Tensoranalysis formula collection

${\displaystyle {\sqrt[{n}]{x}}}$

This article is a collection of formulas for the topic of tensor analysis. Math symbols are used, which are explained in the article List of Math Symbols .

This formula collection summarizes formulas and definitions of analysis with vector and tensor fields of the second level in continuum mechanics .

General

See also

Formula collection tensor algebra

nomenclature

• Operators like " " are not written in italics. ${\displaystyle \operatorname {grad} }$
• Letters in the middle of the alphabet are used as indices: ${\displaystyle i,j,k,l\in \{1,2,3\}}$
• The Einstein sums convention applies regardless of the index position.
  • Comes in a formula in a product , an index twice before as summed over this index is from one to three: . ${\displaystyle c=a_{i}b^{i}}$
    ${\displaystyle c=a_{i}b^{i}=\sum _{i=1}^{3}a_{i}b^{i}}$
  • If several indexes twice before in summing over this: . ${\displaystyle c=A_{ij}B_{j}^{i}}$
    ${\displaystyle c=A_{ij}B_{j}^{i}=\sum _{i=1}^{3}\sum _{j=1}^{3}A_{ij}B_{j}^{i}}$
  • An index that appears simply, as in , is a free index. The formula applies to all values of the free indexes: . ${\displaystyle i}$ ${\displaystyle v_{i}=A_{ij}b_{j}}$
    ${\displaystyle v_{i}=A_{ij}b_{j}\quad \leftrightarrow \quad v_{i}=\sum _{j=1}^{3}A_{ij}b_{j}\quad \forall \;i\in \{1,2,3\}}$
• Vectors:
  • All vectors used here are geometric vectors in three-dimensional Euclidean vector space . ${\displaystyle \mathbb {V} }$
  • Vectors are designated with lower case letters.
  • Unit vectors of length one are hatched as in FIG. ${\displaystyle {\hat {e}}}$
  • Vectors of indefinite length are marked with an arrow as in FIG. ${\displaystyle {\vec {a}}}$
  • Standard base ${\displaystyle {\hat {e}}_{1},{\hat {e}}_{2},{\hat {e}}_{3}}$
  • Any base with a dual base ${\displaystyle {\vec {b}}_{1},{\vec {b}}_{2},{\vec {b}}_{3}}$ ${\displaystyle {\vec {b}}^{1},{\vec {b}}^{2},{\vec {b}}^{3}}$
  • The vector is called the position vector throughout. ${\displaystyle {\vec {x}}=x_{i}{\hat {e}}_{i}}$
• Second order tensors are noted in bold capital letters as in FIG. ${\displaystyle \mathbf {T} }$
• Coordinates:
  • Cartesian coordinates ${\displaystyle x_{1},x_{2},x_{3}\in \mathbb {R} }$
  • Cylindrical coordinates : ${\displaystyle \rho ,\varphi ,z}$
  • Spherical coordinates : ${\displaystyle r,\vartheta ,\varphi }$
  • Curvilinear coordinates ${\displaystyle y_{1},y_{2},y_{3}\in \mathbb {R} }$
• Constants: ${\displaystyle c,{\vec {c}},\mathbf {C} }$
• time ${\displaystyle t\in \mathbb {R} }$
• Variables: scalar or vector valued ${\displaystyle r,s\in \mathbb {R} }$ ${\displaystyle {\vec {r}},{\vec {s}}\in \mathbb {V} ^{3}}$
• Functions:
  • Scalar or vector valued ${\displaystyle f,g\in \mathbb {R} }$ ${\displaystyle {\vec {f}},{\vec {g}}\in \mathbb {V} ^{3}}$
  • Tensor valued: or ${\displaystyle \mathbf {T} ({\vec {x}},t)}$ ${\displaystyle \mathbf {T} ({\vec {y}},t)}$
• Operators:
  • Track (math) : ${\displaystyle \operatorname {Sp} }$
  • Transposed matrix : ${\displaystyle \mathbf {T} ^{\top }}$
  • Inverse matrix : ${\displaystyle \mathbf {T} ^{-1}}$
  • Transposed inverse matrix: ${\displaystyle \mathbf {T} ^{\top -1}}$
• Differential operators :
  • Laplace operator : ${\displaystyle \Delta }$
  • Divergence of a vector field : ${\displaystyle \operatorname {div} }$
  • Gradient (math) : ${\displaystyle \operatorname {grad} }$
  • Rotation of a vector field : ${\displaystyle \operatorname {rot} }$
  • An index after a comma denotes the derivation according to a coordinate:
    ${\displaystyle f_{,i}:={\frac {\partial f}{\partial x_{i}}}\,,\quad f_{i,jk}={\frac {\partial ^{2}f_{i}}{\partial x_{j}\partial x_{k}}}}$
• Continuum Mechanics:
  • shift ${\displaystyle {\vec {u}}=u_{i}{\hat {e}}_{i}}$
  • speed ${\displaystyle {\vec {v}}=v_{i}{\hat {e}}_{i}}$
  • Deformation gradient ${\displaystyle \mathbf {F} }$
  • Spatial velocity gradient ${\displaystyle \mathbf {l} }$
  • the differential operator D / Dt and the added point stand for the substantial time derivative

Kronecker Delta

→ Main article : Kronecker Delta

${\displaystyle \delta _{ij}=\delta ^{ij}=\delta _{i}^{j}=\delta _{j}^{i}=\left\{{\begin{array}{ll}1&\mathrm {falls} \ i=j\\0&\mathrm {sonst} \end{array}}\right.}$

Basis vectors

Cartesian coordinates: ${\displaystyle {\hat {e}}_{1},{\hat {e}}_{2},{\hat {e}}_{3}}$

Cylindrical coordinates :

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

Spherical coordinates :

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

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

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

Derivatives according to the location

Gâteaux differential

→ Main article : Gâteaux differential

${\displaystyle \,\mathrm {D} f(x)[h]:=\left.{\frac {\mathrm {d} }{\mathrm {d} s}}f(x+sh)\right|_{s=0}=\lim _{s\rightarrow 0}{\frac {f(x+sh)-f(x)}{s}}}$

with , scalar, vector or tensor valued but and similar. ${\displaystyle s\in \mathbb {R} }$ ${\displaystyle f,x,h}$ ${\displaystyle x}$ ${\displaystyle h}$

Product rule:

${\displaystyle \mathrm {D} (f(x)\cdot g(x))[h]=\mathrm {D} f(x)[h]\cdot g(x)+f(x)\cdot \mathrm {D} g(x)[h]}$

Chain rule:

${\displaystyle \,\mathrm {D} (f{\circ }g)(x)[h]=\,\mathrm {D} f(g)[Dg(x)[h]]}$

Fréchet derivation

→ Main article : Fréchet derivation

Does a bounded linear operator exist such that ${\displaystyle {\mathcal {A}}}$

${\displaystyle {\mathcal {A}}[h]={Df}(x)[h]{\quad \forall \;}h}$

holds, then the Fréchet derivative from to is called. You then also write ${\displaystyle {\mathcal {A}}}$ ${\displaystyle f}$ ${\displaystyle x}$

${\displaystyle {\frac {\partial f}{\partial x}}={\mathcal {A}}}$.

Nabla operator

→ Main article : Nabla operator

Cartesian coordinates  : ${\displaystyle {\vec {x}}}$ ${\displaystyle \nabla ={\hat {e}}_{i}{\frac {\partial }{\partial x_{i}}}}$

Curvilinear coordinates  :     with    . ${\displaystyle {\vec {y}}}$ ${\displaystyle \nabla ={\vec {b}}^{j}{\frac {\partial }{\partial y_{j}}}}$ ${\displaystyle {\vec {b}}^{j}={\frac {\partial y_{j}}{\partial x_{i}}}{\hat {e}}_{i}}$

gradient

→ Main article : Gradient (mathematics)

Scalar field  : ${\displaystyle f}$

${\displaystyle \operatorname {grad} (f)=\nabla f={\frac {\partial f}{\partial x_{i}}}{\hat {e}}_{i}=f_{,i}{\hat {e}}_{i}}$

Vector field : ${\displaystyle {\vec {f}}=f_{i}{\hat {e}}_{i}}$

${\displaystyle \operatorname {grad} ({\vec {f}})=(\nabla \otimes {\vec {f}})^{\top }={\frac {\partial {\vec {f}}}{\partial x_{j}}}\otimes {\hat {e}}_{j}={\frac {\partial f_{i}}{\partial x_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=f_{i,j}{\hat {e}}_{i}\otimes {\hat {e}}_{j}}$

