Formelsammlung Tensoralgebra

Diese Formelsammlung f​asst Formeln u​nd Definitionen d​er Tensoralgebra für Tensoren zweiter Stufe i​n der Kontinuumsmechanik zusammen. Es w​ird der dreidimensionale Raum zugrunde gelegt.

Allgemeines

Notation

• Operatoren wie ${\displaystyle \mathrm {I} _{1}}$ werden nicht kursiv geschrieben.
• Buchstaben die als Indizes benutzt werden:
  • ${\displaystyle i,j,k,l,m,n\in \{1,2,3\}}$.
    Ausnahme:
    Die imaginäre Einheit ${\displaystyle \mathrm {i} ^{2}=-1}$ und die #Vektorinvariante ${\displaystyle {\vec {\mathrm {i} }}}$ werden in Abgrenzung zu den Indizes nicht kursiv geschrieben.
  • ${\displaystyle p,q,r,s\in \{1,2,\ldots ,9\}}$
  • ${\displaystyle u,v\in \{1,2,\ldots ,6\}}$
• Alle anderen Buchstaben stehen für reelle Zahlen oder komplexe Zahlen.
• Vektoren:
  • Alle hier verwendeten Vektoren sind geometrische Vektoren im dreidimensionalen euklidischen Vektorraum ${\displaystyle \mathbb {V} }$.
  • Vektoren werden mit Kleinbuchstaben bezeichnet.
    Ausnahme #Dualer axialer Vektor ${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}}$
  • Einheitsvektoren mit Länge eins werden wie in ê mit einem Hut versehen. Die Standardbasis von ${\displaystyle \mathbb {V} }$ ist ê_1,2,3.
  • Vektoren mit unbestimmter Länge werden wie in ${\displaystyle {\vec {a}}}$ mit einem Pfeil versehen.
  • Dreiergruppen von Vektoren wie in ${\displaystyle {\vec {h}}_{1},{\vec {h}}_{2},{\vec {h}}_{3}}$ oder ${\displaystyle {\vec {g}}^{1},{\vec {g}}^{2},{\vec {g}}^{3}}$ bezeichnen eine rechtshändige Basis von ${\displaystyle \mathbb {V} }$.
  • Gleichnamige Basisvektoren mit unterem und oberem Index sind dual zueinander, z. B. ${\displaystyle {\vec {g}}_{1},{\vec {g}}_{2},{\vec {g}}_{3}}$ ist dual zu ${\displaystyle {\vec {g}}^{1},{\vec {g}}^{2},{\vec {g}}^{3}}$.
• Tensoren zweiter Stufe werden wie in A mit fetten Großbuchstaben notiert. Die Menge aller Tensoren wird mit ${\displaystyle {\mathcal {L}}:=\mathrm {Lin} (\mathbb {V} ,\mathbb {V} )}$ bezeichnet. Tensoren höherer Stufe werden mit einer hochgestellten Zahl wie in ${\displaystyle {\stackrel {4}{\mathbf {C} }}}$ geschrieben. Tensoren vierter Stufe sind Elemente der Menge ${\displaystyle {\stackrel {4}{\mathcal {L}}}:=\mathrm {Lin} ({\mathcal {L}},{\mathcal {L}})}$.
• Es gilt die Einstein'sche Summenkonvention ohne Beachtung der Indexstellung.
  • Kommt in einer Formel in einem Produkt ein Index doppelt vor wie in ${\displaystyle c=a_{i}b^{i}}$ wird über diesen Index summiert:
    ${\displaystyle c=a_{i}b^{i}=\sum _{i=1}^{3}a_{i}b^{i}}$.
  • Kommen mehrere Indizes doppelt vor wie in ${\displaystyle c=A_{pq}B_{q}^{p}}$ wird über diese summiert:
    ${\displaystyle c=A_{pq}B_{q}^{p}=\sum _{p=1}^{9}\sum _{q=1}^{9}A_{pq}B_{q}^{p}}$.
  • Ein Index, der nur einfach vorkommt wie ${\displaystyle u}$ in ${\displaystyle a_{u}=A_{uv}b_{v}}$, ist ein freier Index. Die Formel gilt dann für alle Werte der freien Indizes:
    ${\displaystyle a_{u}=A_{uv}b_{v}\quad \leftrightarrow \quad a_{u}=\sum _{v=1}^{6}A_{uv}b_{v}\quad \forall \;u\in \{1,\ldots ,6\}}$.

Reservierte und besondere Symbole

Formelzeichen	Abschnitt in der Formelsammlung	Wikipedia-Artikel
${\displaystyle \mathbf {I,1} }$	#Einheitstensor	Einheitstensor
${\displaystyle \mathbf {Q,R} }$	#Orthogonale Tensoren	Orthogonaler Tensor
${\displaystyle \lambda }$	#Eigenwerte	Eigenwertproblem
${\displaystyle \delta _{ij}}$	#Kronecker-Delta	Kronecker-Delta
${\displaystyle \epsilon _{ijk}}$	#Permutationssymbol	Permutationssymbol
${\displaystyle {\stackrel {3}{\mathbf {E} }}}$	#Fundamentaltensor 3. Stufe	Epsilon-Tensor
${\displaystyle [{\vec {a}}]_{\times }}$	#Axialer Tensor oder Kreuzproduktmatrix	Kreuzprodukt
${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}},\mathbf {A} _{\times }}$	#Dualer axialer Vektor	Kreuzprodukt
${\displaystyle {\vec {\mathrm {i} }}}$	#Vektorinvariante	Vektorinvariante
${\displaystyle \mathrm {i} }$		Imaginäre Einheit

Zeichen für Operatoren

Formelzeichen	Abschnitt in der Formelsammlung	Wikipedia-Artikel
${\displaystyle (\cdot )\cdot (\cdot )}$	Skalarprodukt von Vektoren, #Vektortransformation, #Tensorprodukt	Skalarprodukt
${\displaystyle (\cdot )\times (\cdot )}$	#Kreuzprodukt eines Vektors mit einem Tensor, #Kreuzprodukt von Tensoren	Kreuzprodukt
${\displaystyle (\cdot ):(\cdot )}$	#Skalarprodukt von Tensoren	Frobenius-Skalarprodukt
${\displaystyle (\cdot )\otimes (\cdot )}$	#Dyadisches Produkt	Dyadisches Produkt
${\displaystyle (\cdot )\cdot \!\!\times (\cdot )}$	#Skalarkreuzprodukt von Tensoren
${\displaystyle (\cdot )\times \!\!\times (\cdot )}$	#Doppeltes Kreuzprodukt von Tensoren
${\displaystyle (\cdot )\#(\cdot )}$	#Äußeres Tensorprodukt	Äußeres Tensorprodukt
${\displaystyle \parallel (\cdot )\parallel }$	#Betrag	Frobeniusnorm
${\displaystyle |x|,|{\vec {v}}|,|\mathbf {A} |}$	Betrag der Zahl x oder des Vektors ${\displaystyle {\vec {v}}}$, #Determinante des Tensors A	Determinante

Tensorfunktionen

Formelzeichen	Abschnitt in der Formelsammlung	Wikipedia-Artikel
${\displaystyle \mathrm {Sp,tr,I} _{1}}$	#Spur	Spur (Mathematik), Hauptinvariante
${\displaystyle \mathrm {I} _{2}}$	#Zweite Hauptinvariante	Hauptinvariante
${\displaystyle \mathrm {det,I} _{3},|\mathbf {A} |}$	#Determinante	Determinante, Hauptinvariante
sym	#Symmetrischer Anteil	Symmetrische Matrix
skw, skew	#Schiefsymmetrischer Anteil	Schiefsymmetrische Matrix
adj	#Adjunkte	Adjunkte
cof	#Kofaktor	Minor (Mathematik)#Kofaktormatrix
dev	#Deviator	Deviator, Spannungsdeviator
sph	#Kugelanteil	Kugeltensor

Indizes

Formelzeichen	Abschnitt in der Formelsammlung	Wikipedia-Artikel
${\displaystyle (\cdot )_{ij},(\cdot )^{ij},(\cdot )_{j}^{i}}$	#Tensorkomponenten
${\displaystyle (\cdot )^{\top }}$	#Transposition	Transponierte Matrix
${\displaystyle (\cdot )^{\stackrel {mn}{\top }}}$	Transpositionen von Tensoren vierter Stufe
${\displaystyle (\cdot )^{-1}}$	#Inverse	Inverse Matrix
${\displaystyle (\cdot )^{-\top },(\cdot )^{\top -1}}$	#Transposition der #Inverse
${\displaystyle (\cdot )^{\mathrm {S} }}$	#Symmetrischer Anteil	Symmetrische Matrix
${\displaystyle (\cdot )^{\mathrm {A} }}$	#Schiefsymmetrischer Anteil	Schiefsymmetrische Matrix
${\displaystyle (\cdot )^{\mathrm {D} }}$	#Deviator	Deviator, Spannungsdeviator
${\displaystyle (\cdot )^{\mathrm {K} }}$	#Kugelanteil	Kugeltensor
${\displaystyle {\stackrel {n}{(\cdot )}}}$	Tensor n-ter Stufe
${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}},\mathbf {A} _{\times }}$	#Dualer axialer Vektor	Kreuzprodukt

Mengen

Formelzeichen	Elemente
${\displaystyle \mathbb {R} }$	Reelle Zahlen
${\displaystyle \mathbb {C} }$	Komplexe Zahlen
${\displaystyle \mathbb {V} }$	Vektoren
${\displaystyle {\mathcal {L}}=\mathrm {Lin} (\mathbb {V,V} )}$	Tensoren zweiter Stufe
${\displaystyle {\stackrel {4}{\mathcal {L}}}=\mathrm {Lin} ({\mathcal {L,L}})}$	#Tensoren vierter Stufe

Kronecker-Delta

Siehe auch: Kronecker-Delta

${\displaystyle \delta _{ij}=\delta ^{ij}=\delta _{i}^{j}=\delta _{j}^{i}={\begin{cases}1&\mathrm {falls} \quad i=j\\0&\mathrm {sonst} \end{cases}}}$

Für Summen g​ilt dann z. B.

${\displaystyle v_{i}\delta _{ij}=v_{j}}$

${\displaystyle A_{ij}\delta _{ij}=A_{ii}}$

Dies g​ilt für d​ie anderen Indexgruppen entsprechend.

Permutationssymbol

Siehe auch: Permutationssymbol

${\displaystyle \epsilon _{ijk}={\hat {e}}_{i}\cdot ({\hat {e}}_{j}\times {\hat {e}}_{k})={\begin{cases}1&{\text{falls}}\;(i,j,k)\in \{(1,2,3),(2,3,1),(3,1,2)\}\\-1&{\text{falls}}\;(i,j,k)\in \{(1,3,2),(2,1,3),(3,2,1)\}\\0&{\text{sonst, d.h. bei doppeltem Index}}\end{cases}}}$

${\displaystyle \epsilon _{ijk}\epsilon _{lmn}={\begin{vmatrix}\delta _{il}&\delta _{jl}&\delta _{kl}\\\delta _{im}&\delta _{jm}&\delta _{km}\\\delta _{in}&\delta _{jn}&\delta _{kn}\end{vmatrix}}}$

${\displaystyle \epsilon _{ijk}\epsilon _{klm}=\delta _{il}\delta _{jm}-\delta _{im}\delta _{jl}}$

${\displaystyle \epsilon _{ijk}\epsilon _{jkl}=2\delta _{il}}$

${\displaystyle \epsilon _{ijk}\epsilon _{ijk}=6}$

Kreuzprodukt:

${\displaystyle a_{i}{\hat {e}}_{i}\times b_{j}{\hat {e}}_{j}=\epsilon _{ijk}a_{i}b_{j}{\hat {e}}_{k}=\epsilon _{ijk}a_{j}b_{k}{\hat {e}}_{i}=\epsilon _{ijk}a_{k}b_{i}{\hat {e}}_{j}}$

${\displaystyle \epsilon _{ijk}{\hat {e}}_{k}={\hat {e}}_{i}\times {\hat {e}}_{j}}$

Spaltenvektoren und Matrizen

Siehe auch: Matrix (Mathematik)

Die h​ier verwendeten Vektoren s​ind Spaltenvektoren

${\displaystyle {\vec {a}}=a_{i}{\hat {e}}_{i}={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}}$

Drei Vektoren ${\displaystyle {\vec {a}},{\vec {b}},{\vec {c}}}$ können spaltenweise in einer 3×3-Matrix ${\displaystyle M}$ arrangiert werden:

${\displaystyle M={\begin{pmatrix}{\vec {a}}&{\vec {b}}&{\vec {c}}\end{pmatrix}}={\begin{pmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{pmatrix}}}$

Die Determinante d​er Matrix

${\displaystyle |M|={\begin{vmatrix}{\vec {a}}&{\vec {b}}&{\vec {c}}\end{vmatrix}}}$

ist

• ungleich null, wenn die Spaltenvektoren linear unabhängig sind und
• größer null, wenn die Spaltenvektoren zusätzlich ein Rechtssystem bilden.

Also gewährleistet ${\displaystyle {\begin{vmatrix}{\vec {a}}&{\vec {b}}&{\vec {c}}\end{vmatrix}}>0}$, dass die Vektoren ${\displaystyle {\vec {a}},{\vec {b}},{\vec {c}}}$ eine rechtshändige Basis bilden.

Die Spaltenvektoren bilden e​ine Orthonormalbasis, wenn

${\displaystyle M^{\top }M={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}$

worin ${\displaystyle M^{\top }}$ die transponierte Matrix ist. Bei der hier vorausgesetzten Rechtshändigkeit gilt dann zusätzlich ${\displaystyle |M|=+1}$.

Basis und Duale Basis

Siehe auch: Vektorraumbasis

Basisvektoren ${\displaystyle {\vec {g}}_{1},{\vec {g}}_{2},{\vec {g}}_{3}}$

Duale Basisvektoren ${\displaystyle {\vec {g}}^{1},{\vec {g}}^{2},{\vec {g}}^{3}}$

Beziehungen zwischen d​en Basisvektoren

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

${\displaystyle {\vec {g}}^{1}={\frac {{\vec {g}}_{2}\times {\vec {g}}_{3}}{({\vec {g}}_{1},{\vec {g}}_{2},{\vec {g}}_{3})}},\quad g^{2}={\frac {{\vec {g}}_{3}\times {\vec {g}}_{1}}{({\vec {g}}_{1},{\vec {g}}_{2},{\vec {g}}_{3})}},\quad g^{3}={\frac {{\vec {g}}_{1}\times {\vec {g}}_{2}}{({\vec {g}}_{1},{\vec {g}}_{2},{\vec {g}}_{3})}}}$

${\displaystyle {\vec {g}}_{1}={\frac {{\vec {g}}^{2}\times {\vec {g}}^{3}}{({\vec {g}}^{1},{\vec {g}}^{2},{\vec {g}}^{3})}},\quad g_{2}={\frac {{\vec {g}}^{3}\times {\vec {g}}^{1}}{({\vec {g}}^{1},{\vec {g}}^{2},{\vec {g}}^{3})}},\quad g_{3}={\frac {{\vec {g}}^{1}\times {\vec {g}}^{2}}{({\vec {g}}^{1},{\vec {g}}^{2},{\vec {g}}^{3})}}}$

mit d​em Spatprodukt

${\displaystyle ({\vec {a}},{\vec {b}},{\vec {c}}):={\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})={\vec {c}}\cdot ({\vec {a}}\times {\vec {b}})={\vec {b}}\cdot ({\vec {c}}\times {\vec {a}})={\begin{vmatrix}{\vec {a}}&{\vec {b}}&{\vec {c}}\end{vmatrix}}}$

Trägt man die Basisvektoren spaltenweise in eine Matrix ein, dann finden sich die dualen Basisvektoren in den Zeilen der Inversen oder den Spalten der #transponiert #Inversen ${\displaystyle ()^{\top -1}}$:

${\displaystyle {\begin{pmatrix}{\vec {g}}^{1}&{\vec {g}}^{2}&{\vec {g}}^{3}\end{pmatrix}}={\begin{pmatrix}{\vec {g}}_{1}&{\vec {g}}_{2}&{\vec {g}}_{3}\end{pmatrix}}^{\top -1}}$

In der Standardbasis wie in jeder Orthonormalbasis sind die Basisvektoren ${\displaystyle {\hat {e}}_{1},{\hat {e}}_{2},{\hat {e}}_{3}}$ zu sich selbst dual:

${\displaystyle {\hat {e}}_{i}={\hat {e}}^{i}}$

Berechnung von Vektorkomponenten

${\displaystyle {\vec {v}}=v_{i}{\hat {e}}_{i}\quad \rightarrow \;v_{i}={\vec {v}}\cdot {\hat {e}}_{i}}$

${\displaystyle {\vec {v}}=v^{i}{\vec {g}}_{i}\quad \rightarrow \;v^{i}={\vec {v}}\cdot {\vec {g}}^{i}}$

${\displaystyle {\vec {v}}=v_{i}{\vec {g}}^{i}\quad \rightarrow \;v_{i}={\vec {v}}\cdot {\vec {g}}_{i}}$

