T} )\times {\vec {f}}-(\mathbf {T} \cdot {\hat {e}}_{i})\times {\hat {e}}_{k}\otimes {\hat {e}}_{i}\times {\vec {f}}_{,k}\\=&(\nabla \times \mathbf {T} )\times {\vec {f}}-\mathbf {T} \#(\nabla \otimes {\vec {f}})\end{aligned}}$

Beliebige Basis:

\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:

{\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: $\mathrm {rot} ({\vec {f}}\times \mathbf {1} )=\mathrm {grad} ({\vec {f}})$

Laplace-Operator

Definition/Allgemeines

\Delta :=\nabla \cdot \nabla =\nabla ^{2}

Zusammenhang mit anderen Differentialoperatoren:

{\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“:

\Delta {\vec {f}}=\mathrm {grad{\big (}div} ({\vec {f}}){\big )}-\mathrm {rot{\big (}rot} ({\vec {f}}){\big )}

Laplace-Operator in verschiedenen Koordinatensystemen

{\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}}

{\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}}