${\displaystyle \operatorname {grad} ({\vec {x}})=\mathbf {I} }$

Scalar or vector valued function f:

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

Cylindrical coordinates :

${\displaystyle \operatorname {grad} (f)={\frac {\partial f}{\partial \rho }}{\hat {e}}_{\rho }+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}{\hat {e}}_{\varphi }+{\frac {\partial f}{\partial z}}{\hat {e}}_{z}}$

${\displaystyle \operatorname {grad} ({\vec {f}})={\hat {e}}_{\rho }\otimes \operatorname {grad} (f_{\rho })+{\frac {f_{\rho }}{\rho }}{\hat {e}}_{\varphi }\otimes {\hat {e}}_{\varphi }+{\hat {e}}_{\varphi }\otimes \operatorname {grad} (f_{\varphi })-{\frac {f_{\varphi }}{\rho }}{\hat {e}}_{\rho }\otimes {\hat {e}}_{\varphi }+{\hat {e}}_{z}\otimes \operatorname {grad} (f_{z})}$

Spherical coordinates :

${\displaystyle \operatorname {grad} (f)={\frac {\partial f}{\partial r}}{\hat {e}}_{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \vartheta }}{\hat {e}}_{\vartheta }+{\frac {1}{r\sin \vartheta }}{\frac {\partial f}{\partial \varphi }}{\hat {e}}_{\varphi }}$

${\displaystyle {\begin{array}{rcl}\operatorname {grad} ({\vec {f}})&=&{\hat {e}}_{r}\otimes \operatorname {grad} (f_{r})+{\frac {f_{r}}{r}}\mathbf {I} -{\frac {f_{r}}{r}}{\hat {e}}_{r}\otimes {\hat {e}}_{r}+{\hat {e}}_{\vartheta }\otimes \operatorname {grad} (f_{\vartheta })+{\frac {f_{\vartheta }}{r\tan \vartheta }}{\hat {e}}_{\varphi }\otimes {\hat {e}}_{\varphi }-{\frac {f_{\vartheta }}{r}}{\hat {e}}_{r}\otimes {\hat {e}}_{\vartheta }\\&&+{\hat {e}}_{\varphi }\otimes \operatorname {grad} (f_{\varphi })-{\frac {f_{\varphi }}{r\tan \vartheta }}{\hat {e}}_{\vartheta }\otimes {\hat {e}}_{\varphi }-{\frac {f_{\varphi }}{r}}{\hat {e}}_{r}\otimes {\hat {e}}_{\varphi }\end{array}}}$

Integrability condition : Every rotation-free vector field is the gradient field of a scalar potential :

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

Coordinate-free representation as a volume derivative:

• Volume with ${\displaystyle v}$
• Surface with an outer vectorial surface element ${\displaystyle a}$ ${\displaystyle \mathrm {d} {\vec {a}}}$

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

Relationship with the other differential operators:

${\displaystyle \operatorname {grad} (f)=\operatorname {div} (f\mathbf {I} )}$

Product rule:

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

Any base:

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

Product with constants:

${\displaystyle \operatorname {grad} (cf)=c\,\operatorname {grad} (f)}$

${\displaystyle \operatorname {grad} (f{\vec {c}})={\vec {c}}\otimes \operatorname {grad} (f)}$

${\displaystyle \operatorname {grad} (c{\vec {f}})=c\,\operatorname {grad} ({\vec {f}})}$

${\displaystyle \operatorname {grad} ({\vec {c}}\cdot {\vec {f}})=\operatorname {grad} {({\vec {f}})}^{\top }\cdot {\vec {c}}}$

${\displaystyle \operatorname {grad} ({\vec {c}}\times {\vec {f}})={\vec {c}}\times \operatorname {grad} ({\vec {f}})=-\operatorname {grad} ({\vec {f}}\times {\vec {c}})}$

${\displaystyle \operatorname {grad} (\mathbf {C} \cdot {\vec {f}})=\mathbf {C} \cdot \operatorname {grad} ({\vec {f}})}$

divergence

→ Main article : Divergence of a vector field

Vector field  : ${\displaystyle {\vec {f}}=f_{i}{\hat {e}}_{i}}$

${\displaystyle \operatorname {div} ({\vec {f}})=\nabla \cdot {\vec {f}}={\frac {\partial {\vec {f}}}{\partial x_{i}}}\cdot {\hat {e}}_{i}={\frac {\partial f_{i}}{\partial x_{i}}}=\operatorname {Sp} (\operatorname {grad} ({\vec {f}}))}$

${\displaystyle \operatorname {div} {\vec {x}}=3}$

Tensor field : ${\displaystyle \mathbf {T} =T_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}}$

${\displaystyle \operatorname {div} (\mathbf {T} )=\nabla \cdot \mathbf {T} ={\hat {e}}_{i}\cdot {\frac {\partial \mathbf {T} }{\partial x_{i}}}={\frac {\partial T_{ij}}{\partial x_{i}}}{\hat {e}}_{j}}$

Cylindrical coordinates :

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

Spherical coordinates :

${\displaystyle {\begin{aligned}\operatorname {div} ({\vec {f}})=&{\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}f_{r})+{\frac {1}{r\sin \vartheta }}{\frac {\partial }{\partial \vartheta }}(f_{\vartheta }\sin \vartheta )+{\frac {1}{r\sin \vartheta }}{\frac {\partial f_{\varphi }}{\partial \varphi }}\\\operatorname {div} (\mathbf {T} )=&\left({\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}T_{rr})+{\frac {1}{r\sin \vartheta }}T_{\varphi r,\varphi }+{\frac {1}{r\sin \vartheta }}{\frac {\partial }{\partial \vartheta }}(T_{\vartheta r}\sin \vartheta )-{\frac {1}{r}}(T_{\vartheta \vartheta }+T_{\varphi \varphi })\right){\hat {e}}_{r}\\&+\left({\frac {1}{r^{3}}}{\frac {\partial }{\partial r}}(r^{3}T_{r\vartheta })+{\frac {1}{r\sin \vartheta }}T_{\varphi \vartheta ,\varphi }+{\frac {1}{r\sin \vartheta }}{\frac {\partial }{\partial \vartheta }}(T_{\vartheta \vartheta }\sin \vartheta )+{\frac {1}{r}}(T_{\vartheta r}-T_{r\vartheta }-T_{\varphi \varphi }\cot \vartheta )\right){\hat {e}}_{\vartheta }\\&+\left({\frac {1}{r^{3}}}{\frac {\partial }{\partial r}}(r^{3}T_{r\varphi })+{\frac {1}{r\sin \vartheta }}T_{\varphi \varphi ,\varphi }+{\frac {1}{r\sin \vartheta }}{\frac {\partial }{\partial \vartheta }}(T_{\vartheta \varphi }\sin \vartheta )+{\frac {1}{r}}(T_{\varphi r}-T_{r\varphi }+T_{\varphi \vartheta }\cot \vartheta )\right){\hat {e}}_{\varphi }\end{aligned}}}$

Coordinate-free representation:

• Volume with ${\displaystyle v}$
• Surface with an outer vectorial surface element ${\displaystyle a}$ ${\displaystyle \mathrm {d} {\vec {a}}}$

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

Relationship with the other differential operators:

${\displaystyle {\begin{array}{rcl}\operatorname {div} ({\vec {f}})&=&\operatorname {Sp(grad} ({\vec {f}}))\\\operatorname {div} (f\mathbf {I} )&=&\operatorname {grad} (f)\\\operatorname {div} (f_{i}{\hat {e}}_{i})&=&\operatorname {grad} (f_{i})\cdot {\hat {e}}_{i}\\\operatorname {div} ({\vec {f}}\times {\hat {e}}_{i})&=&\operatorname {rot} (f)\cdot {\hat {e}}_{i}\end{array}}}$

Product rule:

${\displaystyle {\begin{array}{rclcl}\operatorname {div} (f{\vec {g}})&=&\displaystyle {\hat {e}}_{i}\cdot \left(f_{,i}{\vec {g}}+f{\vec {g}}_{,i}\right)&=&\operatorname {grad} (f)\cdot {\vec {g}}+f\operatorname {div} ({\vec {g}})\\\operatorname {div} (f\mathbf {T} )&=&{\hat {e}}_{i}\cdot (f_{,i}\mathbf {T} +f\mathbf {T} _{,i})&=&\operatorname {grad} (f)\cdot \mathbf {T} +f\operatorname {div} (\mathbf {T} )\\\operatorname {div} (\mathbf {T} \cdot {\vec {f}})&=&\displaystyle {\hat {e}}_{i}\cdot \left(\mathbf {T} _{,i}\cdot {\vec {f}}+\mathbf {T} \cdot {\vec {f}}_{,i}\right)\\&=&\operatorname {div} (\mathbf {T} )\cdot {\vec {f}}+\operatorname {Sp} \left(\mathbf {T} \cdot {\vec {f}}_{,i}\otimes {\hat {e}}_{i}\right)&=&\operatorname {div} (\mathbf {T} )\cdot {\vec {f}}+\mathbf {T} ^{\top }:\operatorname {grad} ({\vec {f}})\\\operatorname {div} ({\vec {f}}\otimes {\vec {g}})&=&\displaystyle {\hat {e}}_{i}\cdot \left({\vec {f}}_{,i}\otimes {\vec {g}}+{\vec {f}}\otimes {\vec {g}}_{,i}\right)&=&\operatorname {div} ({\vec {f}}){\vec {g}}+\operatorname {grad} ({\vec {g}})\cdot {\vec {f}}\\\operatorname {div} ({\vec {f}}\times {\vec {g}})&=&\displaystyle {\hat {e}}_{i}\cdot \left({\vec {f}}_{,i}\times {\vec {g}}+{\vec {f}}\times {\vec {g}}_{,i}\right)&=&{\vec {g}}\cdot \operatorname {rot} ({\vec {f}})-{\vec {f}}\cdot \operatorname {rot} ({\vec {g}})\\\operatorname {div} (\mathbf {T} \times {\vec {f}})&=&\displaystyle {\hat {e}}_{i}\cdot (\mathbf {T} _{,i}\times {\vec {f}}+\mathbf {T} \times {\vec {f}}_{,i})\\&=&({\hat {e}}_{i}\cdot \mathbf {T} _{,i})\times {\vec {f}}+\mathbf {I} \cdot \!\!\times (\mathbf {T} ^{\top }\cdot {\hat {e}}_{i}\otimes {\vec {f}}_{,i})&=&\operatorname {div} (\mathbf {T} )\times {\vec {f}}-\operatorname {grad} ({\vec {f}})\cdot \!\!\times \mathbf {T} \end{array}}}$

Any base:

${\displaystyle \operatorname {div} (f_{i}{\vec {b}}_{i})=\operatorname {grad} (f_{i})\cdot {\vec {b}}_{i}+f_{i}\,\operatorname {div} ({\vec {b}}_{i})}$

${\displaystyle \operatorname {div} (T^{ij}{\vec {b}}_{i}\otimes {\vec {b}}_{j})=(\operatorname {grad} (T^{ij})\cdot {\vec {b}}_{i}){\vec {b}}_{j}+T^{ij}\,\operatorname {div} ({\vec {b}}_{i}){\vec {b}}_{j}+T^{ij}\,\operatorname {grad} ({\vec {b}}_{j})\cdot {\vec {b}}_{i}}$

Product with constants:

${\displaystyle \operatorname {div} (c{\vec {f}})=c\,\operatorname {div} ({\vec {f}})}$

${\displaystyle \operatorname {div} (f{\vec {c}})=\operatorname {grad} (f)\cdot {\vec {c}}}$

${\displaystyle \operatorname {div} (c\mathbf {T} )=c\,\operatorname {div} (\mathbf {T} )}$

${\displaystyle \operatorname {div} (f\mathbf {C} )=\operatorname {grad} (f)\cdot \mathbf {C} \quad \rightarrow \quad \operatorname {div} (f\mathbf {I} )=\operatorname {grad} (f)}$

${\displaystyle \operatorname {div} (\mathbf {C} \cdot {\vec {f}})=\mathbf {C} ^{\top }:\operatorname {grad} ({\vec {f}})\quad \rightarrow \quad \operatorname {div} (\mathbf {I} \cdot {\vec {f}})=\operatorname {div} ({\vec {f}})=\mathbf {I} :\operatorname {grad} ({\vec {f}})=\operatorname {Sp} (\operatorname {grad} ({\vec {f}}))}$

${\displaystyle \operatorname {div} (\mathbf {T} \cdot {\vec {c}})=\operatorname {div} (\mathbf {T} )\cdot {\vec {c}}}$

${\displaystyle \operatorname {div} ({\vec {c}}\otimes {\vec {f}})=\operatorname {grad} ({\vec {f}})\cdot {\vec {c}}}$

${\displaystyle \operatorname {div} ({\vec {f}}\otimes {\vec {c}})=\operatorname {div} ({\vec {f}}){\vec {c}}}$

rotation

→ Main article : Rotation of a vector field

Vector field  : ${\displaystyle {\vec {f}}=f_{i}{\hat {e}}_{i}}$

${\displaystyle \operatorname {rot} ({\vec {f}})=\nabla \times {\vec {f}}={\frac {\partial f_{j}}{\partial x_{i}}}{\hat {e}}_{i}\times {\hat {e}}_{j}=\varepsilon _{ijk}{\frac {\partial f_{j}}{\partial x_{i}}}{\hat {e}}_{k}=\left({\frac {\partial f_{3}}{\partial x_{2}}}-{\frac {\partial f_{2}}{\partial x_{3}}}\right){\hat {e}}_{1}+\left({\frac {\partial f_{1}}{\partial x_{3}}}-{\frac {\partial f_{3}}{\partial x_{1}}}\right){\hat {e}}_{2}+\left({\frac {\partial f_{2}}{\partial x_{1}}}-{\frac {\partial f_{1}}{\partial x_{2}}}\right){\hat {e}}_{3}}$

${\displaystyle \operatorname {rot} {\vec {x}}={\vec {0}}}$

Tensor field : ${\displaystyle \mathbf {T} =T_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}}$

${\displaystyle \operatorname {rot} (\mathbf {T} )=\nabla \times \mathbf {T} ={\hat {e}}_{i}\times {\frac {\partial \mathbf {T} }{\partial x_{i}}}={\hat {e}}_{i}\times T_{jl,i}{\hat {e}}_{j}\otimes {\hat {e}}_{l}=\varepsilon _{ijk}T_{jl,i}{\hat {e}}_{k}\otimes {\hat {e}}_{l}}$

${\displaystyle \mathbf {T=T} ^{\top }\quad \rightarrow \quad \operatorname {Sp(rot} (\mathbf {T} ))=0}$

Cylindrical coordinates :

${\displaystyle \operatorname {rot} {\vec {f}}=\left[{\frac {1}{\rho }}{\frac {\partial f_{z}}{\partial \varphi }}-{\frac {\partial f_{\varphi }}{\partial z}}\right]{\hat {e}}_{\rho }+\left[{\frac {\partial f_{\rho }}{\partial z}}-{\frac {\partial f_{z}}{\partial \rho }}\right]{\hat {e}}_{\varphi }+{\frac {1}{\rho }}\left[{\frac {\partial }{\partial \rho }}\left(\rho \,f_{\varphi }\right)-{\frac {\partial f_{\rho }}{\partial \varphi }}\right]{\hat {e}}_{z}}$

Spherical coordinates :

${\displaystyle \operatorname {rot} ({\vec {f}})={\frac {1}{r\sin \vartheta }}\left[{\frac {\partial }{\partial \vartheta }}\left(f_{\varphi }\sin \vartheta \right)-{\frac {\partial f_{\vartheta }}{\partial \varphi }}\right]{\hat {e}}_{r}+\left[{\frac {1}{r\sin \vartheta }}{\frac {\partial f_{r}}{\partial \varphi }}-{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(rf_{\varphi }\right)\right]{\hat {e}}_{\vartheta }+{\frac {1}{r}}\left[{\frac {\partial }{\partial r}}\left(rf_{\vartheta }\right)-{\frac {\partial f_{r}}{\partial \vartheta }}\right]{\hat {e}}_{\varphi }}$

Integrability condition: Every divergence-free vector field is the rotation of a vector field:

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

Coordinate-free representation:

• Volume with ${\displaystyle v}$
• Surface with an outer vectorial surface element ${\displaystyle a}$ ${\displaystyle \mathrm {d} {\vec {a}}}$

${\displaystyle \mathrm {rot} ({\vec {f}})=-\lim _{v\rightarrow 0}\left({\frac {1}{v}}\int _{a}{\vec {f}}\times \mathrm {d} {\vec {a}}\right)}$

Relationship with the other differential operators:

${\displaystyle {\begin{array}{rcl}\operatorname {rot} (f{\vec {c}})&=&\operatorname {grad} (f)\times {\vec {c}}\\\operatorname {rot} ({\vec {f}})&=&-\mathbf {I} \cdot \!\!\times \operatorname {grad} ({\vec {f}})\end{array}}}$

Product rule:

${\displaystyle {\begin{array}{rcl}\operatorname {rot} (f{\vec {g}})&=&\displaystyle {\hat {e}}_{i}\times (f_{,i}{\vec {g}}+f{\vec {g}}_{,i})=\operatorname {grad} (f)\times {\vec {g}}+f\operatorname {rot} ({\vec {g}})\\\operatorname {rot} ({\vec {f}}\times {\vec {g}})&=&\displaystyle {\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}\\&=&\displaystyle \operatorname {grad} ({\vec {f}})\cdot {\vec {g}}-\operatorname {div} ({\vec {f}}){\vec {g}}+\operatorname {div} ({\vec {g}}){\vec {f}}-\operatorname {grad} ({\vec {g}})\cdot {\vec {f}}\\&=&\displaystyle \operatorname {div} ({\vec {g}}\otimes {\vec {f}})-\operatorname {div} ({\vec {f}}\otimes {\vec {g}})\\\operatorname {rot} ({\vec {f}}\otimes {\vec {g}})&=&\displaystyle {\hat {e}}_{i}\times \left({\vec {f}}_{,i}\otimes {\vec {g}}+{\vec {f}}\otimes {\vec {g}}_{,i}\right)=\operatorname {rot} ({\vec {f}})\otimes {\vec {g}}-{\vec {f}}\times \operatorname {grad} ({\vec {g}})^{\top }\\\operatorname {rot} (\mathbf {T} \cdot {\vec {f}})&=&\displaystyle {\hat {e}}_{k}\times (\mathbf {T} _{,k}\cdot {\vec {f}}+(\mathbf {T} \cdot {\hat {e}}_{i})\otimes {\hat {e}}_{i}\cdot {\vec {f}}_{,k})=\operatorname {rot} (\mathbf {T} )\cdot {\vec {f}}-\left({\hat {e}}_{i}\cdot {\vec {f}}_{,k}\right)(\mathbf {T} \cdot {\hat {e}}_{i})\times {\hat {e}}_{k}\\&=&\operatorname {rot} (\mathbf {T} )\cdot {\vec {f}}-\mathbf {T} \cdot \!\!\times \operatorname {grad} ({\vec {f}})\\\operatorname {rot} (f\mathbf {T} )&=&\displaystyle {\hat {e}}_{k}\times (f_{,k}\mathbf {T} +f\mathbf {T} _{,k})=\operatorname {grad} (f)\times \mathbf {T} +f\operatorname {rot} (\mathbf {T} )\\\operatorname {rot} (\mathbf {T} \times {\vec {f}})&=&\displaystyle {\hat {e}}_{k}\times (\mathbf {T} _{,k}\times {\vec {f}}+(\mathbf {T} \cdot {\hat {e}}_{i})\otimes {\hat {e}}_{i}\times {\vec {f}}_{,k})=\operatorname {rot} (\mathbf {T} )\times {\vec {f}}-{\hat {e}}_{k}\times (\mathbf {T} \cdot {\hat {e}}_{i})\otimes {\vec {f}}_{,k}\times {\hat {e}}_{i}\\&=&\operatorname {rot} (\mathbf {T} )\times {\vec {f}}-\operatorname {grad} ({\vec {f}})\times \!\!\times \mathbf {T} \\\operatorname {rot} ({\vec {f}}\times \mathbf {C} )&=&\displaystyle {\hat {e}}_{k}\times \left({\vec {f}}_{,k}\times (\mathbf {C} \cdot {\hat {e}}_{i})\otimes {\hat {e}}_{i}\right)=({\hat {e}}_{k}\cdot \mathbf {C} \cdot {\hat {e}}_{i}){\vec {f}}_{,k}\otimes {\hat {e}}_{i}-\left({\hat {e}}_{k}\cdot {\vec {f}}_{,k}\right)\mathbf {C} \cdot {\hat {e}}_{i}\otimes {\hat {e}}_{i}\\&=&\operatorname {grad} ({\vec {f}})\cdot \mathbf {C} -\operatorname {div} ({\vec {f}})\mathbf {C} \end{array}}}$

Any base:

${\displaystyle \operatorname {rot} (f^{i}{\vec {b}}_{i})=\operatorname {grad} (f^{i})\times {\vec {b}}_{i}+f^{i}\,\operatorname {rot} ({\vec {b}}_{i})}$

Product with constants:

${\displaystyle {\begin{array}{rcl}\operatorname {rot} (c{\vec {f}})&=&\displaystyle c\,\operatorname {rot} ({\vec {f}})\\\operatorname {rot} (f{\vec {c}})&=&\displaystyle \operatorname {grad} (f)\times {\vec {c}}\\\operatorname {rot} ({\vec {c}}\times {\vec {f}})&=&\displaystyle -\operatorname {rot} ({\vec {f}}\times {\vec {c}})=\operatorname {div} ({\vec {f}}){\vec {c}}-\operatorname {grad} ({\vec {f}})\cdot {\vec {c}}=\operatorname {div} ({\vec {f}}\otimes {\vec {c}})-\operatorname {div} ({\vec {c}}\otimes {\vec {f}})\\\operatorname {rot} ({\vec {c}}\otimes {\vec {f}})&=&\displaystyle -{\vec {c}}\times \operatorname {grad} ({\vec {f}})^{\top }\\\operatorname {rot} ({\vec {f}}\otimes {\vec {c}})&=&\displaystyle \operatorname {rot} ({\vec {f}})\otimes {\vec {c}}\\\operatorname {rot} (\mathbf {C} \cdot {\vec {f}})&=&\displaystyle -\mathbf {C} \cdot \!\!\times \operatorname {grad} ({\vec {f}})\quad \rightarrow \quad \operatorname {rot} ({\vec {f}})=\operatorname {rot} (\mathbf {I} \cdot {\vec {f}})=-\mathbf {I} \cdot \!\!\times \operatorname {grad} ({\vec {f}})\\\operatorname {rot} (\mathbf {T} \cdot {\vec {c}})&=&\displaystyle \operatorname {rot} (\mathbf {T} )\cdot {\vec {c}}\\\operatorname {rot} (c\mathbf {T} )&=&c\operatorname {rot} (\mathbf {T} )\\\operatorname {rot} (f\mathbf {C} )&=&\operatorname {grad} (f)\times \mathbf {C} \\\operatorname {rot} (\mathbf {C} \times {\vec {f}})&=&-\operatorname {grad} ({\vec {f}})\times \!\!\times \mathbf {C} \\\operatorname {rot} (\mathbf {T} \times {\vec {c}})&=&\operatorname {rot} (\mathbf {T} )\times {\vec {c}}\\\operatorname {rot} ({\vec {f}}\times \mathbf {I} )&=&\operatorname {grad} ({\vec {f}})-\operatorname {div} ({\vec {f}})\mathbf {I} \end{array}}}$

Theorem about non-rotating fields

→ Main article : Poincaré lemma

${\displaystyle {\begin{array}{rrcll}{\textsf {I}}:&\operatorname {rot} ({\vec {u}}):={\hat {e}}_{k}\times {\vec {u}}_{,k}={\vec {0}}&\rightarrow &\exists f\colon &{\vec {u}}=\operatorname {grad} (f)\\{\textsf {II}}:&\operatorname {rot} (\mathbf {T} ^{\top })=\mathbf {0} &\rightarrow &\exists {\vec {u}}\colon &\mathbf {T} =\operatorname {grad} ({\vec {u}})\\{\textsf {III}}:&\operatorname {rot} (\mathbf {T} ^{\top })=\mathbf {0} \wedge \operatorname {Sp} (\mathbf {T} )=0&\rightarrow &\exists \mathbf {W} \colon &\mathbf {T} =\operatorname {rot} (\mathbf {W} )\,\wedge \;\mathbf {W} =-\mathbf {W} ^{\top }\end{array}}}$

Laplace operator

→ Main article : Laplace operator

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

Cartesian coordinates:

${\displaystyle {\begin{aligned}\Delta f=&{\frac {{\partial f}^{2}}{\partial x_{1}^{2}}}+{\frac {{\partial f}^{2}}{\partial x_{2}^{2}}}+{\frac {{\partial f}^{2}}{\partial x_{3}^{2}}}=f_{,kk}\\\Delta {\vec {f}}=&\nabla ^{2}{\vec {f}}=(\nabla \cdot \nabla ){\vec {f}}=\Delta f_{i}{\hat {e}}_{i}=f_{i,kk}{\hat {e}}_{i}\\\Delta \mathbf {T} =&\nabla ^{2}\mathbf {T} =(\nabla \cdot \nabla )\mathbf {T} =\Delta T_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=T_{ij,kk}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\end{aligned}}}$

Cylindrical coordinates :

${\displaystyle {\begin{aligned}\Delta f=&{\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho \,{\frac {\partial f}{\partial \rho }}\right)+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}\\\Delta {\vec {f}}=&\left(\Delta f_{\rho }-{\frac {1}{\rho ^{2}}}f_{\rho }-{\frac {2}{\rho ^{2}}}f_{\varphi ,\varphi }\right){\hat {e}}_{\rho }+\left(\Delta f_{\varphi }-{\frac {1}{\rho ^{2}}}f_{\varphi }+{\frac {2}{\rho ^{2}}}f_{\rho ,\varphi }\right){\hat {e}}_{\varphi }+\Delta f_{z}{\hat {e}}_{z}\end{aligned}}}$

Spherical coordinates :

${\displaystyle {\begin{aligned}\Delta f=&{\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \vartheta }}{\frac {\partial }{\partial \vartheta }}\left(\sin \vartheta \,{\frac {\partial f}{\partial \vartheta }}\right)+{\frac {1}{r^{2}\sin ^{2}\vartheta }}{\frac {\partial ^{2}f}{\partial \phi ^{2}}}\\\Delta {\vec {f}}=&\left(\Delta f_{r}-{\frac {2}{r^{2}}}f_{r}-{\frac {2}{r^{2}\sin \vartheta }}f_{\varphi ,\varphi }-{\frac {2}{r^{2}}}f_{\vartheta ,\vartheta }-{\frac {2}{r^{2}}}f_{\vartheta }\cot \vartheta \right){\hat {e}}_{r}\\&+\left(\Delta f_{\vartheta }+{\frac {2}{r^{2}}}f_{r,\vartheta }-{\frac {2}{r^{2}\sin \vartheta }}f_{\varphi ,\varphi }-{\frac {f_{\vartheta }}{r^{2}\sin ^{2}\vartheta }}\right){\hat {e}}_{\vartheta }\\&+\left(\Delta f_{\varphi }-{\frac {f_{\varphi }}{r^{2}\sin ^{2}\vartheta }}+{\frac {2}{r^{2}\sin ^{2}\vartheta }}f_{r,\varphi }+{\frac {2\cos \vartheta }{r^{2}\sin ^{2}\vartheta }}f_{\vartheta ,\varphi }\right){\hat {e}}_{\varphi }\end{aligned}}}$

Relationship with other differential operators:

${\displaystyle {\begin{array}{rclcl}\Delta f&=&\nabla \cdot (\nabla f)&=&\operatorname {div(grad} (f))\\\Delta {\vec {f}}&=&\nabla \cdot (\nabla \otimes {\vec {f}})&=&\operatorname {div(grad} ({\vec {f}})^{\top })\end{array}}}$

connections

Due to the partly different definitions of the differential operators in the literature, there may be different formulas in the literature. If the definitions of the literature are used here, the formulas here go into those of the literature.

${\displaystyle {\begin{array}{rclcl}\operatorname {div(rot} ({\vec {f}}))&=&\displaystyle \nabla \cdot (\nabla \times {\vec {f}})&=&0\\\operatorname {div(rot} (\mathbf {T} ))&=&\nabla \times \mathbf {T} &=&{\vec {0}}\\\operatorname {div(rot(rot} (\mathbf {T} )^{\top }))&=&\nabla \cdot (\nabla \times (\nabla \times \mathbf {T} )^{\top })&=&{\vec {0}}\\\operatorname {rot(grad} (f))&=&\displaystyle \nabla \times \nabla f&=&{\vec {0}}\\\operatorname {rot(grad} ({\vec {f}})^{\top })&=&\displaystyle \nabla \times (\nabla \otimes {\vec {f}})&=&\mathbf {0} \\\operatorname {div(grad} (f)\times \operatorname {grad} (g))&=&\displaystyle \nabla \cdot (\nabla f\times \nabla g)=\nabla g\cdot (\nabla \times \nabla f)&=&0\\[2ex]\operatorname {div(grad} (f))&=&\nabla \cdot (\nabla f)=(\nabla \cdot \nabla )f&=&\Delta f\\\operatorname {div(grad} ({\vec {f}})^{\top })&=&\nabla \cdot (\nabla \otimes {\vec {f}})=(\nabla \cdot \nabla ){\vec {f}}&=&\Delta {\vec {f}}\\[2ex]\operatorname {div(grad} ({\vec {f}}))&=&\nabla \cdot ({\vec {f}}\otimes \nabla )=(\nabla \cdot {\vec {f}})\nabla &=&\operatorname {grad(div} ({\vec {f}}))\\\operatorname {rot(grad} ({\vec {f}}))&=&\displaystyle \nabla \times ({\vec {f}}\otimes \nabla )=(\nabla \times {\vec {f}})\otimes \nabla &=&\operatorname {grad(rot} ({\vec {f}}))\\[2ex]\operatorname {rot(rot} ({\vec {f}}))&=&\displaystyle \nabla \times (\nabla \times {\vec {f}})=\nabla (\nabla \cdot {\vec {f}})-\Delta {\vec {f}}&=&\operatorname {grad(div} ({\vec {f}}))-\Delta {\vec {f}}\\\operatorname {rot(rot} (\mathbf {T} ))&=&\nabla \times [\nabla \times (\mathbf {T} \cdot {\hat {e}}_{i})\otimes {\hat {e}}_{i}]\\&=&[\operatorname {div} (\mathbf {T} )\cdot {\hat {e}}_{i}]\nabla \otimes {\hat {e}}_{i}-\Delta \mathbf {T} \\&=&(\nabla \otimes \operatorname {div} (\mathbf {T} ))\cdot {\hat {e}}_{i}\otimes {\hat {e}}_{i}-\Delta \mathbf {T} &=&\operatorname {grad(div} (\mathbf {T} ))^{\top }-\Delta \mathbf {T} \end{array}}}$

${\displaystyle {\begin{array}{rcl}\operatorname {rot(rot} (\mathbf {T} )^{\top })&=&-\Delta \mathbf {T} -\operatorname {grad(grad(Sp} (\mathbf {T} )))+\operatorname {grad(div} (\mathbf {T} ))+\operatorname {grad(div} (\mathbf {T} ))^{\top }\\&&+[\Delta \operatorname {Sp} (\mathbf {T} )-\operatorname {div(div} (\mathbf {T} ))]\mathbf {I} \end{array}}}$

In the case of symmetrical, the following also applies: ${\displaystyle \mathbf {T} =\mathbf {G} -\operatorname {Sp} (\mathbf {G} )\mathbf {I} }$

${\displaystyle \operatorname {rot(rot} (\mathbf {T} )^{\top })=-\Delta \mathbf {G} +\operatorname {grad(div} (\mathbf {G} ))+\operatorname {grad(div} (\mathbf {G} ))^{\top }-\operatorname {div(div} (\mathbf {G} ))\mathbf {I} }$

The Laplace operator can be treated like a scalar, i.e. inserted anywhere in the formulas, e.g. B .:

${\displaystyle {\begin{array}{l}\Delta \operatorname {rot(rot} ({\vec {f}}))=\operatorname {rot(\Delta rot} ({\vec {f}}))=\operatorname {rot(rot} (\Delta {\vec {f}}))=\ldots \\\ldots =\Delta \operatorname {grad(div} ({\vec {f}}))-\Delta \Delta {\vec {f}}=\operatorname {grad} (\Delta \operatorname {div} ({\vec {f}}))-\Delta \Delta {\vec {f}}=\operatorname {grad(div} (\Delta {\vec {f}}))-\Delta \Delta {\vec {f}}\end{array}}}$

Grassmann development

${\displaystyle {\vec {f}}\times \operatorname {rot} ({\vec {f}})={\frac {1}{2}}\operatorname {grad} ({\vec {f}}\cdot {\vec {f}})-\operatorname {grad} ({\vec {f}})\cdot {\vec {f}}=(\operatorname {grad} ({\vec {f}})^{\top }-\operatorname {grad} ({\vec {f}}))\cdot {\vec {f}}}$