Beziehung zwischen den Skalarprodukten der Basisvektoren

${\displaystyle ({\vec {g}}_{i}\cdot {\vec {g}}_{k})({\vec {g}}^{k}\cdot {\vec {g}}^{j})={\vec {g}}_{i}\cdot ({\vec {g}}^{j}\cdot {\vec {g}}^{k}){\vec {g}}_{k}={\vec {g}}_{i}\cdot {\vec {g}}^{j}=\delta _{i}^{j}}$

Wechsel der Basis bei Vektoren

Siehe auch: Koordinatentransformation

Wechsel von

Basis ${\displaystyle {\vec {g}}^{1},{\vec {g}}^{2},{\vec {g}}^{3}}$ mit dualer Basis ${\displaystyle {\vec {g}}_{1},{\vec {g}}_{2},{\vec {g}}_{3}}$

nach

Basis ${\displaystyle {\vec {h}}^{1},{\vec {h}}^{2},{\vec {h}}^{3}}$ mit dualer Basis ${\displaystyle {\vec {h}}_{1},{\vec {h}}_{2},{\vec {h}}_{3}}$:

${\displaystyle {\vec {v}}=v_{i}\,{\vec {g}}^{i}=v_{i}^{\ast }\,{\vec {h}}^{i}\quad \rightarrow \;v_{i}^{\ast }=({\vec {h}}_{i}\cdot {\vec {g}}^{j})v_{j}}$

Matrizengleichung:

${\displaystyle {\begin{aligned}{\begin{pmatrix}v_{1}^{\ast }\\v_{2}^{\ast }\\v_{3}^{\ast }\end{pmatrix}}=&{\begin{pmatrix}{\vec {h}}_{1}\cdot {\vec {g}}^{1}&{\vec {h}}_{1}\cdot {\vec {g}}^{2}&{\vec {h}}_{1}\cdot {\vec {g}}^{3}\\{\vec {h}}_{2}\cdot {\vec {g}}^{1}&{\vec {h}}_{2}\cdot {\vec {g}}^{2}&{\vec {h}}_{2}\cdot {\vec {g}}^{3}\\{\vec {h}}_{3}\cdot {\vec {g}}^{1}&{\vec {h}}_{3}\cdot {\vec {g}}^{2}&{\vec {h}}_{3}\cdot {\vec {g}}^{3}\end{pmatrix}}{\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\end{pmatrix}}\\=&{\begin{pmatrix}{\vec {h}}_{1}&{\vec {h}}_{2}&{\vec {h}}_{3}\end{pmatrix}}^{\top }{\begin{pmatrix}{\vec {g}}^{1}&{\vec {g}}^{2}&{\vec {g}}^{3}\end{pmatrix}}{\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\end{pmatrix}}\end{aligned}}}$

Dyadisches Produkt

Siehe auch: Dyadisches Produkt

Die grundlegenden Eigenschaften d​es dyadischen Produkts „⊗“ sind:

Abbildung ${\displaystyle \mathbb {V} \times \mathbb {V} \to {\mathcal {L}}}$

${\displaystyle {\vec {a}}\otimes {\vec {g}}=\mathbf {T} \in {\mathcal {L}}}$

Multiplikation m​it einem Skalar:

${\displaystyle x({\vec {a}}\otimes {\vec {g}})=(x{\vec {a}})\otimes {\vec {g}}={\vec {a}}\otimes (x{\vec {g}})=x{\vec {a}}\otimes {\vec {g}}}$

Distributivität:

${\displaystyle (x+y){\vec {a}}\otimes {\vec {g}}=x{\vec {a}}\otimes {\vec {g}}+y{\vec {a}}\otimes {\vec {g}}}$

${\displaystyle ({\vec {a}}+{\vec {b}})\otimes {\vec {g}}={\vec {a}}\otimes {\vec {g}}+{\vec {b}}\otimes {\vec {g}}}$

${\displaystyle {\vec {a}}\otimes ({\vec {g}}+{\vec {h}})={\vec {a}}\otimes {\vec {g}}+{\vec {a}}\otimes {\vec {h}}}$

Skalarprodukt:

${\displaystyle ({\vec {a}}\otimes {\vec {g}}):({\vec {b}}\otimes {\vec {h}})=({\vec {a}}\cdot {\vec {b}})({\vec {g}}\cdot {\vec {h}})}$

Weitere Eigenschaften v​on Dyaden s​iehe #Dyade u​nd den folgenden Abschnitt.

Tensoren als Elemente eines Vektorraumes

Siehe auch: Dyadisches Produkt und Tensor

Durch die Eigenschaften des dyadischen Produktes wird ${\displaystyle {\mathcal {L}}}$ zu einem euklidischen Vektorraum und entsprechend kann jeder Tensor komponentenweise bezüglich einer Basis von ${\displaystyle {\mathcal {L}}}$ dargestellt werden:

${\displaystyle \mathbf {A} \in {\mathcal {L}}\rightarrow \mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=A^{ij}{\vec {a}}_{i}\otimes {\vec {g}}_{j}}$ mit Komponenten ${\displaystyle A_{ij},A^{ij}\in \mathbb {R} }$.

Die Dyaden ${\displaystyle \{{\hat {e}}_{i}\otimes {\hat {e}}_{j}|i,j=1,2,3\}}$ und ${\displaystyle \{{\vec {a}}_{i}\otimes {\vec {g}}_{j}|i,j=1,2,3\}}$ bilden Basissysteme von ${\displaystyle {\mathcal {L}}}$.

Transposition

Siehe auch: Transponierte Matrix

Abbildung ${\displaystyle {\mathcal {L}}\to {\mathcal {L}}}$

${\displaystyle ({\vec {a}}\otimes {\vec {g}})^{\top }:={\vec {g}}\otimes {\vec {a}}}$

${\displaystyle (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})^{\top }=A_{ij}({\hat {e}}_{j}\otimes {\hat {e}}_{i})=A_{ji}({\hat {e}}_{i}\otimes {\hat {e}}_{j})}$

${\displaystyle (A^{ij}{\vec {a}}_{i}\otimes {\vec {g}}_{j})^{\top }=A^{ij}({\vec {g}}_{j}\otimes {\vec {a}}_{i})=A^{ji}({\vec {g}}_{i}\otimes {\vec {a}}_{j})}$

${\displaystyle \left(\mathbf {A} ^{\top }\right)^{\top }=\mathbf {A} }$

${\displaystyle (\mathbf {A+B} )^{\top }=\mathbf {A} ^{\top }+\mathbf {B} ^{\top }}$

${\displaystyle (\mathbf {A\cdot B} )^{\top }=\mathbf {B} ^{\top }\cdot \mathbf {A} ^{\top }}$

Vektortransformation

Abbildung ${\displaystyle {\mathcal {L}}\times \mathbb {V} \to \mathbb {V} }$ oder ${\displaystyle \mathbb {V} \times {\mathcal {L}}\to \mathbb {V} }$

Dyaden:

${\displaystyle ({\vec {a}}\otimes {\vec {g}})\cdot {\vec {h}}:=({\vec {g}}\cdot {\vec {h}}){\vec {a}}}$

${\displaystyle {\vec {b}}\cdot ({\vec {a}}\otimes {\vec {g}}):=({\vec {a}}\cdot {\vec {b}}){\vec {g}}}$

${\displaystyle ({\vec {a}}\otimes {\vec {g}})\cdot {\vec {h}}={\vec {h}}\cdot ({\vec {a}}\otimes {\vec {g}})^{\top }}$

${\displaystyle {\vec {b}}\cdot ({\vec {a}}\otimes {\vec {g}})=({\vec {a}}\otimes {\vec {g}})^{\top }\cdot {\vec {b}}}$

Allgemeine Tensoren:

${\displaystyle A_{ij}({\hat {e}}_{i}\otimes {\hat {e}}_{j})\cdot {\vec {v}}=A_{ij}({\vec {v}}\cdot {\hat {e}}_{j}){\hat {e}}_{i}}$

${\displaystyle A^{ij}({\vec {a}}_{i}\otimes {\vec {g}}_{j})\cdot {\vec {v}}=A^{ij}({\vec {v}}\cdot {\vec {g}}_{j}){\vec {a}}_{i}}$

${\displaystyle {\vec {v}}\cdot A_{ij}({\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{ij}({\vec {v}}\cdot {\hat {e}}_{i}){\hat {e}}_{j}}$

${\displaystyle {\vec {v}}\cdot A^{ij}({\vec {a}}_{i}\otimes {\vec {g}}_{j})=A^{ij}({\vec {v}}\cdot {\hat {a}}_{i}){\vec {g}}_{j}}$

Symbolisch:

${\displaystyle \mathbf {A} \cdot {\vec {v}}={\vec {v}}\cdot \mathbf {A} ^{\top }}$

${\displaystyle {\vec {v}}\cdot \mathbf {A} =\mathbf {A} ^{\top }\cdot {\vec {v}}}$

Tensorprodukt

Siehe auch: Matrizenmultiplikation

Abbildung ${\displaystyle {\mathcal {L}}\times {\mathcal {L}}\to {\mathcal {L}}}$

${\displaystyle ({\vec {a}}\otimes {\vec {g}})\cdot ({\vec {h}}\otimes {\vec {u}}):=({\vec {g}}\cdot {\vec {h}}){\vec {a}}\otimes {\vec {u}}}$

${\displaystyle ({\vec {a}}\otimes {\vec {g}})\cdot \mathbf {A} ={\vec {a}}\otimes ({\vec {g}}\cdot \mathbf {A} )={\vec {a}}\otimes {\vec {g}}\cdot \mathbf {A} ={\vec {a}}\otimes (\mathbf {A} ^{\top }\cdot {\vec {g}})}$

${\displaystyle \mathbf {A} \cdot ({\vec {a}}\otimes {\vec {g}})=(\mathbf {A} \cdot {\vec {a}})\otimes {\vec {g}}=\mathbf {A} \cdot {\vec {a}}\otimes {\vec {g}}}$

${\displaystyle (A_{ik}{\hat {e}}_{i}\otimes {\hat {e}}_{k})\cdot (B_{lj}{\hat {e}}_{l}\otimes {\hat {e}}_{j})=A_{ik}B_{kj}{\hat {e}}_{i}\otimes {\hat {e}}_{j}}$

${\displaystyle \left(A^{ij}{\vec {a}}_{i}\otimes {\vec {g}}_{j}\right)\cdot \left(B^{kl}{\vec {h}}_{k}\otimes {\vec {u}}_{l}\right)=A^{ij}({\vec {g}}_{j}\cdot {\vec {h}}_{k})B^{kl}{\vec {a}}_{i}\otimes {\vec {u}}_{l}}$

Skalarprodukt von Tensoren

Siehe auch: Frobenius-Skalarprodukt

Abbildung ${\displaystyle {\mathcal {L}}\times {\mathcal {L}}\to \mathbb {R} }$

Definition über d​ie #Spur:

${\displaystyle ({\vec {a}}\otimes {\vec {g}}):({\vec {b}}\otimes {\vec {h}}):=\mathrm {Sp} (({\vec {a}}\otimes {\vec {g}})^{\top }\cdot ({\vec {b}}\otimes {\vec {h}}))=({\vec {a}}\cdot {\vec {b}})({\vec {g}}\cdot {\vec {h}})}$

${\displaystyle \mathbf {A}$ :\mathbf {B} :=\mathrm {Sp} (\mathbf {A} ^{\top }\cdot \mathbf {B} )}

Eigenschaften:

${\displaystyle \mathbf {A}$ :\mathbf {B} =\mathbf {B} :\mathbf {A} =\mathbf {A} ^{\top }:\mathbf {B} ^{\top }=\mathbf {B} ^{\top }:\mathbf {A} ^{\top }}

${\displaystyle \mathbf {A} ^{\top }:\mathbf {B} =\mathbf {A}$ :\mathbf {B} ^{\top }}

${\displaystyle \mathbf {A}$ :(\mathbf {B\cdot C} )=(\mathbf {B} ^{\top }\cdot \mathbf {A} ):\mathbf {C} =(\mathbf {A\cdot C} ^{\top }):\mathbf {B} }

${\displaystyle (\mathbf {A\cdot B} ):\mathbf {C} =\mathbf {B}$ :(\mathbf {A} ^{\top }\cdot \mathbf {C} )=\mathbf {A} :(\mathbf {C\cdot B} ^{\top })}

${\displaystyle ({\vec {u}}\otimes {\vec {v}}):\mathbf {A} ={\vec {u}}\cdot \mathbf {A} \cdot {\vec {v}}}$

Kreuzprodukt eines Vektors mit einem Tensor

Abbildung ${\displaystyle \mathbb {V} \times {\mathcal {L}}\to {\mathcal {L}}}$ oder ${\displaystyle {\mathcal {L}}\times \mathbb {V} \to {\mathcal {L}}}$

Dyaden:

${\displaystyle {\vec {a}}\times ({\vec {b}}\otimes {\vec {g}})=({\vec {a}}\times {\vec {b}})\otimes {\vec {g}}={\vec {a}}\times {\vec {b}}\otimes {\vec {g}}}$

${\displaystyle ({\vec {a}}\otimes {\vec {g}})\times {\vec {h}}={\vec {a}}\otimes ({\vec {g}}\times {\vec {h}})={\vec {a}}\otimes {\vec {g}}\times {\vec {h}}}$

${\displaystyle {\vec {a}}\times {\vec {b}}\otimes {\vec {g}}=-[({\vec {b}}\otimes {\vec {g}})^{\top }\times {\vec {a}}]^{\top }}$

${\displaystyle {\vec {a}}\otimes {\vec {g}}\times {\vec {h}}=-[{\vec {h}}\times ({\vec {a}}\otimes {\vec {g}})^{\top }]^{\top }}$

${\displaystyle a_{j}{\hat {e}}_{j}\times (A_{kl}{\hat {e}}_{k}\otimes {\hat {e}}_{l})=a_{j}A_{kl}({\hat {e}}_{j}\times {\hat {e}}_{k})\otimes {\hat {e}}_{l}=\epsilon _{ijk}a_{j}A_{kl}{\hat {e}}_{i}\otimes {\hat {e}}_{l}}$

${\displaystyle (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})\times a_{k}{\hat {e}}_{k}=A_{ij}a_{k}{\hat {e}}_{i}\otimes ({\hat {e}}_{j}\times {\hat {e}}_{k})=\epsilon _{jkl}A_{ij}a_{k}{\hat {e}}_{i}\otimes {\hat {e}}_{l}}$

Allgemeine Tensoren:

${\displaystyle ({\vec {a}}\times \mathbf {A} )\cdot {\vec {g}}:={\vec {a}}\times (\mathbf {A} \cdot {\vec {g}})={\vec {a}}\times ({\vec {g}}\cdot \mathbf {A} ^{\top })}$

${\displaystyle {\vec {b}}\cdot ({\vec {a}}\times \mathbf {A} ):=({\vec {b}}\times {\vec {a}})\cdot \mathbf {A} }$

${\displaystyle {\vec {g}}\cdot (\mathbf {A} \times {\vec {a}}):=({\vec {g}}\cdot \mathbf {A} )\times {\vec {a}}=(\mathbf {A} ^{\top }\cdot {\vec {g}})\times {\vec {a}}}$

${\displaystyle (\mathbf {A} \times {\vec {a}})\cdot {\vec {b}}=\mathbf {A} \cdot ({\vec {a}}\times {\vec {b}})}$

${\displaystyle {\vec {a}}\times \mathbf {A} =-\left(\mathbf {A} ^{\top }\times {\vec {a}}\right)^{\top }}$

${\displaystyle \mathbf {A} \times {\vec {a}}=-\left({\vec {a}}\times \mathbf {A} ^{\top }\right)^{\top }}$

Symmetrische Tensoren: ${\displaystyle {\vec {a}}\times \mathbf {A} ^{\mathrm {S} }=-\left(\mathbf {A} ^{\mathrm {S} }\times {\vec {a}}\right)^{\top }}$

Insbesondere Kugeltensoren: ${\displaystyle {\vec {a}}\times \mathbf {A} ^{\mathrm {K} }=\mathbf {A} ^{\mathrm {K} }\times {\vec {a}}=-({\vec {a}}\times \mathbf {A} ^{\mathrm {K} })^{\top }}$

Schiefsymmetrische Tensoren: ${\displaystyle {\vec {a}}\times \mathbf {A} ^{\mathrm {A} }=\left(\mathbf {A} ^{\mathrm {A} }\times {\vec {a}}\right)^{\top }}$

#Axialer Tensor o​der Kreuzproduktmatrix m​it dem #Einheitstensor:

${\displaystyle ({\vec {a}}\times \mathbf {1} )\cdot {\vec {g}}={\vec {a}}\cdot ({\vec {g}}\times \mathbf {1} )={\vec {a}}\cdot (\mathbf {1} \times {\vec {g}})={\vec {a}}\times {\vec {g}}}$

Mehrfach:

${\displaystyle ({\vec {a}}\times ({\vec {b}}\times \mathbf {A} ))\cdot {\vec {g}}={\vec {a}}\times ({\vec {b}}\times (\mathbf {A} \cdot {\vec {g}}))=({\vec {a}}\cdot \mathbf {A} \cdot {\vec {g}}){\vec {b}}-({\vec {a}}\cdot {\vec {b}})\mathbf {A} \cdot {\vec {g}}}$

${\displaystyle {\vec {a}}\times ({\vec {b}}\times \mathbf {A} )={\vec {b}}\otimes {\vec {a}}\cdot \mathbf {A} -({\vec {a}}\cdot {\vec {b}})\mathbf {A} }$

Meistens i​st aber:

${\displaystyle (\mathbf {A} \cdot {\vec {a}})\times {\vec {g}}\neq \mathbf {A} \cdot ({\vec {a}}\times {\vec {g}})=(\mathbf {A} \times {\vec {a}})\cdot {\vec {g}}}$

${\displaystyle {\vec {a}}\times ({\vec {g}}\cdot \mathbf {A} )\neq ({\vec {a}}\times {\vec {g}})\cdot \mathbf {A} ={\vec {a}}\cdot ({\vec {g}}\times \mathbf {A} )}$

Kreuzprodukt von Tensoren

Abbildung ${\displaystyle {\mathcal {L}}\times {\mathcal {L}}\to \mathbb {V} }$

${\displaystyle \mathbf {A\times B} ={\stackrel {3}{\mathbf {E} }}:(\mathbf {A\cdot B} ^{\top })=-{\stackrel {3}{\mathbf {E} }}:(\mathbf {B\cdot A} ^{\top })=-\mathbf {B\times A} \in \mathbb {V} }$

mit #Fundamentaltensor 3. Stufe ${\displaystyle {\stackrel {3}{\mathbf {E} }}}$.

${\displaystyle ({\vec {a}}\otimes {\vec {g}})\times ({\vec {b}}\otimes {\vec {h}})=({\vec {g}}\cdot {\vec {h}}){\vec {a}}\times {\vec {b}}}$

${\displaystyle {\begin{aligned}&A_{ik}({\hat {e}}_{i}\otimes {\hat {e}}_{k})\times [B_{jl}({\hat {e}}_{j}\otimes {\hat {e}}_{l})]=A_{ik}B_{jk}({\hat {e}}_{i}\times {\hat {e}}_{j})=\ldots \\&\ldots ={\begin{pmatrix}A_{21}B_{31}-A_{31}B_{21}+A_{22}B_{32}-A_{32}B_{22}+A_{23}B_{33}-A_{33}B_{23}\\A_{31}B_{11}-A_{11}B_{31}+A_{32}B_{12}-A_{12}B_{32}+A_{33}B_{13}-A_{13}B_{33}\\A_{11}B_{21}-A_{21}B_{11}+A_{12}B_{22}-A_{22}B_{12}+A_{13}B_{23}-A_{23}B_{13}\end{pmatrix}}\end{aligned}}}$

Zusammenhang m​it #Dualer axialer Vektor u​nd #Vektorinvariante:

${\displaystyle \mathbf {A\times B} =-2{\stackrel {A}{\overrightarrow {\mathbf {A\cdot B} ^{\top }}}}={\vec {\mathrm {i} }}(\mathbf {A\cdot B} ^{\top })}$

Mit #Einheitstensor:

${\displaystyle \mathbf {1\times A} =2{\stackrel {A}{\overrightarrow {\mathbf {A} }}}=-{\vec {\mathrm {i} }}(\mathbf {A} )}$

Mehrfachprodukte:

${\displaystyle (\mathbf {A\cdot B} )\times \mathbf {C} =\mathbf {A} \times (\mathbf {C\cdot B} ^{\top })}$

${\displaystyle \mathbf {A} \times (\mathbf {B\cdot C} )=(\mathbf {A\cdot C} ^{\top })\times \mathbf {B} }$

Zusammenhang m​it dem #Skalarkreuzprodukt v​on Tensoren:

${\displaystyle \mathbf {A\times B} =\mathbf {A} \cdot \!\!\times (\mathbf {B} ^{\top })}$

Skalarkreuzprodukt von Tensoren

Abbildung ${\displaystyle {\mathcal {L}}\times {\mathcal {L}}\to \mathbb {V} }$

${\displaystyle ({\vec {a}}\otimes {\vec {g}})\cdot \!\!\times ({\vec {h}}\otimes {\vec {u}})=-({\vec {u}}\otimes {\vec {h}})\cdot \!\!\times ({\vec {g}}\otimes {\vec {a}}):=({\vec {g}}\cdot {\vec {h}}){\vec {a}}\times {\vec {u}}}$

${\displaystyle {\begin{aligned}&A_{ik}({\hat {e}}_{i}\otimes {\hat {e}}_{k})\cdot \!\!\times [B_{lj}({\hat {e}}_{l}\otimes {\hat {e}}_{j})]=A_{ik}B_{kj}({\hat {e}}_{i}\times {\hat {e}}_{j})=\ldots \\&\ldots ={\begin{pmatrix}A_{21}B_{13}-A_{31}B_{12}+A_{22}B_{23}-A_{32}B_{22}+A_{23}B_{33}-A_{33}B_{32}\\A_{31}B_{11}-A_{11}B_{13}+A_{32}B_{21}-A_{12}B_{23}+A_{33}B_{31}-A_{13}B_{33}\\A_{11}B_{12}-A_{21}B_{11}+A_{12}B_{22}-A_{22}B_{21}+A_{13}B_{32}-A_{23}B_{31}\end{pmatrix}}\end{aligned}}}$

Das Skalarkreuzprodukt m​it dem #Einheitstensor vertauscht d​as dyadische Produkt d​urch das Kreuzprodukt:

${\displaystyle \mathbf {1} \cdot \!\!\times ({\vec {a}}\otimes {\vec {b}})={\vec {a}}\times {\vec {b}}}$

Allgemein:

${\displaystyle \mathbf {A} \cdot \!\!\times \mathbf {B} =-(\mathbf {B} ^{\top })\cdot \!\!\times (\mathbf {A} ^{\top })}$

${\displaystyle \mathbf {A} \cdot \!\!\times (\mathbf {B\cdot C} )=(\mathbf {A\cdot B} )\cdot \!\!\times \mathbf {C} }$

${\displaystyle (\mathbf {A\cdot B} )\cdot \!\!\times \mathbf {C} =\mathbf {A} \cdot \!\!\times (\mathbf {B\cdot C} )}$

Zusammenhang m​it dem #Kreuzprodukt v​on Tensoren:

${\displaystyle \mathbf {S} \cdot \!\!\times \mathbf {T} =\mathbf {S\times (T^{\top })} }$

Zusammenhang m​it #Vektorinvariante u​nd #Dualer axialer Vektor:

${\displaystyle \mathbf {A} \cdot \!\!\times \mathbf {B} ={\vec {\mathrm {i} }}(\mathbf {A} \cdot \mathbf {B} )=-2{\stackrel {A}{\overrightarrow {\mathbf {A} \cdot \mathbf {B} }}}}$

Doppeltes Kreuzprodukt von Tensoren

Siehe a​uch #Äußeres Tensorprodukt #

Abbildung ${\displaystyle {\mathcal {L}}\times {\mathcal {L}}\to {\mathcal {L}}}$

${\displaystyle ({\vec {a}}\otimes {\vec {g}})\times \!\!\times ({\vec {h}}\otimes {\vec {b}}):=({\vec {g}}\times {\vec {h}})\otimes ({\vec {a}}\times {\vec {b}})=({\vec {g}}\otimes {\vec {a}})\#({\vec {h}}\otimes {\vec {b}})}$

${\displaystyle A_{ij}({\hat {e}}_{i}\otimes {\hat {e}}_{j})\times \!\!\times [B_{kl}({\hat {e}}_{k}\otimes {\hat {e}}_{l})]:=A_{ij}B_{kl}({\hat {e}}_{j}\times {\hat {e}}_{k})\otimes ({\hat {e}}_{i}\times {\hat {e}}_{l})}$

${\displaystyle \mathbf {A} \times \!\!\times \mathbf {B} =\mathbf {A} ^{\top }\#\mathbf {B} }$

Äußeres Tensorprodukt

Siehe auch: Äußeres Tensorprodukt

Abbildung ${\displaystyle {\mathcal {L}}\times {\mathcal {L}}\to {\mathcal {L}}}$

${\displaystyle ({\vec {a}}\otimes {\vec {g}})\#({\vec {b}}\otimes {\vec {h}}):=({\vec {a}}\times {\vec {b}})\otimes ({\vec {g}}\times {\vec {h}})=({\vec {g}}\otimes {\vec {a}})\times \!\!\times ({\vec {b}}\otimes {\vec {h}})}$

${\displaystyle {\begin{aligned}&(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})\#(B_{kl}{\hat {e}}_{k}\otimes {\hat {e}}_{l})=A_{ij}B_{kl}({\hat {e}}_{i}\times {\hat {e}}_{k})\otimes ({\hat {e}}_{j}\times {\hat {e}}_{l})\\&\qquad \qquad \qquad \qquad \qquad \quad \;\;\;=\epsilon _{ikm}\epsilon _{jln}A_{ij}B_{kl}{\hat {e}}_{m}\otimes {\hat {e}}_{n}\end{aligned}}}$

Mit d​er Formel für d​as Produkt zweier #Permutationssymbole:

${\displaystyle {\begin{aligned}\mathbf {A} \#\mathbf {B} =&[\mathrm {Sp} (\mathbf {A} )\mathrm {Sp} (\mathbf {B} )-\mathrm {Sp} (\mathbf {A\cdot B} )]\mathbf {1} \\&+[\mathbf {A\cdot B} +\mathbf {B\cdot A} -\mathrm {Sp} (\mathbf {A} )\mathbf {B} -\mathrm {Sp} (\mathbf {B} )\mathbf {A} ]^{\top }\end{aligned}}}$

Grundlegende Eigenschaften:

${\displaystyle \mathbf {A} \#\mathbf {B} =\mathbf {B} \#\mathbf {A} =(\mathbf {A} ^{\top }\#\mathbf {B} ^{\top })^{\top }}$

${\displaystyle (\mathbf {A+B} )\#\mathbf {C} =\mathbf {A} \#\mathbf {C} +\mathbf {B} \#\mathbf {C} }$

${\displaystyle \mathbf {A} \#(\mathbf {B+C} )=\mathbf {A} \#\mathbf {B} +\mathbf {A} \#\mathbf {C} }$

Kreuzprodukt u​nd #Kofaktor:

${\displaystyle (\mathbf {A} \#\mathbf {B} )\cdot ({\vec {u}}\times {\vec {v}})=(\mathbf {A} \cdot {\vec {u}})\times (\mathbf {B} \cdot {\vec {v}})-(\mathbf {A} \cdot {\vec {v}})\times (\mathbf {B} \cdot {\vec {u}})}$

${\displaystyle {\frac {1}{2}}(\mathbf {A} \#\mathbf {A} )\cdot ({\vec {u}}\times {\vec {v}})=\mathrm {cof} (\mathbf {A} )\cdot ({\vec {u}}\times {\vec {v}})=(\mathbf {A} \cdot {\vec {u}})\times (\mathbf {A} \cdot {\vec {v}})}$

#Hauptinvarianten:

${\displaystyle {\frac {1}{2}}(\mathbf {A\#1} ):\mathbf {1} =\mathrm {Sp} (\mathbf {A} )}$

${\displaystyle {\frac {1}{2}}(\mathbf {A\#A} ):\mathbf {1} =\mathrm {I} _{2}(\mathbf {A} )}$

${\displaystyle {\frac {1}{6}}(\mathbf {A\#A} ):\mathbf {A} =\det(\mathbf {A} )}$

Weitere Eigenschaften:

${\displaystyle \mathbf {1} \#\mathbf {1} =2\,\mathbf {1} }$

${\displaystyle \mathbf {A} \#\mathbf {1} =\mathrm {Sp} (\mathbf {A} )\mathbf {1} -\mathbf {A} ^{\top }}$

${\displaystyle (\mathbf {A} \#\mathbf {B} ):\mathbf {C} =(\mathbf {B} \#\mathbf {C} ):\mathbf {A} =(\mathbf {C} \#\mathbf {A} ):\mathbf {B} }$

${\displaystyle \mathrm {Sp} (\mathbf {A} \#\mathbf {B} )=\mathrm {Sp} (\mathbf {A} )\mathrm {Sp} (\mathbf {B} )-\mathrm {Sp} (\mathbf {A\cdot B} )}$

${\displaystyle (\mathbf {A} \#\mathbf {B} )\cdot (\mathbf {C} \#\mathbf {D} )=(\mathbf {A\cdot C} )\#(\mathbf {B\cdot D} )+(\mathbf {A\cdot D} )\#(\mathbf {B\cdot C} )}$

Aber meistens:

${\displaystyle (\mathbf {A} \#\mathbf {B} )\#\mathbf {C} \neq \mathbf {A} \#(\mathbf {B} \#\mathbf {C} )}$

.

Produkte von Tensoren, Dyaden und Vektoren

${\displaystyle \mathbf {A} \cdot ({\vec {a}}\otimes {\vec {g}})=(\mathbf {A} \cdot {\vec {a}})\otimes {\vec {g}}}$

${\displaystyle {\vec {a}}\otimes (\mathbf {A} \cdot {\vec {g}})=({\vec {a}}\otimes {\vec {g}})\cdot \mathbf {A} ^{\top }}$

${\displaystyle {\vec {a}}\cdot \mathbf {A} \cdot {\vec {g}}=\mathbf {A}$ :({\vec {a}}\otimes {\vec {g}})}

Spatprodukt u​nd #Determinante e​ines Tensors:

${\displaystyle (\mathbf {A} \cdot {\vec {a}})\cdot [(\mathbf {A} \cdot {\vec {b}})\times (\mathbf {A} \cdot {\vec {c}})]=\mathrm {det} (\mathbf {A} )\;{\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})}$

Kreuzprodukt u​nd #Kofaktor:

${\displaystyle (\mathbf {A} \cdot {\vec {a}})\times (\mathbf {A} \cdot {\vec {b}})=\mathrm {cof} (\mathbf {A} )\cdot ({\vec {a}}\times {\vec {b}})}$

${\displaystyle \mathbf {A} ^{\top }\cdot [(\mathbf {A} \cdot {\vec {a}})\times (\mathbf {A} \cdot {\vec {b}})]=\mathrm {det} (\mathbf {A} )\;{\vec {a}}\times {\vec {b}}}$

#Axialer Tensor o​der Kreuzproduktmatrix, #Kreuzprodukt v​on Tensoren, #Skalarkreuzprodukt v​on Tensoren, #Dualer axialer Vektor u​nd #Vektorinvariante:

${\displaystyle ({\vec {u}}\times \mathbf {1} )\cdot {\vec {v}}=({\vec {u}}\otimes {\vec {v}})\times \mathbf {1} =({\vec {u}}\otimes {\vec {v}})\cdot \!\!\times \mathbf {1} ={\stackrel {A}{\overrightarrow {({\vec {u}}\times {\vec {v}})\times \mathbf {1} }}}={\vec {\mathrm {i} }}({\vec {u}}\otimes {\vec {v}})={\vec {u}}\times {\vec {v}}}$

Tensorkomponenten

${\displaystyle \mathbf {A} =A_{ij}\,{\hat {e}}_{i}\otimes {\hat {e}}_{j}={\begin{pmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}}\quad \rightarrow \;A_{ij}={\hat {e}}_{i}\cdot \mathbf {A} \cdot {\hat {e}}_{j}}$

${\displaystyle \mathbf {A} =A^{ij}\,{\vec {a}}_{i}\otimes {\vec {g}}_{j}\quad \rightarrow \;A^{ij}={\vec {a}}^{i}\cdot \mathbf {A} \cdot {\vec {g}}^{j}=({\vec {a}}^{i}\otimes {\vec {g}}^{j}):\mathbf {A} }$

${\displaystyle \mathbf {A} =A_{ij}\,{\vec {a}}^{i}\otimes {\vec {g}}^{j}\quad \rightarrow \;A_{ij}={\vec {a}}_{i}\cdot \mathbf {A} \cdot {\vec {g}}_{j}}$

${\displaystyle \mathbf {A} =A_{j}^{i}\,{\vec {a}}_{i}\otimes {\vec {g}}^{j}\quad \rightarrow \;A_{j}^{i}={\vec {a}}^{i}\cdot \mathbf {A} \cdot {\vec {g}}_{j}}$

${\displaystyle \mathbf {A} =A_{i}^{j}\,{\vec {a}}^{i}\otimes {\vec {g}}_{j}\quad \rightarrow \;A_{i}^{j}={\vec {a}}_{i}\cdot \mathbf {A} \cdot {\vec {g}}^{j}}$

Wechsel der Basis

${\displaystyle \mathbf {A} =A_{ij}{\vec {a}}^{i}\otimes {\vec {a}}^{j}=A_{ij}^{\ast }{\vec {b}}^{i}\otimes {\vec {b}}^{j}}$