${\displaystyle \operatorname {grad} ({\vec {f}})\cdot {\vec {f}}={\frac {1}{2}}\operatorname {grad} ({\vec {f}}\cdot {\vec {f}})-{\vec {f}}\times \operatorname {rot} ({\vec {f}})}$

Theorems about gradient, divergence and rotation

• Helmholtz theorem : Each vector field can be clearly broken down into a divergence-free and a rotation-free part. According to the integrability conditions for rotations and gradients, the first part is a rotation field and the second is a gradient field.
  ${\displaystyle {\begin{array}{rclccl}{\vec {f}}={\vec {f}}_{1}+{\vec {f}}_{2}:&&&\operatorname {div} ({\vec {f}}_{1})=0&\wedge &\operatorname {rot} ({\vec {f}}_{2})={\vec {0}}\\\leftrightarrow \exists g,{\vec {g}}:&&{\vec {f}}=&\operatorname {rot} ({\vec {g}})&+&\operatorname {grad} (g)\end{array}}}$
• A vector field whose divergence and curl disappears, is harmonious: .
  ${\displaystyle \operatorname {div} ({\vec {f}})=0\;\wedge \;\operatorname {rot} ({\vec {f}})={\vec {0}}\quad \rightarrow \quad \Delta {\vec {f}}={\vec {0}}}$
• See also theorem on non-rotation fields

Integral sentences

Gaussian integral theorem

→ Main article : Gaussian integral theorem

• Volume with volume shape and ${\displaystyle v}$ ${\displaystyle \mathrm {d} v}$
• Surface with an outer vectorial surface element ${\displaystyle a}$ ${\displaystyle \mathrm {d} {\vec {a}}}$
• Position vectors ${\displaystyle {\vec {x}}\in v}$
• Scalar, vector or tensor-valued function of the location  : ${\displaystyle f,{\vec {f}},\mathbf {T} }$ ${\displaystyle {\vec {x}}}$

${\displaystyle {\begin{array}{rcl}\int _{v}\operatorname {grad} (f)\,\mathrm {d} v&=&\int _{a}f\,\mathrm {d} {\vec {a}}\\\int _{v}\operatorname {grad} ({\vec {f}})\,\mathrm {d} v&=&\int _{a}{\vec {f}}\otimes \mathrm {d} {\vec {a}}\\\int _{v}\operatorname {div} ({\vec {f}})\,\mathrm {d} v&=&\int _{a}{\vec {f}}\cdot \mathrm {d} {\vec {a}}\\\int _{v}\operatorname {rot} ({\vec {f}})\,\mathrm {d} v&=&-\int _{a}{\vec {f}}\times \mathrm {d} {\vec {a}}\\\int _{v}\operatorname {div} (\mathbf {T} )\,\mathrm {d} v&=&\int _{a}\mathbf {T} ^{\top }\cdot \mathrm {d} {\vec {a}}\end{array}}}$

Classic integral theorem from Stokes

→ Main article : Stokes theorem

Given:

• Area with an outer vectorial surface element ${\displaystyle a}$ ${\displaystyle \mathrm {d} {\vec {a}}}$
• Boundary curve of the surface with line element ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle \mathrm {d} {\vec {b}}}$
• Position vectors ${\displaystyle {\vec {x}}\in a}$

Vector-valued function  : ${\displaystyle {\vec {f}}({\vec {x}},t)}$

${\displaystyle \int _{a}\operatorname {rot} ({\vec {f}})\cdot \mathrm {d} {\vec {a}}=\oint _{b}{\vec {f}}\cdot \mathrm {d} {\vec {b}}}$

Continuum mechanics

Small deformations

Engineering expansion:

${\displaystyle {\boldsymbol {\varepsilon }}=\varepsilon _{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}={\frac {1}{2}}(u_{i,j}+u_{j,i}){\hat {e}}_{i}\otimes {\hat {e}}_{j}}$

Compatibility Conditions:

${\displaystyle {\begin{array}{rcl}2\varepsilon _{12,12}-\varepsilon _{22,11}-\varepsilon _{11,22}&=&0\\2\varepsilon _{13,13}-\varepsilon _{33,11}-\varepsilon _{11,33}&=&0\\2\varepsilon _{23,23}-\varepsilon _{33,22}-\varepsilon _{22,33}&=&0\\\varepsilon _{11,23}+\varepsilon _{23,11}-\varepsilon _{12,13}-\varepsilon _{13,12}&=&0\\\varepsilon _{22,13}+\varepsilon _{13,22}-\varepsilon _{12,23}-\varepsilon _{23,12}&=&0\\\varepsilon _{12,33}+\varepsilon _{33,12}-\varepsilon _{13,23}-\varepsilon _{23,13}&=&0\end{array}}}$

Rigid body motion

Orthogonal tensor describes the rotation. ${\displaystyle \mathbf {Q} }$

${\displaystyle \mathbf {\Omega } :={\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }={(\mathbf {Q} \cdot {\dot {\mathbf {Q} }}^{\top })}^{\top }=-\mathbf {Q} \cdot {\dot {\mathbf {Q} }}^{\top }}$

Vector invariant or axial vector of the skew-symmetric tensor  : ${\displaystyle {\vec {\omega }}}$ ${\displaystyle \mathbf {\Omega } }$

${\displaystyle \mathbf {\Omega } {\vec {r}}={\vec {\omega }}\times {\vec {r}}{\quad \forall \;}{\vec {r}}}$

Rigid body motion with  : ${\displaystyle {\vec {r}}=\mathrm {const.} }$

${\displaystyle {\vec {x}}={\vec {f}}+\mathbf {Q} \cdot {\vec {r}}\quad \rightarrow \quad {\vec {r}}=\mathbf {Q} ^{\top }\cdot ({\vec {x}}-{\vec {f}})}$

${\displaystyle {\vec {v}}={\dot {\vec {f}}}+{\dot {\mathbf {Q} }}\cdot {\vec {r}}={\dot {\vec {f}}}+{\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }\cdot ({\vec {x}}-{\vec {f}})={\dot {\vec {f}}}+\mathbf {\Omega } \cdot ({\vec {x}}-{\vec {f}})={\dot {\vec {f}}}+{\vec {\omega }}\times ({\vec {x}}-{\vec {f}})}$

Derivatives of the invariants

${\displaystyle {\frac {\partial \operatorname {I} _{1}(\mathbf {T} )}{\partial \mathbf {T} }}={\frac {\partial \mathrm {Sp} (\mathbf {T} )}{\partial \mathbf {T} }}=\mathbf {I} }$

${\displaystyle {\frac {\partial \operatorname {I} _{2}(\mathbf {T} )}{\partial \mathbf {T} }}=\operatorname {I} _{1}(\mathbf {T} )\mathbf {I} -\mathbf {T} ^{\top }}$

${\displaystyle {\frac {\partial \operatorname {I} _{3}(\mathbf {T} )}{\partial \mathbf {T} }}={\frac {\partial \mathrm {det} (\mathbf {T} )}{\partial \mathbf {T} }}=\mathrm {det} (\mathbf {T} )\mathbf {T} ^{\mathrm {T} -1}}$

with the transposed inverse of the tensor . ${\displaystyle \mathbf {T} ^{\mathrm {T} -1}}$ ${\displaystyle \mathbf {T} }$

${\displaystyle {\frac {\partial \parallel \mathbf {T} \parallel }{\partial \mathbf {T} }}={\frac {\mathbf {T} }{\parallel \mathbf {T} \parallel }}}$

Eigenvalues ​​(no sum over ): ${\displaystyle i}$

${\displaystyle \mathbf {T} \cdot {\vec {v}}_{i}=\lambda _{i}{\vec {v}}_{i}\quad \rightarrow \quad {\frac {\partial \lambda _{i}}{\partial \mathbf {T} }}={\vec {v}}_{i}\otimes {\vec {v}}_{i}}$

Function of the invariants: ${\displaystyle f}$

${\displaystyle {\frac {\partial f}{\partial \mathbf {T} }}(\operatorname {I} _{1}(\mathbf {T} ),\,\operatorname {I} _{2}(\mathbf {T} ),\,\operatorname {I} _{3}(\mathbf {T} ))=\left({\frac {\partial f}{\partial \operatorname {I} _{1}}}+\operatorname {I} _{1}{\frac {\partial f}{\partial \operatorname {I} _{2}}}+\operatorname {I} _{2}{\frac {\partial f}{\partial \operatorname {I} _{3}}}\right)\mathbf {I} -\left({\frac {\partial f}{\partial \operatorname {I} _{2}}}+\operatorname {I} _{1}{\frac {\partial f}{\partial \operatorname {I} _{3}}}\right)\mathbf {T} ^{\top }+{\frac {\partial f}{\partial \operatorname {I} _{3}}}\mathbf {T} ^{\top }\cdot \mathbf {T} ^{\top }}$