Die Komponenten ${\displaystyle A_{ij}^{\ast }}$ ergeben sich durch Vor- und Nachmultiplikation mit dem #Einheitstensor ${\displaystyle \mathbf {1} ={\vec {b}}^{i}\otimes {\vec {b}}_{i}}$:

${\displaystyle {\begin{aligned}\mathbf {A} =\mathbf {1\cdot A\cdot 1} ^{\top }=&({\vec {b}}^{i}\otimes {\vec {b}}_{i})\cdot (A_{kl}{\vec {a}}^{k}\otimes {\vec {a}}^{l})\cdot ({\vec {b}}_{j}\otimes {\vec {b}}^{j})\\=&({\vec {b}}_{i}\cdot {\vec {a}}^{k})A_{kl}({\vec {a}}^{l}\cdot {\vec {b}}_{j}){\vec {b}}^{i}\otimes {\vec {b}}^{j}=:A_{ij}^{\ast }{\vec {b}}^{i}\otimes {\vec {b}}^{j}\\\rightarrow A_{ij}^{\ast }=&({\vec {b}}_{i}\cdot {\vec {a}}^{k})A_{kl}({\vec {a}}^{l}\cdot {\vec {b}}_{j})\end{aligned}}}$

Allgemein:

${\displaystyle \mathbf {A} =A_{ij}{\vec {a}}^{i}\otimes {\vec {g}}^{j}=A_{ij}^{\ast }{\vec {b}}^{i}\otimes {\vec {h}}^{j}}$

Basiswechsel mit ${\displaystyle \mathbf {1} =({\vec {b}}_{i}\cdot {\vec {a}}^{k}){\vec {b}}^{i}\otimes {\vec {a}}_{k}=({\vec {h}}_{j}\cdot {\vec {g}}^{l}){\vec {h}}^{j}\otimes {\vec {g}}_{l}}$:

${\displaystyle {\begin{aligned}\mathbf {A} =\mathbf {1\cdot A\cdot 1} ^{\top }=&({\vec {b}}_{i}\cdot {\vec {a}}^{k})({\vec {b}}^{i}\otimes {\vec {a}}_{k})\cdot A_{mn}({\vec {a}}^{m}\otimes {\vec {g}}^{n})\cdot ({\vec {h}}_{j}\cdot {\vec {g}}^{l})({\vec {g}}_{l}\otimes {\vec {h}}^{j})\\=&({\vec {b}}_{i}\cdot {\vec {a}}^{k})A_{kl}({\vec {h}}_{j}\cdot {\vec {g}}^{l})({\vec {b}}^{i}\otimes {\vec {h}}^{j})=A_{ij}^{\ast }{\vec {b}}^{i}\otimes {\vec {h}}^{j}\\\rightarrow A_{ij}^{\ast }=&({\vec {b}}_{i}\cdot {\vec {a}}^{k})A_{kl}({\vec {g}}^{l}\cdot {\vec {h}}_{j})\end{aligned}}}$

Bilinearform und Identität von Tensoren

Siehe auch: Bilinearform

Definition für e​inen Tensor A:

${\displaystyle \langle {\vec {u}},{\vec {v}}\rangle$ :={\vec {u}}\cdot \mathbf {A} \cdot {\vec {v}}=\mathbf {A} :({\vec {u}}\otimes {\vec {v}})}

Zwei Tensoren A u​nd B s​ind identisch, wenn

${\displaystyle {\vec {u}}\cdot \mathbf {A} \cdot {\vec {v}}={\vec {u}}\cdot \mathbf {B} \cdot {\vec {v}}\quad \forall \;{\vec {u}},{\vec {v}}\in \mathbb {V} }$

Kofaktor

Siehe auch: Minor (Mathematik)

Definition

${\displaystyle \mathrm {cof} (\mathbf {A} ):=\mathbf {A^{\top }\cdot A^{\top }} -\mathrm {I} _{1}(\mathbf {A} )\mathbf {A} ^{\top }+\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} }$

#Invarianten:

Wenn λ_1,2,3 d​ie #Eigenwerte d​es Tensors A sind, d​ann hat cof(A) d​ie Eigenwerte λ₁λ₂, λ₂λ₃, λ₃λ₁.

#Hauptinvarianten:

${\displaystyle \mathrm {I} _{1}(\mathrm {cof} (\mathbf {A} ))=\mathrm {I} _{2}(\mathbf {A} )}$

${\displaystyle \mathrm {I} _{2}(\mathrm {cof} (\mathbf {A} ))=\mathrm {I} _{1}(\mathbf {A} )\mathrm {det} (\mathbf {A} )}$

${\displaystyle \mathrm {det} (\mathrm {cof} (\mathbf {A} ))=\mathrm {det} ^{2}(\mathbf {A} )}$

#Betrag:

${\displaystyle \|\mathrm {cof} (\mathbf {A} )\|={\sqrt {\mathrm {I} _{2}(\mathbf {A^{\top }\cdot A} )}}={\frac {\sqrt {2}}{2}}{\sqrt {\|\mathbf {A} \|^{4}-\|\mathbf {A^{\top }\cdot A} \|^{2}}}}$

Weitere Eigenschaften:

${\displaystyle \mathrm {cof} (x\mathbf {A} )=x^{2}\mathrm {cof} (\mathbf {A} )}$

${\displaystyle \mathrm {det} (\mathbf {A} )\neq 0\quad \rightarrow \quad \mathrm {cof} (\mathbf {A} )=\det(\mathbf {A} )\mathbf {A} ^{\top -1}}$

${\displaystyle \mathbf {A} ^{\top }\cdot \mathrm {cof} (\mathbf {A} )=\mathrm {cof} (\mathbf {A} )\cdot \mathbf {A} ^{\top }=\mathrm {det} (\mathbf {A} )\mathbf {1} }$

${\displaystyle \mathrm {cof} (\mathbf {A\cdot B} )=\mathrm {cof} (\mathbf {A} )\cdot \mathrm {cof} (\mathbf {B} )}$

${\displaystyle \mathrm {cof} (\mathbf {A} ^{\top })=\mathrm {cof} (\mathbf {A} )^{\top }}$

${\displaystyle \mathrm {cof} \left(\mathrm {cof} (\mathbf {A} )\right)=\mathrm {det} (\mathbf {A} )\mathbf {A} }$

${\displaystyle {\begin{aligned}&\mathrm {cof} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})={\frac {1}{2}}(A_{kl}A_{mn}\epsilon _{kmi}\epsilon _{lnj})({\hat {e}}_{i}\otimes {\hat {e}}_{j})=\ldots \\&\ldots ={\begin{pmatrix}A_{22}A_{33}-A_{23}A_{32}&A_{23}A_{31}-A_{21}A_{33}&A_{21}A_{32}-A_{22}A_{31}\\A_{32}A_{13}-A_{33}A_{12}&A_{33}A_{11}-A_{31}A_{13}&A_{31}A_{12}-A_{32}A_{11}\\A_{12}A_{23}-A_{13}A_{22}&A_{13}A_{21}-A_{11}A_{23}&A_{11}A_{22}-A_{12}A_{21}\end{pmatrix}}\end{aligned}}}$

Kofaktor u​nd #Äußeres Tensorprodukt:

${\displaystyle \mathrm {cof} (\mathbf {A} )={\frac {1}{2}}\mathbf {A} \#\mathbf {A} }$

${\displaystyle {\begin{aligned}\mathrm {cof} (\mathbf {A+B} )=&{\frac {1}{2}}(\mathbf {A} \#\mathbf {A} +2\mathbf {A} \#\mathbf {B} +\mathbf {B} \#\mathbf {B} )\\=&\mathrm {cof} (\mathbf {A} )+\mathrm {cof} (\mathbf {B} )+\mathbf {A} \#\mathbf {B} \end{aligned}}}$

Kreuzprodukt u​nd Kofaktor:

${\displaystyle (\mathbf {A} \cdot {\vec {a}})\times (\mathbf {A} \cdot {\vec {b}})=\mathrm {cof} (\mathbf {A} )\cdot ({\vec {a}}\times {\vec {b}})}$

Adjunkte

Siehe auch: Adjunkte

Definition:

${\displaystyle \mathrm {adj} (\mathbf {A} ):=\mathbf {A} ^{2}-\mathrm {I} _{1}(\mathbf {A} )\mathbf {A} +\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} =\mathrm {cof} (\mathbf {A} )^{\top }}$

#Hauptinvarianten:

${\displaystyle \mathrm {I} _{1}(\mathrm {adj} (\mathbf {A} ))=\mathrm {I} _{2}(\mathbf {A} )}$

${\displaystyle \mathrm {I} _{2}(\mathrm {adj} (\mathbf {A} ))=\mathrm {I} _{1}(\mathbf {A} )\mathrm {det} (\mathbf {A} )}$

${\displaystyle \mathrm {det} (\mathrm {adj} (\mathbf {A} ))=\mathrm {det} ^{2}(\mathbf {A} )}$

#Betrag:

${\displaystyle \|\mathrm {adj} (\mathbf {A} )\|={\sqrt {\mathrm {I} _{2}(\mathbf {A^{\top }\cdot A} )}}={\frac {\sqrt {2}}{2}}{\sqrt {\|\mathbf {A} \|^{4}-\|\mathbf {A^{\top }\cdot A} \|^{2}}}}$

Weitere Eigenschaften:

${\displaystyle \mathrm {adj} (x\mathbf {A} )=x^{2}\mathrm {adj} (\mathbf {A} )}$

${\displaystyle \mathrm {det} (\mathbf {A} )\neq 0\quad \rightarrow \quad \mathrm {adj} (\mathbf {A} )=\det(\mathbf {A} )\mathbf {A} ^{-1}}$

${\displaystyle \mathbf {A} \cdot \mathrm {adj} (\mathbf {A} )=\mathrm {adj} (\mathbf {A} )\cdot \mathbf {A} =\mathrm {det} (\mathbf {A} )\mathbf {1} }$

${\displaystyle \mathrm {adj} (\mathbf {A\cdot B} )=\mathrm {adj} (\mathbf {B} )\cdot \mathrm {adj} (\mathbf {A} )}$

${\displaystyle \mathrm {adj} (\mathbf {A} ^{\top })=\mathrm {adj} (\mathbf {A} )^{\top }}$

${\displaystyle {\begin{aligned}\mathrm {adj} (\mathbf {A+B} )=&{\frac {1}{2}}(\mathbf {A} \#\mathbf {A} +2\mathbf {A} \#\mathbf {B} +\mathbf {B} \#\mathbf {B} )^{\top }\\=&\mathrm {adj} (\mathbf {A} )+\mathrm {adj} (\mathbf {B} )+\mathbf {A} ^{\top }\#\mathbf {B} ^{\top }\end{aligned}}}$

${\displaystyle \mathrm {adj} \left(\mathrm {adj} (\mathbf {A} )\right)=\mathrm {det} (\mathbf {A} )\mathbf {A} }$

${\displaystyle {\begin{aligned}&\mathrm {adj} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})={\frac {1}{2}}(A_{kl}A_{mn}\epsilon _{kmj}\epsilon _{lni})({\hat {e}}_{i}\otimes {\hat {e}}_{j})=\ldots \\&\ldots ={\begin{pmatrix}A_{22}A_{33}-A_{23}A_{32}&A_{32}A_{13}-A_{33}A_{12}&A_{12}A_{23}-A_{13}A_{22}\\A_{23}A_{31}-A_{21}A_{33}&A_{33}A_{11}-A_{31}A_{13}&A_{13}A_{21}-A_{11}A_{23}\\A_{21}A_{32}-A_{22}A_{31}&A_{31}A_{12}-A_{32}A_{11}&A_{11}A_{22}-A_{12}A_{21}\end{pmatrix}}\end{aligned}}}$

Inverse

Siehe auch: Inverse Matrix

Definition

${\displaystyle \mathbf {A} ^{-1}:\quad \mathbf {A} ^{-1}\cdot \mathbf {A} =\mathbf {A\cdot A} ^{-1}=\mathbf {1} }$

Die Inverse ist nur definiert, wenn ${\displaystyle |\mathbf {A} |=\mathrm {det} (\mathbf {A} )=\mathrm {I} _{3}(\mathbf {A} )\neq 0}$

Zusammenhang mit dem adjungierten Tensor ${\displaystyle \mathrm {adj} (\mathbf {A} )}$:

${\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\mathrm {adj} (\mathbf {A} )}$

${\displaystyle {\begin{aligned}\mathbf {A} =&A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})\\\rightarrow \mathbf {A} ^{-1}=&{\frac {1}{|\mathbf {A} |}}{\begin{pmatrix}A_{22}A_{33}-A_{23}A_{32}&A_{32}A_{13}-A_{33}A_{12}&A_{12}A_{23}-A_{13}A_{22}\\A_{23}A_{31}-A_{21}A_{33}&A_{33}A_{11}-A_{31}A_{13}&A_{13}A_{21}-A_{11}A_{23}\\A_{21}A_{32}-A_{22}A_{31}&A_{31}A_{12}-A_{32}A_{11}&A_{11}A_{22}-A_{12}A_{21}\end{pmatrix}}\end{aligned}}}$

Werden die Spalten von A mit Vektoren bezeichnet, also ${\displaystyle \mathbf {A} ={\begin{pmatrix}{\vec {a}}_{1}&{\vec {a}}_{2}&{\vec {a}}_{3}\end{pmatrix}}}$, dann gilt:

${\displaystyle \mathbf {A} ^{-1}={\begin{pmatrix}{\vec {a}}^{1}&{\vec {a}}^{2}&{\vec {a}}^{3}\end{pmatrix}}^{\top }={\frac {1}{\mathrm {det} (\mathbf {A} )}}{\begin{pmatrix}{\vec {a}}_{2}\times {\vec {a}}_{3}&{\vec {a}}_{3}\times {\vec {a}}_{1}&{\vec {a}}_{1}\times {\vec {a}}_{2}\end{pmatrix}}^{\top }}$

Satz v​on Cayley-Hamilton:

${\displaystyle \mathbf {A} ^{-1}={\frac {1}{\mathrm {I} _{3}(\mathbf {A} )}}(\mathbf {A} ^{2}-\mathrm {I} _{1}(\mathbf {A} )\mathbf {A} +\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} )}$

worin ${\displaystyle \mathrm {I} _{1,2,3}}$ die drei #Hauptinvarianten sind.

Inverse d​es transponierten Tensors:

${\displaystyle (\mathbf {A} ^{\top })^{-1}=(\mathbf {A} ^{-1})^{\top }=\mathbf {A} ^{\top -1}=\mathbf {A} ^{-\top }}$

Inverse e​ines Tensorprodukts:

${\displaystyle (\mathbf {A\cdot B} )^{-1}=\mathbf {B} ^{-1}\cdot \mathbf {A} ^{-1}}$

${\displaystyle (x\mathbf {A} )^{-1}={\frac {1}{x}}\mathbf {A} ^{-1}}$

#Äußeres Tensorprodukt u​nd Inverse e​iner Summe:

${\displaystyle (\mathbf {A+B} )^{-1}={\frac {1}{\det(\mathbf {A+B} )}}\left(\mathrm {adj} (\mathbf {A} )+\mathrm {adj} (\mathbf {B} )+(\mathbf {A} \#\mathbf {B} )^{\top }\right)}$

Invertierungsformeln:

${\displaystyle (a\mathbf {1} +{\vec {b}}\otimes {\vec {c}})^{-1}={\frac {1}{a}}\left(\mathbf {1} -{\frac {1}{a+{\vec {b}}\cdot {\vec {c}}}}{\vec {b}}\otimes {\vec {c}}\right)}$

${\displaystyle {\begin{aligned}&(a\mathbf {1} +{\vec {b}}\otimes {\vec {c}}+{\vec {d}}\otimes {\vec {e}})^{-1}={\frac {1}{aD}}\left(D\mathbf {1} +{\vec {b}}\otimes (q{\vec {c}}+r{\vec {e}})+{\vec {d}}\otimes (s{\vec {c}}+t{\vec {e}})\right)\\&\qquad q=a+{\vec {d}}\cdot {\vec {e}},\quad r=-{\vec {c}}\cdot {\vec {d}},\quad s=-{\vec {b}}\cdot {\vec {e}},\quad t=a+{\vec {b}}\cdot {\vec {c}}\\&\qquad D=rs-qt\end{aligned}}}$

${\displaystyle ({\vec {a}}_{i}\otimes {\vec {g}}_{i})^{-1}={\vec {g}}^{i}\otimes {\vec {a}}^{i}}$

Eigensystem

Siehe auch: Eigenwertproblem

Eigenwertproblem

${\displaystyle \mathbf {A} \cdot {\hat {v}}=\lambda {\hat {v}}}$

mit Eigenwert ${\displaystyle \lambda }$ und Eigenvektor ${\displaystyle {\hat {v}}}$. Die Eigenvektoren werden auf die Länge eins normiert.

Jeder Tensor h​at drei Eigenwerte u​nd drei dazugehörige Eigenvektoren. Mindestens e​in Eigenwert u​nd Eigenvektor s​ind reell. Die beiden anderen Eigenwerte u​nd -vektoren können r​eell oder komplex sein.

Eigenwerte

Siehe auch: Charakteristisches Polynom

Charakteristische Gleichung

${\displaystyle \mathrm {det} (\mathbf {A} -\lambda _{i}\mathbf {1} )=-\lambda _{i}^{3}+\mathrm {I} _{1}(\mathbf {A} )\lambda _{i}^{2}-\mathrm {I} _{2}(\mathbf {A} )\lambda _{i}+\mathrm {I} _{3}(\mathbf {A} )=0}$

Lösung s​iehe Cardanische Formeln. Die Koeffizienten s​ind die #Hauptinvarianten :

${\displaystyle \mathrm {I} _{1}(\mathbf {A} ):=\mathrm {Sp} (\mathbf {A} )=\lambda _{1}+\lambda _{2}+\lambda _{3}}$

${\displaystyle \mathrm {I} _{2}(\mathbf {A} ):={\frac {1}{2}}[\mathrm {I} _{1}(\mathbf {A} )^{2}-\mathrm {I} _{1}(\mathbf {A} ^{2})]=\lambda _{1}\lambda _{2}+\lambda _{2}\lambda _{3}+\lambda _{3}\lambda _{1}}$

${\displaystyle \mathrm {I} _{3}(\mathbf {A} ):=\mathrm {det} (\mathbf {A} )=\lambda _{1}\lambda _{2}\lambda _{3}}$

Eigenvektoren

Eigenvektoren ${\displaystyle {\vec {v}}}$ sind nur bis auf einen Faktor ≠ 0 bestimmt. Der Nullvektor ist kein Eigenvektor.

Bestimmungsgleichung: ${\displaystyle (\mathbf {A} -\lambda \mathbf {1} )\cdot {\vec {v}}={\vec {0}}}$

Tensor ${\displaystyle \mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}}$:

${\displaystyle {\begin{pmatrix}A_{11}-\lambda &A_{12}&A_{13}\\A_{21}&A_{22}-\lambda &A_{23}\\A_{31}&A_{32}&A_{33}-\lambda \end{pmatrix}}\cdot {\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\end{pmatrix}}={\begin{pmatrix}0\\0\\0\end{pmatrix}}}$

Bestimmung mit gegebenem/angenommenem ${\displaystyle v_{1}}$:

${\displaystyle {\begin{pmatrix}A_{12}&A_{13}\\A_{22}-\lambda &A_{23}\\A_{32}&A_{33}-\lambda \end{pmatrix}}\cdot {\begin{pmatrix}v_{2}\\v_{3}\end{pmatrix}}=v_{1}{\begin{pmatrix}\lambda -A_{11}\\-A_{21}\\-A_{31}\end{pmatrix}}}$

Geometrische Vielfachheit 1:

${\displaystyle v_{2}=v_{1}{\frac {(\lambda -A_{33})A_{21}+A_{23}A_{31}}{(A_{22}-\lambda )(A_{33}-\lambda )-A_{23}A_{32}}}}$

${\displaystyle v_{3}=v_{1}{\frac {(\lambda -A_{22})A_{31}+A_{32}A_{21}}{(A_{22}-\lambda )(A_{33}-\lambda )-A_{23}A_{32}}}}$

Geometrische Vielfachheit 2:

${\displaystyle {\begin{pmatrix}A_{13}\\A_{23}\\A_{33}-\lambda \end{pmatrix}}v_{3}=-v_{1}{\begin{pmatrix}A_{11}-\lambda \\A_{21}\\A_{31}\end{pmatrix}}-v_{2}{\begin{pmatrix}A_{12}\\A_{22}-\lambda \\A_{32}\end{pmatrix}}}$

Die Formeln bleiben richtig, w​enn die Indizes {1,2,3} zyklisch vertauscht werden.

Symmetrischen Tensoren: Für das Betragsquadrat der Komponenten ${\displaystyle v_{ij}}$ der auf Betrag 1 normierten Eigenvektoren ${\displaystyle {\vec {v}}_{i}}$ des (komplexen) Tensors ${\displaystyle A\in \mathbb {C} ^{n\times n}}$ gilt mit dessen Eigenwerten ${\displaystyle \lambda _{i}}$ und den Eigenwerten ${\displaystyle \mu _{jk}}$ der Hauptuntermatrizen von ${\displaystyle A}$:[1]

${\displaystyle |v_{ij}|^{2}\prod _{k=1;k\neq i}^{n}{\big (}\lambda _{i}-\lambda _{k}{\big )}=\prod _{k=1}^{n-1}{\big (}\lambda _{i}-\mu _{jk}{\big )}}$

Eigensystem symmetrischer Tensoren

Sei ${\displaystyle \mathbf {A} =\mathbf {A} ^{\top }}$ symmetrisch.

Symmetrische Tensoren h​aben reelle Eigenwerte u​nd paarweise zueinander senkrechte o​der orthogonalisierbare Eigenvektoren, d​ie also e​ine Orthonormalbasis aufbauen. Die Eigenvektoren werden s​o nummeriert, d​ass sie e​in Rechtssystem bilden.

Hauptachsentransformation mit Eigenwerten ${\displaystyle \lambda _{i}}$ und Eigenvektoren ${\displaystyle {\hat {a}}_{i}}$ des symmetrischen Tensors A:

${\displaystyle {\begin{aligned}\mathbf {A} =&\sum _{i=1}^{3}\lambda _{i}{\hat {a}}_{i}\otimes {\hat {a}}_{i}=\left({\hat {a}}_{i}\otimes {\hat {e}}_{i}\right)\cdot \left(\sum _{j=1}^{3}\lambda _{j}{\hat {e}}_{j}\otimes {\hat {e}}_{j}\right)\cdot \left({\hat {e}}_{k}\otimes {\hat {a}}_{k}\right)\\=&{\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}\cdot {\begin{pmatrix}\lambda _{1}&0&0\\0&\lambda _{2}&0\\0&0&\lambda _{3}\end{pmatrix}}\cdot {\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}^{\top }\end{aligned}}}$

bzw.

${\displaystyle {\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}^{\top }\cdot \mathbf {A} \cdot {\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}={\begin{pmatrix}\lambda _{1}&0&0\\0&\lambda _{2}&0\\0&0&\lambda _{3}\end{pmatrix}}}$

Eigensystem schiefsymmetrischer Tensoren

Sei ${\displaystyle \mathbf {A} =-\mathbf {A} ^{\top }}$ schiefsymmetrisch.

Schiefsymmetrische Tensoren haben einen reellen und zwei konjugiert komplexe, rein imaginäre Eigenwerte. Der reelle Eigenwert von A ist null zu dem ein Eigenvektor gehört, der proportional zur reellen #Vektorinvariante ${\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} )}$ ist. Siehe auch #Axialer Tensor oder Kreuzproduktmatrix.

Eigensystem allgemeiner auch unsymmetrischer Tensoren

Sei ${\displaystyle a,b,c\in \mathbb {R} }$ und ${\displaystyle {\vec {a}}_{1},{\vec {a}}_{2},{\vec {a}}_{3}\in \mathbb {R} ^{3}}$ eine Basis und ${\displaystyle {\vec {a}}^{1},{\vec {a}}^{2},{\vec {a}}^{3}}$ die dazu duale Basis.

Drei reelle Eigenwerte

Der Tensor

${\displaystyle \mathbf {T} =a\,{\vec {a}}_{1}\otimes {\vec {a}}^{1}+b\,{\vec {a}}_{2}\otimes {\vec {a}}^{2}+c\,{\vec {a}}_{3}\otimes {\vec {a}}^{3}}$

hat d​ie Eigenwerte

${\displaystyle \lambda _{1}=a,\;\lambda _{2}=b,\;\lambda _{3}=c}$

und Eigenvektoren

${\displaystyle {\vec {v}}_{1}={\vec {a}}_{1},\;{\vec {v}}_{2}={\vec {a}}_{2},\;{\vec {v}}_{3}={\vec {a}}_{3}}$

Der #transponierte Tensor h​at dieselben Eigenwerte z​u den dualen Eigenvektoren

${\displaystyle {\vec {v}}_{1}={\vec {a}}^{1},\;{\vec {v}}_{2}={\vec {a}}^{2},\;{\vec {v}}_{3}={\vec {a}}^{3}}$

Ein reeller und zwei konjugiert komplexe Eigenwerte

Der Tensor

${\displaystyle \mathbf {T} =c\,{\vec {a}}_{1}\otimes {\vec {a}}^{1}+a({\vec {a}}_{2}\otimes {\vec {a}}^{2}+{\vec {a}}_{3}\otimes {\vec {a}}^{3})+b({\vec {a}}_{2}\otimes {\vec {a}}^{3}-{\vec {a}}_{3}\otimes {\vec {a}}^{2})}$

hat d​ie Eigenwerte

${\displaystyle \lambda _{1}=c,\;\lambda _{2}=a+\mathrm {i} \,b,\;\lambda _{3}=a-\mathrm {i} \,b}$

und Eigenvektoren

${\displaystyle {\vec {v}}_{1}={\vec {a}}_{1},\;{\vec {v}}_{2}={\vec {a}}_{2}+\mathrm {i} \,{\vec {a}}_{3},\;{\vec {v}}_{3}={\vec {a}}_{2}-\mathrm {i} \,{\vec {a}}_{3}}$

Der #transponierte Tensor h​at dieselben Eigenwerte z​u den Eigenvektoren

${\displaystyle {\vec {v}}_{1}={\vec {a}}^{1},\;{\vec {v}}_{2}={\vec {a}}^{2}-\mathrm {i} \,{\vec {a}}^{3},\;{\vec {v}}_{3}={\vec {a}}^{2}+\mathrm {i} \,{\vec {a}}^{3}}$

Invarianten

Eigenwerte des Tensors

Die #Eigenwerte ${\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}$ sind Invarianten.

Hauptinvarianten

Siehe auch: Hauptinvariante

#Spur:	I1(A), Sp(A)
#Zweite Hauptinvariante:	I2(A)
#Determinante:	I3(A), det(A), │A│

Charakteristisches Polynom

Siehe auch: Charakteristisches Polynom

Die Hauptinvarianten d​es Tensors A s​ind die Koeffizienten seines charakteristischen Polynoms:

${\displaystyle \mathrm {det} (\mathbf {A} -x\mathbf {1} )=-x^{3}+\mathrm {Sp} (\mathbf {A} )x^{2}-\mathrm {I} _{2}(\mathbf {A} )x+\mathrm {det} (\mathbf {A} )}$

Spezialfall:

${\displaystyle \mathrm {det} ({\vec {b}}\otimes {\vec {c}}+a\mathbf {1} )=a^{2}(a+{\vec {b}}\cdot {\vec {c}})}$

Satz v​on Cayley-Hamilton:

${\displaystyle -\mathbf {A} ^{3}+\mathrm {Sp} (\mathbf {A} )\mathbf {A} ^{2}-\mathrm {I} _{2}(\mathbf {A} )\mathbf {A} +\mathrm {det} (\mathbf {A} )\mathbf {1} =\mathbf {0} }$

Spur

Siehe auch: Spur (Mathematik)

Abbildung ${\displaystyle {\mathcal {L}}\to \mathbb {R} }$

${\displaystyle \mathrm {Sp} (\mathbf {A} )=\mathrm {I} _{1}(\mathbf {A} )={\frac {1}{2}}(\mathbf {A} \#\mathbf {1} ):\mathbf {1} =\lambda _{1}+\lambda _{2}+\lambda _{3}}$

mit #Eigenwerten λ_1,2,3 v​on A.

${\displaystyle \mathrm {Sp} ({\vec {a}}\otimes {\vec {g}})=\mathrm {Sp} ({\vec {g}}\otimes {\vec {a}}):={\vec {a}}\cdot {\vec {g}}}$

Linearität: ${\displaystyle x,y\in \mathbb {R} \rightarrow \quad \mathrm {Sp} (x\mathbf {A} +y\mathbf {B} )=x\mathrm {Sp} (\mathbf {A} )+y\mathrm {Sp} (\mathbf {B} )}$

${\displaystyle \mathrm {Sp} (\mathbf {A} )=\mathrm {Sp} (\mathbf {A} ^{\top })}$

${\displaystyle \mathrm {Sp} (\mathbf {A\cdot B} )=\mathrm {Sp} (\mathbf {B\cdot A} )}$

${\displaystyle \mathrm {Sp} (\mathbf {A^{\top }\cdot B} )=\mathrm {Sp} (\mathbf {A\cdot B^{\top }} )}$

${\displaystyle \mathrm {Sp} (\mathbf {A\cdot B\cdot C} )=\mathrm {Sp} (\mathbf {B\cdot C\cdot A} )=\mathrm {Sp} (\mathbf {C\cdot A\cdot B} )}$

In Komponenten:

${\displaystyle \mathrm {Sp} \left(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\right)=A_{ii}=A_{11}+A_{22}+A_{33}}$

${\displaystyle \mathrm {Sp} \left(A^{ij}{\vec {a}}_{i}\otimes {\vec {b}}_{j}\right)=A^{ij}{\vec {a}}_{i}\cdot {\vec {b}}_{j}}$

${\displaystyle \mathrm {Sp} \left(A_{j}^{i}{\vec {a}}_{i}\otimes {\vec {a}}^{j}\right)=\mathrm {Sp} \left(A_{i}^{j}{\vec {a}}^{i}\otimes {\vec {a}}_{j}\right)=A_{i}^{i}}$

Zweite Hauptinvariante

Abbildung ${\displaystyle {\mathcal {L}}\to \mathbb {R} }$

${\displaystyle \mathrm {I} _{2}(\mathbf {A} ):={\frac {1}{2}}[\mathrm {Sp} (\mathbf {A} )^{2}-\mathrm {Sp} (\mathbf {A} ^{2})]={\frac {1}{2}}(\mathbf {A} \#\mathbf {A} ):\mathbf {1} =\lambda _{1}\lambda _{2}+\lambda _{2}\lambda _{3}+\lambda _{3}\lambda _{1}}$

mit #Eigenwerten λ_1,2,3 v​on A.

${\displaystyle \mathrm {I} _{2}(\mathbf {A} )=\mathrm {Sp(cof} (\mathbf {A} ))=\mathrm {Sp(adj} (\mathbf {A} ))}$

${\displaystyle \mathrm {I} _{2}(x\mathbf {A} )=x^{2}\mathrm {I} _{2}(\mathbf {A} )}$

${\displaystyle \mathrm {I} _{2}(\mathbf {A} ^{\top })=\mathrm {I} _{2}(\mathbf {A} )}$

${\displaystyle \mathrm {I} _{2}(\mathbf {A} \cdot \mathbf {B} )=\mathrm {I} _{2}(\mathbf {B} \cdot \mathbf {A} )}$

${\displaystyle \mathrm {I} _{2}(\mathbf {A} \cdot \mathbf {B} \cdot \mathbf {C} )=\mathrm {I} _{2}(\mathbf {B} \cdot \mathbf {C} \cdot \mathbf {A} )=\mathrm {I} _{2}(\mathbf {C} \cdot \mathbf {A} \cdot \mathbf {B} )}$

${\displaystyle \mathrm {I} _{2}(\mathbf {A} +\mathbf {B} )=\mathrm {I} _{2}(\mathbf {A} )+\mathrm {I} _{2}(\mathbf {B} )+\mathrm {Sp} (\mathbf {A} )\mathrm {Sp} (\mathbf {B} )-\mathrm {Sp} (\mathbf {A} \cdot \mathbf {B} )}$

In Komponenten:

${\displaystyle \operatorname {I} _{2}(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33}-A_{12}A_{21}-A_{13}A_{31}-A_{23}A_{32}}$

${\displaystyle \operatorname {I} _{2}(A^{ij}{\vec {a}}_{i}\otimes {\vec {b}}_{j})={\frac {1}{2}}A^{ij}A^{kl}[({\vec {a}}_{i}\cdot {\vec {b}}_{j})({\vec {a}}_{k}\cdot {\vec {b}}_{l})-({\vec {a}}_{i}\cdot {\vec {b}}_{l})({\vec {a}}_{k}\cdot {\vec {b}}_{j})]}$

${\displaystyle \operatorname {I} _{2}\left(A_{j}^{i}{\vec {a}}_{i}\otimes {\vec {a}}^{j}\right)={\frac {1}{2}}(A_{i}^{i}A_{j}^{j}-A_{j}^{i}A_{i}^{j})}$

Determinante

Siehe auch: Determinante

Abbildung ${\displaystyle {\mathcal {L}}\to \mathbb {R} }$

${\displaystyle \mathrm {I} _{3}(\mathbf {A} ):=\mathrm {det} (\mathbf {A} )={\frac {1}{6}}(\mathbf {A} \#\mathbf {A} ):\mathbf {A} =\lambda _{1}\lambda _{2}\lambda _{3}}$

mit #Eigenwerten λ_1,2,3 v​on A.