Time derivatives of the invariants

${\displaystyle {\frac {\mathrm {D} \operatorname {I} _{1}(\mathbf {T} )}{\mathrm {D} t}}=\operatorname {Sp} ({\dot {\mathbf {T} }})}$

${\displaystyle {\frac {\mathrm {D} \operatorname {I} _{2}(\mathbf {T} )}{\mathrm {D} t}}=\operatorname {Sp} (\mathbf {T} )\,\operatorname {Sp} ({\dot {\mathbf {T} }})-\operatorname {Sp} (\mathbf {T} \cdot {\dot {\mathbf {T} }})}$

${\displaystyle {\frac {\mathrm {D} \operatorname {I} _{3}(\mathbf {T} )}{\mathrm {D} t}}={\frac {\mathrm {D} \mathrm {det} (\mathbf {T} )}{\mathrm {D} t}}=\mathrm {det} (\mathbf {T} )\,\operatorname {Sp} ({\dot {\mathbf {T} }}\cdot \mathbf {T} ^{-1})}$

${\displaystyle {\frac {\mathrm {D} \parallel \mathbf {T} \parallel }{\mathrm {D} t}}={\frac {\mathbf {T} \cdot {\dot {\mathbf {T} }}}{\parallel \mathbf {T} \parallel }}}$

Time derivative of inverse tensors

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}(\mathbf {T} ^{-1})=-\mathbf {T} ^{-1}\cdot {\dot {\mathbf {T} }}\cdot \mathbf {T} ^{-1}}$

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}(\mathbf {T} ^{\top -1})=-\mathbf {T} ^{\top -1}\cdot {\dot {\mathbf {T} }}^{\top }\cdot \mathbf {T} ^{\top -1}}$

Orthogonal tensors:

${\displaystyle {\dot {\mathbf {Q} }}^{\top }=-\mathbf {Q} ^{\top }\cdot {\dot {\mathbf {Q} }}\cdot \mathbf {Q} ^{\top }}$

Convective coordinates

→ Main article : Convective coordinates

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

Covariant basis vectors ,    ${\displaystyle {\vec {B}}_{i}={\frac {\mathrm {d} {\vec {X}}}{\mathrm {d} y_{i}}}}$ ${\displaystyle {\vec {b}}_{i}={\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} y_{i}}}}$

Contravariant basis vectors ,    ${\displaystyle {\vec {B}}^{i}={\frac {\mathrm {d} y_{i}}{\mathrm {d} {\vec {X}}}}=\operatorname {grad} (y_{i})}$ ${\displaystyle {\vec {b}}^{i}={\frac {\mathrm {d} y_{i}}{\mathrm {d} {\vec {x}}}}=\operatorname {grad} (y_{i})}$

${\displaystyle {\vec {B}}_{i}\cdot {\vec {B}}^{j}={\vec {b}}_{i}\cdot {\vec {b}}^{j}=\delta _{i}^{j}}$

Deformation gradient ${\displaystyle \mathbf {F} ={\vec {b}}_{i}\otimes {\vec {B}}^{i}}$

Spatial velocity gradient ${\displaystyle \mathbf {l} ={\dot {\vec {b}}}_{i}\otimes {\vec {b}}^{i}=-{\vec {b}}_{i}\otimes {\dot {\vec {b}}}^{i}}$

Covariant tensor ${\displaystyle \mathbf {T} =T_{ij}{\vec {b}}^{i}\otimes {\vec {b}}^{j}}$

Contravariant tensor ${\displaystyle \mathbf {T} =T^{ij}{\vec {b}}_{i}\otimes {\vec {b}}_{j}}$

Speed ​​gradient

→ Main article : Velocity gradient

Spatial velocity gradient: ${\displaystyle \mathbf {l} =\operatorname {grad} ({\vec {v}})}$

Speed ​​divergence: ${\displaystyle \operatorname {div} ({\vec {v}})=\operatorname {Sp} (\mathbf {l} )}$

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\mathrm {det} (\mathbf {F} )=\mathrm {det} (\mathbf {F} )\,\operatorname {div} ({\vec {v}})}$

${\displaystyle \mathbf {l} ={\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}=-\mathbf {F} \cdot {\frac {\mathrm {D} }{\mathrm {D} t}}(\mathbf {F} ^{-1})}$

Objective time derivations

→ Main article : Euclidean transformation

Designations as in #Convective coordinates .

Spatial velocity gradient ${\displaystyle \mathbf {l} ={\dot {\vec {b}}}_{i}\otimes {\vec {b}}^{i}=-{\vec {b}}_{i}\otimes {\dot {\vec {b}}}\,^{i}=\mathbf {d} +\mathbf {w} }$

Spatial distortion speed ${\displaystyle \mathbf {d} ={\frac {1}{2}}(\mathbf {l} +\mathbf {l} ^{\top })}$

Vortex or spin tensor ${\displaystyle \mathbf {w} ={\frac {1}{2}}(\mathbf {l} -\mathbf {l} ^{\top })}$

Objective time derivatives of vectors

Given: : ${\displaystyle {\vec {v}}=v_{i}{\vec {b}}^{i}=v^{i}{\vec {b}}_{i}}$

${\displaystyle {\begin{array}{rclcl}{\stackrel {\Delta }{\vec {v}}}&=&{\dot {\vec {v}}}+\mathbf {l} ^{\top }\cdot {\vec {v}}&=&{\dot {v}}_{i}{\vec {b}}^{i}\\{\stackrel {\nabla }{\vec {v}}}&=&{\dot {\vec {v}}}-\mathbf {l} \cdot {\vec {v}}&=&{\dot {v}}^{i}{\vec {b}}_{i}\\{\stackrel {\circ }{\vec {v}}}&=&{\dot {\vec {v}}}-\mathbf {w} \cdot {\vec {v}}\end{array}}}$

Objective time derivatives of tensors

Given: ${\displaystyle \mathbf {T} =T_{ij}{\vec {b}}^{i}\otimes {\vec {b}}^{j}=T^{ij}{\vec {b}}_{i}\otimes {\vec {b}}_{j}}$

${\displaystyle {\begin{array}{rclcl}{\stackrel {\Delta }{\mathbf {T} }}&=&{\dot {\mathbf {T} }}+\mathbf {T\cdot l} +\mathbf {l} ^{\top }\cdot \mathbf {T} &=&{\dot {T}}_{ij}{\vec {b}}^{i}\otimes {\vec {b}}^{j}\\{\stackrel {\nabla }{\mathbf {T} }}&=&{\dot {\mathbf {T} }}-\mathbf {l\cdot T} -\mathbf {T\cdot l} ^{\top }&=&{\dot {T}}^{ij}{\vec {b}}_{i}\otimes {\vec {b}}_{j}\\{\stackrel {\circ }{\mathbf {T} }}&=&{\dot {\mathbf {T} }}+\mathbf {T\cdot w} -\mathbf {w\cdot T} \\{\stackrel {\diamond }{\mathbf {T} }}&=&{\dot {\mathbf {T} }}+\operatorname {Sp} (\mathbf {l} )\mathbf {T} -\mathbf {l\cdot T} -\mathbf {T\cdot l} ^{\top }\end{array}}}$

Material time derivative

→ Main article : Substantive derivative

${\displaystyle {\dot {f}}({\vec {x}},t)={\frac {\mathrm {D} f}{\mathrm {D} t}}={\frac {\partial f}{\partial t}}+\operatorname {grad} (f)\cdot {\vec {v}}}$

${\displaystyle {\dot {\vec {f}}}({\vec {x}},t)={\frac {\mathrm {D} {\vec {f}}}{\mathrm {D} t}}={\frac {\partial {\vec {f}}}{\partial t}}+\operatorname {grad} ({\vec {f}})\cdot {\vec {v}}}$

Cartesian coordinates: ${\displaystyle {\frac {\mathrm {D} f}{\mathrm {D} t}}:={\frac {\partial f}{\partial t}}+v_{x}{\frac {\partial f}{\partial x}}+v_{y}{\frac {\partial f}{\partial y}}+v_{z}{\frac {\partial f}{\partial z}}}$