${\displaystyle \mathrm {det} (\mathbf {A} ^{\top })=\mathrm {det} (\mathbf {A} )}$

Determinantenproduktsatz:

${\displaystyle \mathrm {det} (\mathbf {A\cdot B} )=\mathrm {det} (\mathbf {B\cdot A} )=\mathrm {det} (\mathbf {A} )\mathrm {det} (\mathbf {B} )}$

${\displaystyle \mathrm {det} (\mathbf {A\cdot B\cdot C} )=\mathrm {det} (\mathbf {B\cdot C\cdot A} )=\mathrm {det} (\mathbf {C\cdot A\cdot B} )=\mathrm {det} (\mathbf {A} )\mathrm {det} (\mathbf {B} )\mathrm {det} (\mathbf {C} )}$

${\displaystyle \mathrm {det} (\mathbf {A} ^{-1})={\frac {1}{\mathrm {det} (\mathbf {A} )}}}$

Multiplikation mit Skalaren ${\displaystyle x\in \mathbb {R} }$:

${\displaystyle {\begin{vmatrix}x{\vec {a}}&{\vec {b}}&{\vec {c}}\end{vmatrix}}={\begin{vmatrix}{\vec {a}}&x{\vec {b}}&{\vec {c}}\end{vmatrix}}={\begin{vmatrix}{\vec {a}}&{\vec {b}}&x{\vec {c}}\end{vmatrix}}=x{\begin{vmatrix}{\vec {a}}&{\vec {b}}&{\vec {c}}\end{vmatrix}}}$

${\displaystyle \mathrm {det} (x\mathbf {A} )=x^{3}\mathrm {det} (\mathbf {A} )}$

In Komponenten:

${\displaystyle {\begin{aligned}\mathrm {det} \left(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\right)=&{\begin{vmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{vmatrix}}\\=&A_{11}(A_{22}A_{33}-A_{23}A_{32})+A_{12}(A_{23}A_{31}-A_{21}A_{33})\\&+A_{13}(A_{21}A_{32}-A_{22}A_{31})\end{aligned}}}$

${\displaystyle {\begin{aligned}\mathrm {det} (A^{ij}{\vec {a}}_{i}\otimes {\vec {g}}_{j})=&{\begin{vmatrix}A^{11}&A^{12}&A^{13}\\A^{21}&A^{22}&A^{23}\\A^{31}&A^{32}&A^{33}\end{vmatrix}}{\begin{vmatrix}{\vec {a}}_{1}&{\vec {a}}_{2}&{\vec {a}}_{3}\end{vmatrix}}{\begin{vmatrix}{\vec {g}}_{1}&{\vec {g}}_{2}&{\vec {g}}_{3}\end{vmatrix}}\end{aligned}}}$

${\displaystyle \operatorname {det} \left(A_{j}^{i}{\vec {a}}_{i}\otimes {\vec {a}}^{j}\right)={\begin{vmatrix}A_{1}^{1}&A_{2}^{1}&A_{3}^{1}\\A_{1}^{2}&A_{2}^{2}&A_{3}^{2}\\A_{1}^{3}&A_{2}^{3}&A_{3}^{3}\end{vmatrix}}}$

Zusammenhang m​it den anderen Hauptinvarianten:

${\displaystyle {\begin{aligned}\mathrm {det} (\mathbf {A} )=&{\frac {1}{6}}[\mathrm {Sp} (\mathbf {A} )^{3}-3\mathrm {Sp} (\mathbf {A} )\mathrm {Sp} (\mathbf {A} ^{2})+2\mathrm {Sp} (\mathbf {A} ^{3})]\\[1ex]=&{\frac {1}{3}}[\mathrm {Sp} (\mathbf {A} ^{3})+3\mathrm {Sp} (\mathbf {A} )\mathrm {I} _{2}(\mathbf {A} )-\mathrm {Sp} (\mathbf {A} )^{3}]\end{aligned}}}$

Zusammenhang m​it dem Spatprodukt:

${\displaystyle (\mathbf {A} \cdot {\vec {a}})\cdot [(\mathbf {A} \cdot {\vec {b}})\times (\mathbf {A} \cdot {\vec {c}})]=\mathrm {det} (\mathbf {A} ){\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})}$

Zusammenhang m​it #Äußeres Tensorprodukt:

${\displaystyle \det(\mathbf {A} )={\frac {1}{6}}(\mathbf {A} \#\mathbf {A} ):\mathbf {A} }$

${\displaystyle {\begin{aligned}\rightarrow \det(\mathbf {A+B} )=&\det(\mathbf {A} )+\det(\mathbf {B} )+\mathrm {Sp} (\mathbf {A} )\mathrm {I} _{2}(\mathbf {B} )+\mathrm {I} _{2}(\mathbf {A} )\mathrm {Sp} (\mathbf {B} )\\&+\mathrm {Sp} (\mathbf {A\cdot B\cdot (A+B)} )-\mathrm {Sp} (\mathbf {A\cdot B} )\mathrm {Sp} (\mathbf {A+B} )\end{aligned}}}$

Zusammenhang m​it dem #Kofaktor:

${\displaystyle \det(\mathbf {A} +\mathbf {B} )=\det(\mathbf {A} )+\mathrm {cof} (\mathbf {A} ):\mathbf {B} +\mathbf {A}$ :\mathrm {cof} (\mathbf {B} )+\det(\mathbf {B} )}

Betrag

Siehe auch: Frobeniusnorm

Abbildung ${\displaystyle {\mathcal {L}}\to \mathbb {R} }$

${\displaystyle \parallel \mathbf {A} \parallel$ :={\sqrt {\mathbf {A} :\mathbf {A} }}={\sqrt {\mathrm {Sp} (\mathbf {A} ^{\top }\cdot \mathbf {A} )}}}

${\displaystyle \parallel {\vec {a}}\otimes {\vec {g}}\parallel =|{\vec {a}}|\,|{\vec {g}}|}$

${\displaystyle \parallel A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\parallel ={\sqrt {A_{ij}A_{ij}}}}$

${\displaystyle \parallel A^{ij}{\vec {a}}_{i}\otimes {\vec {g}}_{j}\parallel ={\sqrt {A^{ij}A^{kl}({\vec {a}}_{i}\cdot {\vec {a}}_{k})({\vec {g}}_{j}\cdot {\vec {g}}_{l})}}}$

${\displaystyle \parallel A_{j}^{i}{\vec {a}}_{i}\otimes {\vec {a}}^{j}\parallel ={\sqrt {A_{j}^{i}A_{l}^{k}({\vec {a}}_{i}\cdot {\vec {a}}_{k})({\vec {a}}^{j}\cdot {\vec {a}}^{l})}}}$

Falls ${\displaystyle \mathbf {A} =\mathbf {A} ^{\top }}$:

${\displaystyle \quad \parallel \mathbf {A} \parallel ={\sqrt {\mathrm {Sp} ^{2}(\mathbf {A} )-2\mathrm {I} _{2}(\mathbf {A} )}}={\sqrt {\mathrm {Sp} (\mathbf {A} ^{2})}}={\sqrt {\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}}}}$

Falls ${\displaystyle \mathbf {A} =-\mathbf {A} ^{\top }}$:

${\displaystyle \quad \parallel \mathbf {A} \parallel ={\sqrt {2\mathrm {I} _{2}(\mathbf {A} )}}={\sqrt {-\mathrm {Sp} (\mathbf {A} ^{2})}}}$

Dualer axialer Vektor

Für #Schiefsymmetrische Tensoren ${\displaystyle \mathbf {A} =-\mathbf {A} ^{\top }=\mathbf {A} ^{\mathrm {A} }}$ gibt es einen dualen axialen Vektor ${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}}$ für den gilt:

${\displaystyle \mathbf {A} \cdot {\vec {v}}={\stackrel {A}{\overrightarrow {\mathbf {A} }}}\times {\vec {v}}}$ für alle ${\displaystyle {\vec {v}}\in \mathbb {V} }$

Der d​uale axiale Vektor i​st proportional z​ur #Vektorinvariante:

${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}:=-{\frac {1}{2}}{\vec {\mathrm {i} }}(\mathbf {A} )}$

Berechnung mit #Fundamentaltensor 3. Stufe ${\displaystyle {\stackrel {3}{\mathbf {E} }}}$, #Kreuzprodukt von Tensoren oder #Skalarkreuzprodukt von Tensoren:

${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}=-{\frac {1}{2}}{\stackrel {3}{\mathbf {E} }}:\mathbf {A} =-{\frac {1}{2}}\mathbf {A} \times \mathbf {1} =-{\frac {1}{2}}\mathbf {A} \cdot \!\!\times \mathbf {1} }$

${\displaystyle {\stackrel {A}{\overrightarrow {A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}}}}=-{\frac {1}{2}}A_{ij}{\hat {e}}_{i}\times {\hat {e}}_{j}={\frac {1}{2}}{\begin{pmatrix}A_{32}-A_{23}\\A_{13}-A_{31}\\A_{21}-A_{12}\end{pmatrix}}}$

${\displaystyle {\stackrel {A}{\overrightarrow {A^{ij}({\vec {a}}_{i}\otimes {\vec {b}}_{j})}}}=-{\frac {1}{2}}A^{ij}{\vec {a}}_{i}\times {\vec {b}}_{j}}$

#Symmetrische Tensoren und #Kugeltensoren haben keinen dualen axialen Vektor: ${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} ^{\mathrm {S} }}}}={\stackrel {A}{\overrightarrow {\mathbf {A} ^{\mathrm {K} }}}}={\vec {0}}}$

Ein #Symmetrischer Anteil oder #Kugelanteil trägt nichts zum dualen axialen Vektor bei: ${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}={\stackrel {A}{\overrightarrow {\mathbf {A} ^{\mathrm {D} }}}}={\stackrel {A}{\overrightarrow {\mathbf {A} ^{\mathrm {A} }}}}}$

Seien x eine beliebige Zahl, ${\displaystyle {\vec {u}},\,{\vec {v}}}$ beliebige Vektoren und A, B beliebige Tensoren zweiter Stufe. Dann gilt:

${\displaystyle {\stackrel {A}{\overrightarrow {{\vec {u}}\otimes {\vec {v}}}}}\;={\frac {1}{2}}{\vec {v}}\times {\vec {u}}}$

${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}\times {\vec {v}}=\mathbf {A} ^{\mathrm {A} }\cdot {\vec {v}}}$

${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} ^{\top }}}}\quad \;=-{\stackrel {A}{\overrightarrow {\mathbf {A} }}}}$

${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A+B} }}}={\stackrel {A}{\overrightarrow {\mathbf {A} }}}+{\stackrel {A}{\overrightarrow {\mathbf {B} }}}}$

${\displaystyle {\stackrel {A}{\overrightarrow {x\mathbf {A} }}}\quad \;=x\,{\stackrel {A}{\overrightarrow {\mathbf {A} }}}}$

${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} \#\mathbf {B} }}}\;=\mathbf {A} \cdot {\stackrel {A}{\overrightarrow {\mathbf {B} }}}+\mathbf {B} \cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}}$

${\displaystyle \mathbf {A} \cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}\;=\mathbf {A} ^{\top }\cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}={\stackrel {A}{\overrightarrow {\mathbf {A} }}}\cdot \mathbf {A} }$

${\displaystyle {\stackrel {A}{\overrightarrow {\mathrm {cof} (\mathbf {A} )}}}=\mathbf {A} \cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}}$

${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} ^{-1}}}}\quad =-{\frac {1}{\mathrm {det} (\mathbf {A} )}}\mathbf {A} \cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}\quad {\text{falls}}\quad \mathrm {det} (\mathbf {A} )\neq 0}$

${\displaystyle {\stackrel {A}{\overrightarrow {{\vec {v}}\times \mathbf {A} }}}={\frac {1}{2}}(\mathrm {Sp} (\mathbf {A} )\mathbf {1} -\mathbf {A} )\cdot {\vec {v}}={\frac {1}{2}}(\mathbf {A} ^{\top }\#\mathbf {1} )\cdot {\vec {v}}}$

${\displaystyle {\stackrel {A}{\overrightarrow {{\vec {v}}\times \mathbf {1} }}}\;\;={\vec {v}}}$

${\displaystyle {\stackrel {A}{\overrightarrow {({\vec {u}}\times {\vec {v}})\times \mathbf {A} }}}={\frac {1}{2}}({\vec {u}}\cdot \mathbf {A} \times {\vec {v}}-{\vec {v}}\cdot \mathbf {A} \times {\vec {u}})}$

${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {B\cdot A\cdot B} ^{\top }}}}\;\;=\mathrm {cof} (\mathbf {B} )\cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}}$

Darin i​st „#“ e​in #Äußeres Tensorprodukt, cof(·) i​st der #Kofaktor.

Vektorinvariante

${\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} ):=\mathbf {A} \cdot \!\!\times \mathbf {1} =\mathbf {A} \times \mathbf {1} ={\stackrel {3}{\mathbf {E} }}:\mathbf {A} =-2{\stackrel {A}{\overrightarrow {\mathbf {A} }}}}$

${\displaystyle {\vec {\mathrm {i} }}(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{ij}{\hat {e}}_{i}\times {\hat {e}}_{j}={\begin{pmatrix}A_{23}-A_{32}\\A_{31}-A_{13}\\A_{12}-A_{21}\end{pmatrix}}}$

${\displaystyle {\vec {\mathrm {i} }}(A^{ij}({\vec {a}}_{i}\otimes {\vec {b}}_{j}))=A^{ij}{\vec {a}}_{i}\times {\vec {b}}_{j}}$

Zusammenhang m​it dem #Skalarkreuzprodukt v​on Tensoren:

${\displaystyle \mathbf {A} \cdot \!\!\times \mathbf {B} ={\vec {\mathrm {i} }}(\mathbf {A} \cdot \mathbf {B} )}$

#Symmetrische Tensoren haben keine Vektorinvariante: ${\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} ^{\mathrm {S} })={\vec {0}}}$

Die Eigenschaften des dualen axialen Vektors sind hierher übertragbar. Seien x eine beliebige Zahl, ${\displaystyle {\vec {u}},\,{\vec {v}}}$ beliebige Vektoren und A, B beliebige Tensoren zweiter Stufe. Dann gilt:

${\displaystyle {\vec {\mathrm {i} }}({\vec {u}}\otimes {\vec {v}})\;\;={\vec {u}}\times {\vec {v}}}$

${\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} )\times {\vec {v}}\;=-2\mathbf {A} ^{\mathrm {A} }\cdot {\vec {v}}=(\mathbf {A^{\top }-A} )\cdot {\vec {v}}}$

${\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} ^{\top })\quad \;=-{\vec {\mathrm {i} }}(\mathbf {A} )}$

${\displaystyle {\vec {\mathrm {i} }}(\mathbf {A+B} )={\vec {\mathrm {i} }}(\mathbf {A} )+{\vec {\mathrm {i} }}(\mathbf {B} )}$

${\displaystyle {\vec {\mathrm {i} }}(x\mathbf {A} )\quad \;=x\,{\vec {\mathrm {i} }}(\mathbf {A} )}$

${\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} \#\mathbf {B} )\;=\mathbf {A} \cdot {\vec {\mathrm {i} }}(\mathbf {B} )+\mathbf {B} \cdot {\vec {\mathrm {i} }}(\mathbf {A} )}$

${\displaystyle \mathbf {A} \cdot {\vec {\mathrm {i} }}(\mathbf {A} )\;=\mathbf {A} ^{\top }\cdot {\vec {\mathrm {i} }}(\mathbf {A} )={\vec {\mathrm {i} }}(\mathbf {A} )\cdot \mathbf {A} }$

${\displaystyle {\vec {\mathrm {i} }}(\mathrm {cof} (\mathbf {A} ))=\mathbf {A} \cdot {\vec {\mathrm {i} }}(\mathbf {A} )}$

${\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} ^{-1})\quad =-{\frac {1}{\mathrm {det} (\mathbf {A} )}}\mathbf {A} \cdot {\vec {\mathrm {i} }}(\mathbf {A} )\quad {\text{falls}}\quad \mathrm {det} (\mathbf {A} )\neq 0}$

${\displaystyle {\vec {\mathrm {i} }}({\vec {v}}\times \mathbf {A} )\;=(\mathbf {A} -\mathrm {Sp} (\mathbf {A} )\mathbf {1} )\cdot {\vec {v}}=-(\mathbf {A} ^{\top }\#\mathbf {1} )\cdot {\vec {v}}}$

${\displaystyle {\vec {\mathrm {i} }}({\vec {v}}\times \mathbf {1} )\;\;=-2{\vec {v}}}$

${\displaystyle {\vec {\mathrm {i} }}(({\vec {u}}\times {\vec {v}})\times \mathbf {A} )={\vec {v}}\cdot \mathbf {A} \times {\vec {u}}-{\vec {u}}\cdot \mathbf {A} \times {\vec {v}}}$

${\displaystyle {\vec {\mathrm {i} }}(\mathbf {B\cdot A\cdot B} ^{\top })=\mathrm {cof} (\mathbf {B} )\cdot {\vec {\mathrm {i} }}(\mathbf {A} )}$

Darin i​st „#“ e​in #Äußeres Tensorprodukt, cof(·) i​st der #Kofaktor.

Spezielle Tensoren

Dyade

Siehe auch: Dyadisches Produkt

Definition

${\displaystyle \mathbf {A}$ :={\vec {a}}\otimes {\vec {b}}}

Kofaktor: ${\displaystyle \mathrm {cof} (\mathbf {A} )=\mathbf {0} }$

#Invarianten:

${\displaystyle \mathrm {Sp} (\mathbf {A} )={\vec {a}}\cdot {\vec {b}}}$

${\displaystyle \mathrm {I} _{2}(\mathbf {A} )=0}$

${\displaystyle \mathrm {det} (\mathbf {A} )=0}$

${\displaystyle \parallel \mathbf {A} \parallel =|{\vec {a}}|\,|{\vec {b}}|}$

#Eigensystem:

${\displaystyle {\begin{aligned}\lambda _{1}=&{\vec {a}}\cdot {\vec {b}},&{\vec {v}}_{1}=&{\frac {\vec {a}}{|{\vec {a}}|}}\\\lambda _{2}=&0,&{\vec {v}}_{2}=&{\frac {{\vec {a}}\times {\vec {b}}}{|{\vec {a}}\times {\vec {b}}|}}\\\lambda _{3}=&0,&{\vec {v}}_{3}=&{\frac {({\vec {a}}\times {\vec {b}})\times {\vec {b}}}{|({\vec {a}}\times {\vec {b}})\times {\vec {b}}|}}\end{aligned}}}$

Dyadentripel

Gegeben e​in beliebiger Tensor 2. Stufe A. Dieser k​ann immer a​ls Summe dreier Dyaden dargestellt werden:

${\displaystyle {\begin{aligned}\mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=&{\vec {s}}_{j}\otimes {\hat {e}}_{j}={\begin{pmatrix}{\vec {s}}_{1}&{\vec {s}}_{2}&{\vec {s}}_{3}\end{pmatrix}}\\=&{\hat {e}}_{i}\otimes {\vec {z}}_{i}={\begin{pmatrix}{\vec {z}}_{1}&{\vec {z}}_{2}&{\vec {z}}_{3}\end{pmatrix}}^{\top }\\=&{\vec {a}}_{k}\otimes {\vec {g}}_{k}\end{aligned}}}$

mit Spaltenvektoren ${\displaystyle {\vec {s}}_{j}=A_{ij}{\hat {e}}_{i}=\mathbf {A} \cdot {\hat {e}}_{j}}$, Zeilenvektoren ${\displaystyle {\vec {z}}_{i}=A_{ij}{\hat {e}}_{j}={\hat {e}}_{i}\cdot \mathbf {A} }$ und ${\displaystyle {\vec {g}}_{k}=({\vec {a}}^{k}\cdot {\hat {e}}_{i})A_{ij}{\hat {e}}_{j}={\vec {a}}^{k}\cdot \mathbf {A} }$.

#Hauptinvarianten (${\displaystyle x_{m,n}:={\vec {x}}_{m}\cdot {\hat {e}}_{n}}$):

${\displaystyle \mathrm {I} _{1}(\mathbf {A} )=s_{i,i}=z_{i,i}={\vec {a}}_{i}\cdot {\vec {g}}_{i}}$

${\displaystyle {\begin{aligned}\mathrm {I} _{2}(\mathbf {A} )=&{\frac {1}{2}}(s_{i,i}s_{j,j}-s_{i,j}s_{j,i})={\frac {1}{2}}(z_{i,i}z_{j,j}-z_{i,j}z_{j,i})\\=&{\frac {1}{2}}[({\vec {a}}_{i}\cdot {\vec {g}}_{i})({\vec {a}}_{j}\cdot {\vec {g}}_{j})-({\vec {a}}_{i}\cdot {\vec {g}}_{j})({\vec {a}}_{j}\cdot {\vec {g}}_{i})]\end{aligned}}}$

${\displaystyle \mathrm {I} _{3}(\mathbf {A} )={\begin{vmatrix}{\vec {s}}_{1}&{\vec {s}}_{2}&{\vec {s}}_{3}\end{vmatrix}}={\begin{vmatrix}{\vec {z}}_{1}&{\vec {z}}_{2}&{\vec {z}}_{3}\end{vmatrix}}={\begin{vmatrix}{\vec {a}}_{1}&{\vec {a}}_{2}&{\vec {a}}_{3}\end{vmatrix}}{\begin{vmatrix}{\vec {g}}_{1}&{\vec {g}}_{2}&{\vec {g}}_{3}\end{vmatrix}}}$

#Betrag:

${\displaystyle \parallel \mathbf {A} \parallel ={\sqrt {{\vec {s}}_{i}\cdot {\vec {s}}_{i}}}={\sqrt {{\vec {z}}_{i}\cdot {\vec {z}}_{i}}}={\sqrt {({\vec {a}}_{i}\cdot {\vec {a}}_{j})({\vec {g}}_{i}\cdot {\vec {g}}_{j})}}}$

#Dualer axialer Vektor:

${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}={\frac {1}{2}}{\hat {e}}_{i}\times {\vec {s}}_{i}={\frac {1}{2}}{\vec {z}}_{i}\times {\hat {e}}_{i}={\frac {1}{2}}{\vec {g}}_{i}\times {\vec {a}}_{i}}$

#Vektorinvariante:

${\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} )={\vec {s}}_{i}\times {\hat {e}}_{i}={\hat {e}}_{i}\times {\vec {z}}_{i}={\vec {a}}_{i}\times {\vec {g}}_{i}}$

#Kofaktor:

${\displaystyle {\begin{aligned}\mathrm {cof} (\mathbf {A} )=&{\vec {z}}_{i}\otimes {\vec {s}}_{i}-\mathrm {I} _{1}(\mathbf {A} )\mathbf {A} ^{\top }+\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} \\=&({\vec {a}}_{i}\cdot {\vec {g}}_{j}){\vec {g}}_{i}\otimes {\vec {a}}_{j}-({\vec {a}}_{i}\cdot {\vec {g}}_{i}){\vec {g}}_{j}\otimes {\vec {a}}_{j}+\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} \end{aligned}}}$

#Inverse:

${\displaystyle \mathbf {A} ^{-1}={\hat {e}}_{i}\otimes {\vec {s}}^{i}={\vec {z}}^{i}\otimes {\hat {e}}_{i}={\vec {g}}^{i}\otimes {\vec {a}}^{i}}$

Einheitstensor

Siehe auch: Einheitstensor

${\displaystyle \mathbf {1} ={\hat {e}}_{i}\otimes {\hat {e}}_{i}=\delta _{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}$

${\displaystyle \mathbf {1} ={\vec {g}}_{i}\otimes {\vec {g}}^{i}={\vec {g}}^{i}\otimes {\vec {g}}_{i}=g^{ij}{\vec {g}}_{i}\otimes {\vec {g}}_{j}=g_{ij}{\vec {g}}^{i}\otimes {\vec {g}}^{j}}$

mit ${\displaystyle g_{ij}={\vec {g}}_{i}\cdot {\vec {g}}_{j}\,,\;g^{ij}={\vec {g}}^{i}\cdot {\vec {g}}^{j}}$

Allgemein:

${\displaystyle \mathbf {1} =({\vec {a}}^{i}\cdot {\vec {g}}^{j}){\vec {a}}_{i}\otimes {\vec {g}}_{j}}$

#Transposition u​nd #Inverse:

${\displaystyle \mathbf {1} =\mathbf {1} ^{\top }=\mathbf {1} ^{-1}=\mathbf {1} ^{\top -1}}$

Kofaktor: ${\displaystyle \mathrm {cof} (\mathbf {1} )=\mathbf {1} }$

Vektortransformation

${\displaystyle \mathbf {1} \cdot {\vec {v}}={\vec {v}}\cdot \mathbf {1} ={\vec {v}}}$

Tensorprodukt

${\displaystyle \mathbf {A\cdot 1} =\mathbf {1\cdot A} =\mathbf {A} }$

Skalarprodukt

${\displaystyle \mathbf {A}$ :\mathbf {1} =\mathrm {Sp} (\mathbf {A} )}

#Invarianten:

${\displaystyle \mathrm {Sp} (\mathbf {1} )=\mathbf {1}$ :\mathbf {1} =3}

${\displaystyle \mathrm {I} _{2}(\mathbf {1} )=3}$

${\displaystyle \mathrm {det} (\mathbf {1} )=1}$

${\displaystyle \parallel \mathbf {1} \parallel ={\sqrt {3}}}$

#Eigenwerte:

${\displaystyle \lambda _{1,2,3}=1}$

Alle Vektoren s​ind #Eigenvektoren.

Unimodulare Tensoren

Siehe auch: Spezielle lineare Gruppe

Definition

${\displaystyle \mathbf {H}$ :\quad \mathrm {det} (\mathbf {H} )=1}

Kofaktor: ${\displaystyle \mathrm {cof} (\mathbf {H} )=\mathbf {H} ^{\top -1}}$

Determinantenproduktsatz:

${\displaystyle \mathrm {det} (\mathbf {A\cdot H} )=\mathrm {det} (\mathbf {H\cdot A} )=\mathrm {det} (\mathbf {A} )}$

Orthogonale Tensoren

Siehe auch: Orthogonaler Tensor

Definition

${\displaystyle \mathbf {Q}$ :\quad \mathbf {Q} ^{-1}=\mathbf {Q} ^{\top }\quad {\textsf {oder}}\quad \mathbf {Q\cdot Q} ^{\top }=\mathbf {Q} ^{\top }\cdot \mathbf {Q} =\mathbf {1} }

Kofaktor: ${\displaystyle \mathrm {cof} (\mathbf {Q} )=\mathrm {det} (\mathbf {Q} )\mathbf {Q} =\pm \mathbf {Q} }$

#Invarianten (${\displaystyle \alpha }$ ist der Drehwinkel):

${\displaystyle \mathrm {Sp} (\mathbf {Q} )=\mathrm {det} (\mathbf {Q} )+2\cos(\alpha )}$

${\displaystyle \mathrm {I} _{2}(\mathbf {Q} )=\mathrm {det} (\mathbf {Q} )\cdot \mathrm {Sp} (\mathbf {Q} )=1+2\,\mathrm {det} (\mathbf {Q} )\cos(\alpha )}$

${\displaystyle \mathrm {det} (\mathbf {Q} )=\pm 1}$

${\displaystyle \parallel \mathbf {Q} \parallel ={\sqrt {3}}}$

Eigentlich orthogonaler Tensor ${\displaystyle \mathrm {det} (\mathbf {Q} )=+1}$, entspricht einer Drehung.

Uneigentlich orthogonaler Tensor ${\displaystyle \mathrm {det} (\mathbf {Q} )=-1}$, entspricht einer Drehspiegelung.

Spatprodukt:

${\displaystyle (\mathbf {Q} \cdot {\vec {a}})\cdot [(\mathbf {Q} \cdot {\vec {b}})\times (\mathbf {Q} \cdot {\vec {c}})]=\mathrm {det} (\mathbf {Q} ){\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})}$

Kreuzprodukt u​nd #Kofaktor:

${\displaystyle (\mathbf {Q} \cdot {\vec {a}})\times (\mathbf {Q} \cdot {\vec {b}})=\mathrm {det} (\mathbf {Q} )\mathbf {Q} \cdot ({\vec {a}}\times {\vec {b}})}$

${\displaystyle \mathrm {cof} (\mathbf {Q} )=\mathrm {det} (\mathbf {Q} )\mathbf {Q} }$

Gegeben ein Einheitsvektor ${\displaystyle {\hat {n}}={\begin{pmatrix}n_{1}&n_{2}&n_{3}\end{pmatrix}}^{\top }}$ und Drehwinkel α. Dann sind die folgenden Tensoren R zueinander gleich, orthogonal und drehen um die Achse ${\displaystyle {\hat {n}}}$ mit Winkel α:

Rodrigues-Formel:

${\displaystyle {\begin{aligned}\mathbf {R} =&\mathbf {1} +s_{\alpha }{\hat {n}}\times \mathbf {1} +d_{\alpha }({\hat {n}}\times \mathbf {1} )^{2}\\=&\mathbf {1} +s_{\alpha }{\hat {n}}\times \mathbf {1} +d_{\alpha }({\hat {n}}\otimes {\hat {n}}-\mathbf {1} )\end{aligned}}}$

${\displaystyle \mathbf {R} ={\begin{pmatrix}c_{\alpha }+d_{\alpha }n_{1}^{2}&-s_{\alpha }n_{3}+d_{\alpha }n_{1}n_{2}&s_{\alpha }n_{2}+d_{\alpha }n_{1}n_{3}\\s_{\alpha }n_{3}+d_{\alpha }n_{1}n_{2}&c_{\alpha }+d_{\alpha }n_{2}^{2}&-s_{\alpha }n_{1}+d_{\alpha }n_{2}n_{3}\\-s_{\alpha }n_{2}+d_{\alpha }n_{1}n_{3}&s_{\alpha }n_{1}+d_{\alpha }n_{2}n_{3}&c_{\alpha }+d_{\alpha }n_{3}^{2}\end{pmatrix}}}$

mit ${\displaystyle c_{\alpha }=\cos(\alpha ),\;d_{\alpha }=1-\cos(\alpha ),\;s_{\alpha }=\sin(\alpha )}$.

Euler-Rodrigues-Formel: ${\displaystyle a=\cos \left({\tfrac {\alpha }{2}}\right),b=\sin \left({\tfrac {\alpha }{2}}\right)n_{1},c=\sin \left({\tfrac {\alpha }{2}}\right)n_{2},d=\sin \left({\tfrac {\alpha }{2}}\right)n_{3}}$ also ${\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1}$:

${\displaystyle \mathbf {R}$ :={\begin{pmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2(bc-ad)&2(bd+ac)\\2(bc+ad)&a^{2}+c^{2}-b^{2}-d^{2}&2(cd-ab)\\2(bd-ac)&2(cd+ab)&a^{2}+d^{2}-b^{2}-c^{2}\end{pmatrix}}}

Formulierung m​it Drehvektor:

Drehvektor		Orthogonaler Tensor
${\displaystyle {\vec {\alpha }}=\alpha {\vec {n}}}$	→	${\displaystyle \mathbf {R} =\mathbf {1} +{\frac {\sin(\alpha )}{\alpha }}{\vec {\alpha }}\times \mathbf {1} +{\frac {1-\cos(\alpha )}{\alpha ^{2}}}({\vec {\alpha }}\times \mathbf {1} )^{2}}$
${\displaystyle {\vec {\alpha }}=\tan(\alpha ){\vec {n}}}$	→	${\displaystyle \mathbf {R} =\mathbf {1} +\cos(\alpha ){\vec {\alpha }}\times \mathbf {1} +{\frac {\cos ^{2}(\alpha )}{1+\cos(\alpha )}}({\vec {\alpha }}\times \mathbf {1} )^{2}}$
${\displaystyle {\vec {\alpha }}=\tan \left({\frac {\alpha }{2}}\right)\;{\vec {n}}}$	→	${\displaystyle \mathbf {R} =\mathbf {1} +{\frac {2}{1+{\vec {\alpha }}\cdot {\vec {\alpha }}}}({\vec {\alpha }}\times \mathbf {1} +({\vec {\alpha }}\times \mathbf {1} )^{2})}$
${\displaystyle {\vec {\alpha }}=\sin(\alpha )\;{\vec {n}}}$	→	${\displaystyle \mathbf {R} =\mathbf {1} +{\vec {\alpha }}\times \mathbf {1} +{\dfrac {1}{1+\cos(\alpha )}}({\vec {\alpha }}\times \mathbf {1} )^{2}}$
${\displaystyle {\vec {\alpha }}=\sin \left({\frac {\alpha }{2}}\right)\;{\vec {n}}}$	→	${\displaystyle \mathbf {R} =\mathbf {1} +2\cos \left({\frac {\alpha }{2}}\right){\vec {\alpha }}\times \mathbf {1} +2({\vec {\alpha }}\times \mathbf {1} )^{2}}$
${\displaystyle {\vec {\alpha }}=\cos(\alpha )\;{\vec {n}}}$	→	${\displaystyle \mathbf {R} =\mathbf {1} +\tan(\alpha ){\vec {\alpha }}\times \mathbf {1} +{\frac {1-\cos(\alpha )}{{\vec {\alpha }}\cdot {\vec {\alpha }}}}({\vec {\alpha }}\times \mathbf {1} )^{2}}$
${\displaystyle {\vec {\alpha }}=\cos \left({\frac {\alpha }{2}}\right)\;{\vec {n}}}$	→	${\displaystyle \mathbf {R} =\mathbf {1} +2\sin \left({\frac {\alpha }{2}}\right){\vec {\alpha }}\times \mathbf {1} +2{\frac {1-{\vec {\alpha }}\cdot {\vec {\alpha }}}{{\vec {\alpha }}\cdot {\vec {\alpha }}}}({\vec {\alpha }}\times \mathbf {1} )^{2}}$

Darin ist ${\displaystyle ({\vec {\alpha }}\times \mathbf {1} )^{2}=({\vec {\alpha }}\times \mathbf {1} )\cdot ({\vec {\alpha }}\times \mathbf {1} )={\vec {\alpha }}\otimes {\vec {\alpha }}-({\vec {\alpha }}\cdot {\vec {\alpha }})\mathbf {1} }$

Beispiel für Drehspiegelung:

${\displaystyle \mathbf {Q} =-\mathbf {1} +\sin(\alpha ){\hat {n}}\times \mathbf {1} -(1+\cos(\alpha ))({\hat {n}}\times \mathbf {1} )^{2}}$

Drehung von Vektorraumbasis ${\displaystyle {\vec {u}}_{1,2,3}\;{\textsf {nach}}\;{\vec {v}}_{1,2,3}}$ mit Drehachse ${\displaystyle {\hat {n}}}$:

${\displaystyle \mathbf {Q} \cdot {\vec {u}}_{i}={\vec {v}}_{i}\,,\quad \mathbf {Q} \cdot {\vec {u}}^{i}={\vec {v}}^{i}\,,\quad \mathbf {Q} ={\vec {v}}_{i}\otimes {\vec {u}}^{i}={\vec {v}}^{i}\otimes {\vec {u}}_{i}}$

${\displaystyle {\hat {n}}\simeq {\vec {v}}_{i}\times {\vec {u}}^{i}={\vec {v}}^{i}\times {\vec {u}}_{i}=-2{\stackrel {A}{\overrightarrow {\mathbf {Q} }}}={\vec {\mathrm {i} }}(\mathbf {Q} )}$

mit #Dualer axialer Vektor ${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {Q} }}}}$ und #Vektorinvariante ${\displaystyle {\vec {\mathrm {i} }}(\mathbf {Q} )}$.