Cylindrical coordinates: ${\displaystyle {\frac {\mathrm {D} f}{\mathrm {D} t}}:={\frac {\partial f}{\partial t}}+v_{\rho }{\frac {\partial f}{\partial \rho }}+{\frac {v_{\varphi }}{\rho }}{\frac {\partial f}{\partial \varphi }}+v_{z}{\frac {\partial f}{\partial z}}}$

Spherical coordinates: ${\displaystyle {\frac {\mathrm {D} f}{\mathrm {D} t}}:={\frac {\partial f}{\partial t}}+v_{r}{\frac {\partial f}{\partial r}}+{\frac {v_{\varphi }}{r\sin \vartheta }}{\frac {\partial f}{\partial \varphi }}+{\frac {v_{\vartheta }}{r}}{\frac {\partial f}{\partial \vartheta }}}$

Material time derivatives of vectors are put together using them. ${\displaystyle {\tfrac {\mathrm {D} {\vec {f}}}{\mathrm {D} t}}={\tfrac {\mathrm {D} f_{i}}{\mathrm {D} t}}{\hat {e}}_{i}}$

Transport kits

Reynolds transport theorem

→ Main article : Reynolds' transport theorem

Given:

• time ${\displaystyle t}$
• Time-dependent volume with volume shape with ${\displaystyle v}$ ${\displaystyle \mathrm {d} v}$
• The surface of the volume and the outer vectorial surface element ${\displaystyle a}$ ${\displaystyle \mathrm {d} {\vec {a}}}$
• Position vectors ${\displaystyle {\vec {x}}\in v}$
• Speed ​​field: ${\displaystyle {\vec {v}}({\vec {x}},t)}$
• A scalar or vector-valued density function per unit of volume that is transported with the moving points. ${\displaystyle f({\vec {x}},t)}$
• The integral quantity for the volume: ${\displaystyle \int _{v}{\vec {f}}({\vec {x}},t)\,\mathrm {d} v}$

Scalar function  : ${\displaystyle f({\vec {x}},t)}$

${\displaystyle {\begin{array}{rcl}\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\int _{v}f\,\mathrm {d} v&=&\displaystyle \int _{v}{\frac {\partial f}{\partial t}}\,\mathrm {d} v+\int _{a}f({\vec {v}}\cdot \mathrm {d} {\vec {a}})=\int _{v}\left({\frac {\partial f}{\partial t}}+\operatorname {div} (f{\vec {v}})\right)\,\mathrm {d} v\\&=&\displaystyle \int _{v}\left({\frac {\partial f}{\partial t}}+\operatorname {grad} (f)\cdot {\vec {v}}+\operatorname {div} ({\vec {v}})\,f\right)\,\mathrm {d} v=\int _{v}\left({\dot {f}}+\operatorname {div} ({\vec {v}})\,f\right)\,\mathrm {d} v\end{array}}}$

Vector-valued function  : ${\displaystyle {\vec {f}}({\vec {x}},t)}$

${\displaystyle {\begin{array}{rcl}\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\int _{v}{\vec {f}}\,\mathrm {d} v&=&\displaystyle \displaystyle \int _{v}{\frac {\partial {\vec {f}}}{\partial t}}\,\mathrm {d} v+\int _{a}{\vec {f}}({\vec {v}}\cdot \mathrm {d} {\vec {a}})=\int _{v}\left({\frac {\partial {\vec {f}}}{\partial t}}+\operatorname {div} ({\vec {v}}\otimes {\vec {f}})\right)\,\mathrm {d} v\\&=&\displaystyle \int _{v}\left({\frac {\partial {\vec {f}}}{\partial t}}+\operatorname {grad} ({\vec {f}})\cdot {\vec {v}}+\operatorname {div} ({\vec {v}}){\vec {f}}\right)\,\mathrm {d} v=\int _{v}({\dot {\vec {f}}}+\operatorname {div} ({\vec {v}}){\vec {f}})\,\mathrm {d} v\end{array}}}$

Transport set for curve integrals

Given:

• time ${\displaystyle t}$
• Time-dependent curve along which integration takes place with a spatial, vectorial line element in the volume v ${\displaystyle b\colon [0,1)\mapsto v}$ ${\displaystyle \mathrm {d} {\vec {b}}}$
• Position vectors ${\displaystyle {\vec {x}}\in v}$
• Speed ​​field: ${\displaystyle {\vec {v}}({\vec {x}},t)}$
• A scalar or vector-valued field quantity that is transported with the moving points. ${\displaystyle f({\vec {x}},t)}$
• The integral quantity along the way: ${\displaystyle \int _{b}f({\vec {x}},t)\cdot \mathrm {d} {\vec {b}}}$

Scalar function  : ${\displaystyle f({\vec {x}},t)}$

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\oint _{b}f\,\mathrm {d} {\vec {b}}=\oint _{b}({\dot {f}}+f\,\operatorname {grad} {\vec {v}})\cdot \mathrm {d} {\vec {b}}}$

Vector-valued function : ${\displaystyle {\vec {f}}({\vec {x}},t)}$

${\displaystyle {\frac {\mathrm {D} }{\mathrm {D} t}}\oint _{b}{\vec {f}}\cdot \mathrm {d} {\vec {b}}=\oint _{b}({\dot {\vec {f}}}+{\vec {f}}\cdot \operatorname {grad} {\vec {v}})\cdot \mathrm {d} {\vec {b}}}$

Footnotes

1. ↑ In the literature (e.g. Altenbach 2012) the transposed relationship is also used: Then, in order to compare the formulas, and must be swapped.
   ${\displaystyle {\tilde {\operatorname {grad} }}({\vec {f}})=\nabla \otimes {\vec {f}}={\hat {e}}_{i}\otimes {\frac {\partial {\vec {f}}}{\partial x_{i}}}={\frac {\partial f_{j}}{\partial x_{i}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=\operatorname {grad} ({\vec {f}})^{\top }}$
   ${\displaystyle {\tilde {\operatorname {grad} }}({\vec {f}})}$ ${\displaystyle \operatorname {grad} ({\vec {f}})^{\top }}$
2. ↑ In the literature (e.g. Truesdell 1972) the transposed relationship is also used: Then, in order to compare the formulas, and must be swapped.
   ${\displaystyle {\tilde {\operatorname {div} }}(\mathbf {T} )=\mathbf {T} \cdot \nabla ={\frac {\partial \mathbf {T} }{\partial x_{k}}}\cdot {\hat {e}}_{k}=T_{ik,k}{\hat {e}}_{i}=\operatorname {div} (\mathbf {T} ^{\top })}$
   ${\displaystyle {\tilde {\operatorname {div} }}(\mathbf {T} )}$ ${\displaystyle \operatorname {div} (\mathbf {T} ^{\top })}$
3. ↑ In the literature (e.g. Truesdell 1972) the transposed relationship is also used: Then, in order to compare the formulas, and must be swapped.
   ${\displaystyle {\tilde {\operatorname {rot} }}(\mathbf {T} )=\nabla \times \mathbf {T} ^{\top }={\hat {e}}_{k}\times {\frac {\partial \mathbf {T} ^{\top }}{\partial x_{k}}}={\hat {e}}_{k}\times T_{ij,k}{\hat {e}}_{j}\otimes {\hat {e}}_{i}=\operatorname {rot} (\mathbf {T} ^{\top })}$
   ${\displaystyle {\tilde {\operatorname {rot} }}(\mathbf {T} )}$ ${\displaystyle \operatorname {rot} (\mathbf {T} ^{\top })}$
4. ^ R. Greve (2003), p. 111

literature

• H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 .
• M. Bestehorn: hydrodynamics and structure formation . Springer, 2006, ISBN 978-3-540-33796-6 .
• Adolf J. Schwab : Conceptual world of field theory. Practical, clear introduction. Electromagnetic fields, Maxwell's equations, gradient, rotation, divergence . 6th, unchanged edition. Springer, Berlin a. a. 2002, ISBN 3-540-42018-5 .
• Konrad Königsberger : Analysis . revised edition. tape 2. 4 . Springer, Berlin a. a. 2000, ISBN 3-540-43580-8 .
• Ralf Greve: Continuum Mechanics . Springer, 2003, ISBN 3-540-00760-1 .
• C. Truesdell: Solid Mechanics II . In: S. Flügge (Ed.): Handbuch der Physik . Volume VIa / 2. Springer, 1972, ISBN 3-540-05535-5 .