Gegeben Orthonormalbasis ${\displaystyle {\hat {v}}_{1,2,3}}$, Drehwinkel ${\displaystyle \alpha }$ und ${\displaystyle {\hat {v}}_{1}}$ ist Drehachse:

${\displaystyle {\begin{aligned}\mathbf {Q} =&{\color {red}\pm }{\hat {v}}_{1}\otimes {\hat {v}}_{1}+\cos(\alpha )({\hat {v}}_{2}\otimes {\hat {v}}_{2}+{\hat {v}}_{3}\otimes {\hat {v}}_{3})+\sin(\alpha )({\hat {v}}_{3}\otimes {\hat {v}}_{2}-{\hat {v}}_{2}\otimes {\hat {v}}_{3})\\=&{\begin{pmatrix}{\color {red}\pm 1}&0&0\\0&\cos(\alpha )&-\sin(\alpha )\\0&\sin(\alpha )&\cos(\alpha )\end{pmatrix}}_{{\hat {v}}_{i}\otimes {\hat {v}}_{j}}\end{aligned}}}$

${\displaystyle {\color {red}+1}}$: Drehung, ${\displaystyle {\color {red}-1}}$: Drehspiegelung um ${\displaystyle {\hat {v}}_{1}}$

Wenn ${\displaystyle {\hat {v}}_{1,2,3}}$ ein Rechtssystem (Mathematik) bilden, dann dreht Q gegen den Uhrzeigersinn, sonst im Uhrzeigersinn um die Drehachse.

#Eigensystem:

${\displaystyle {\begin{aligned}\lambda _{1}=&\mathrm {det} (\mathbf {Q} )\,,&{\vec {q}}_{1}=&{\hat {v}}_{1}\\\lambda _{2}=&e^{\mathrm {i} \alpha },&{\vec {q}}_{2}=&{\frac {1}{\sqrt {2}}}({\hat {v}}_{2}-\mathrm {i} {\hat {v}}_{3}).\\\lambda _{3}=&e^{-\mathrm {i} \alpha },&{\vec {q}}_{3}=&{\frac {1}{\sqrt {2}}}({\hat {v}}_{2}+\mathrm {i} {\hat {v}}_{3})\end{aligned}}}$

Drehwinkel:

${\displaystyle \cos(\alpha )={\frac {1}{2}}(\mathrm {Sp} (\mathbf {Q} )-\mathrm {det} (\mathbf {Q} ))}$

Drehachse ${\displaystyle {\hat {n}}}$ ist #Vektorinvariante:

${\displaystyle {\hat {n}}\simeq {\vec {\mathrm {i} }}(\mathbf {Q} )=\mathbf {1} \cdot \!\!\times \mathbf {Q} }$

${\displaystyle \mathbf {Q} ={\vec {s}}_{i}\otimes {\vec {e}}_{i}={\vec {e}}_{i}\otimes {\vec {z}}_{i}\quad \rightarrow \quad {\hat {n}}\simeq {\vec {s}}_{i}\times {\vec {e}}_{i}={\vec {e}}_{i}\times {\vec {z}}_{i}}$

${\displaystyle {\frac {1}{2}}(\mathbf {Q} -\mathbf {Q} ^{\top })=\sin(\alpha ){\hat {n}}\times \mathbf {1} =\sin(\alpha ){\begin{pmatrix}0&-n_{3}&n_{2}\\n_{3}&0&-n_{1}\\-n_{2}&n_{1}&0\end{pmatrix}},\quad |{\hat {n}}|=1}$

Positiv definite Tensoren

Definition

${\displaystyle \mathbf {A}$ :\quad {\vec {v}}\cdot \mathbf {A} \cdot {\vec {v}}>0\quad \forall \;{\vec {v}}\in \mathbb {V} \setminus \{{\vec {0}}\}}

Kofaktor: ${\displaystyle \mathrm {cof} (\mathbf {A} )=\mathbf {A^{\top }\cdot A^{\top }} -\mathrm {I} _{1}(\mathbf {A} )\mathbf {A} ^{\top }+\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} =\det(\mathbf {A} )\mathbf {A} ^{\top -1}}$

Notwendige Bedingungen für positive Definitheit:

${\displaystyle \mathrm {det} (\mathbf {A} )>0}$

${\displaystyle \mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\quad \rightarrow \quad A_{11},\,A_{22},\,A_{33}>0}$

${\displaystyle \mathbf {A} =A_{j}^{i}{\vec {a}}_{i}\otimes {\vec {a}}^{j}\quad \rightarrow \quad A_{1}^{1},\,A_{2}^{2},\,A_{3}^{3}>0}$

Notwendige u​nd hinreichende Bedingung für positive Definitheit: Alle #Eigenwerte v​on A s​ind größer a​ls null.

Immer positiv definit f​alls det(A) ≠ 0:

A·A^⊤ und A^⊤·A

Symmetrische Tensoren

Siehe auch: Symmetrische Matrix

Definition

${\displaystyle \mathbf {A}$ :\quad \mathbf {A} =\mathbf {A} ^{\top }}

Kofaktor: ${\displaystyle \mathrm {cof} (\mathbf {A} )=\mathrm {adj} (\mathbf {A} )=\mathbf {A} ^{2}-\mathrm {Sp} (\mathbf {A} )\mathbf {A} +\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} }$

#Betrag:

${\displaystyle \quad \parallel \mathbf {A} \parallel ={\sqrt {\mathrm {Sp} ^{2}(\mathbf {A} )-2\mathrm {I} _{2}(\mathbf {A} )}}={\sqrt {\mathrm {Sp} (\mathbf {A} ^{2})}}={\sqrt {\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}}}}$

Bei Symmetrischen Tensoren verschwinden i​hr #Dualer axialer Vektor u​nd ihre #Vektorinvariante:

${\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} ^{\mathrm {S} }}}}={\vec {\mathrm {i} }}(\mathbf {A} ^{\mathrm {S} })={\vec {0}}}$

Bilinearform:

${\displaystyle {\vec {u}}\cdot \mathbf {A} \cdot {\vec {v}}={\vec {v}}\cdot \mathbf {A} \cdot {\vec {u}}\quad \forall {\vec {u}},{\vec {v}}\in \mathbb {V} }$

Alle #Eigenwerte λ_1,2,3 sind reell. Alle #Eigenvektoren ${\displaystyle {\vec {a}}_{1,2,3}}$ sind reell und paarweise orthogonal zueinander oder orthogonalisierbar. Hauptachsentransformation:

${\displaystyle {\begin{aligned}\mathbf {A} =&\sum _{i=1}^{3}\lambda _{i}{\hat {a}}_{i}\otimes {\hat {a}}_{i}=({\hat {a}}_{i}\otimes {\hat {e}}_{i})\left(\sum _{i=1}^{3}\lambda _{j}{\hat {e}}_{j}\otimes {\hat {e}}_{j}\right)({\hat {e}}_{k}\otimes {\hat {a}}_{k})\\=&{\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}\cdot {\begin{pmatrix}\lambda _{1}&0&0\\0&\lambda _{2}&0\\0&0&\lambda _{3}\end{pmatrix}}\cdot {\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}^{\top }\end{aligned}}}$

Bezüglich d​er Standardbasis:

${\displaystyle \mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}={\begin{pmatrix}A_{11}&A_{12}&A_{13}\\A_{12}&A_{22}&A_{23}\\A_{13}&A_{23}&A_{33}\end{pmatrix}}}$

#Invarianten:

${\displaystyle \mathrm {Sp} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{11}+A_{22}+A_{33}}$

${\displaystyle \mathrm {I} _{2}(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33}-A_{12}^{2}-A_{13}^{2}-A_{23}^{2}}$

${\displaystyle {\begin{aligned}\mathrm {det} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=&A_{11}(A_{22}A_{33}-A_{23}^{2})+A_{12}(A_{23}A_{13}-A_{12}A_{33})\\&+A_{13}(A_{12}A_{23}-A_{13}A_{22})\end{aligned}}}$

${\displaystyle \parallel A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\parallel ={\sqrt {A_{11}^{2}+A_{22}^{2}+A_{33}^{2}+2A_{12}^{2}+2A_{13}^{2}+2A_{23}^{2}}}}$

Symmetrische und positiv definite Tensoren

Definition

${\displaystyle \mathbf {A}$ :\quad \mathbf {A} =\mathbf {A} ^{\top }\quad {\text{und}}\quad {\vec {v}}\cdot \mathbf {A} \cdot {\vec {v}}>0\quad \forall \;{\vec {v}}\in \mathbb {V} \setminus \{{\vec {0}}\}}

Kofaktor: ${\displaystyle \mathrm {cof} (\mathbf {A} )=\mathrm {adj} (\mathbf {A} )=\mathrm {det} (\mathbf {A} )\mathbf {A} ^{-1}=\mathbf {A} ^{2}-\mathrm {Sp} (\mathbf {A} )\mathbf {A} +\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} }$

Mit den #Eigenwerten ${\displaystyle \lambda _{1},\,\lambda _{2},\,\lambda _{3}}$, den #Eigenvektoren ${\displaystyle {\hat {a}}_{1},\,{\hat {a}}_{2},\,{\hat {a}}_{3}}$ und einer reellwertigen Funktion ${\displaystyle f(x)\in \mathbb {R} }$ eines reellen Argumentes ${\displaystyle x\in \mathbb {R} }$ definiert man über das #Eigensystem symmetrischer Tensoren

${\displaystyle {\begin{aligned}\mathbf {A} =&\sum _{i=1}^{3}\lambda _{i}{\hat {a}}_{i}\otimes {\hat {a}}_{i}\\=&{\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}\cdot {\begin{pmatrix}\lambda _{1}&0&0\\0&\lambda _{2}&0\\0&0&\lambda _{3}\end{pmatrix}}\cdot {\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}^{\top }\end{aligned}}}$

den Funktionswert d​es Tensors:

${\displaystyle {\begin{aligned}f(\mathbf {A} ):=&\sum _{i=1}^{3}f(\lambda _{i}){\hat {a}}_{i}\otimes {\hat {a}}_{i}\\=&{\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}\cdot {\begin{pmatrix}f(\lambda _{1})&0&0\\0&f(\lambda _{2})&0\\0&0&f(\lambda _{3})\end{pmatrix}}\cdot {\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}^{\top }\end{aligned}}}$

Ist f e​ine mehrdeutige Funktion, w​ie die Wurzel (Mathematik), m​it n alternativen Werten, d​ann steht f(A) mehrdeutig für n³ alternative Tensoren.

Insbesondere m​it dem Deformationsgradient F:

Rechter Strecktensor

${\displaystyle \mathbf {U} =+{\sqrt {\mathbf {F} ^{\top }\cdot \mathbf {F} }}}$

Linker Strecktensor

${\displaystyle \mathbf {v} =+{\sqrt {\mathbf {F\cdot F} ^{\top }}}}$

Henky-Dehnung

${\displaystyle \mathbf {E} _{H}:=\ln(\mathbf {U} )={\frac {1}{2}}\ln(\mathbf {F} ^{\top }\cdot \mathbf {F} )}$

Voigt-Notation symmetrischer Tensoren zweiter Stufe

Siehe auch: Voigtsche Notation

Die Tensoren

${\displaystyle {\begin{aligned}\mathbf {S} _{1}=&{\vec {e}}_{1}\otimes {\vec {e}}_{1}\\\mathbf {S} _{2}=&{\vec {e}}_{2}\otimes {\vec {e}}_{2}\\\mathbf {S} _{3}=&{\vec {e}}_{3}\otimes {\vec {e}}_{3}\\\mathbf {S} _{4}=&{\vec {e}}_{2}\otimes {\vec {e}}_{3}+{\vec {e}}_{3}\otimes {\vec {e}}_{2}\\\mathbf {S} _{5}=&{\vec {e}}_{1}\otimes {\vec {e}}_{3}+{\vec {e}}_{3}\otimes {\vec {e}}_{1}\\\mathbf {S} _{6}=&{\vec {e}}_{1}\otimes {\vec {e}}_{2}+{\vec {e}}_{2}\otimes {\vec {e}}_{1}\end{aligned}}}$

bilden eine Basis im Vektorraum ${\displaystyle \mathrm {sym} (\mathbb {V} ,\mathbb {V} )\subset {\mathcal {L}}}$ der symmetrischen Tensoren zweiter Stufe. Bezüglich dieser Basis können alle symmetrischen Tensoren zweiter Stufe in Voigt'scher Notation dargestellt werden:

${\displaystyle \mathbf {A} \in \mathrm {sym} (\mathbb {V} ,\mathbb {V} )\quad \rightarrow \quad \mathbf {A} =A_{r}\mathbf {S} _{r}{\hat {=}}{\begin{bmatrix}A_{1}\\A_{2}\\A_{3}\\A_{4}\\A_{5}\\A_{6}\end{bmatrix}}}$

Diese Vektoren dürfen addiert, subtrahiert u​nd mit e​inem Skalar multipliziert werden. Beim Skalarprodukt muss

${\displaystyle \mathbf {A}$ :\mathbf {B} =A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3}+2A_{4}B_{4}+2A_{5}B_{5}+2A_{6}B_{6}}

berücksichtigt werden. Siehe a​uch #Voigt'sche Notation v​on Tensoren vierter Stufe.

Schiefsymmetrische Tensoren

Siehe auch: Schiefsymmetrische Matrix

Definition

${\displaystyle \mathbf {A}$ :\quad \mathbf {A} =-\mathbf {A} ^{\top }}

Kofaktor: ${\displaystyle \mathrm {cof} (\mathbf {A} )=\mathrm {adj} (\mathbf {A} )=\mathbf {A\cdot A} +\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} =\mathbf {A\cdot A} -{\frac {1}{2}}\mathrm {Sp} (\mathbf {A} ^{2})\mathbf {1} }$

#Invarianten:

${\displaystyle \mathrm {Sp} (\mathbf {A} )=0}$

${\displaystyle \mathrm {I} _{2}(\mathbf {A} )=-{\frac {1}{2}}\mathrm {Sp} (\mathbf {A} ^{2})}$

${\displaystyle \mathrm {det} (\mathbf {A} )=0}$

${\displaystyle \quad \parallel \mathbf {A} \parallel ={\sqrt {2\mathrm {I} _{2}(\mathbf {A} )}}={\sqrt {-\mathrm {Sp} (\mathbf {A} ^{2})}}}$

In kartesischen Koordinaten:

${\displaystyle \mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}={\begin{pmatrix}0&A_{12}&A_{13}\\-A_{12}&0&A_{23}\\-A_{13}&-A_{23}&0\end{pmatrix}}}$

#Invarianten:

${\displaystyle \mathrm {Sp} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=0}$

${\displaystyle \mathrm {I} _{2}(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{12}^{2}+A_{13}^{2}+A_{23}^{2}}$

${\displaystyle \mathrm {det} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=0}$

${\displaystyle \parallel A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\parallel ={\sqrt {2}}{\sqrt {A_{12}^{2}+A_{13}^{2}+A_{23}^{2}}}}$

Bilinearform:

${\displaystyle {\vec {u}}\cdot \mathbf {A} \cdot {\vec {v}}=-{\vec {v}}\cdot \mathbf {A} \cdot {\vec {u}}\quad \forall {\vec {u}},{\vec {v}}\in \mathbb {V} }$

${\displaystyle {\vec {v}}\cdot \mathbf {A} \cdot {\vec {v}}=0\quad \forall {\vec {v}}\in \mathbb {V} }$

Ein Eigenwert i​st null, z​wei imaginär konjugiert komplex, s​iehe #Axialer Tensor o​der Kreuzproduktmatrix.

#Dualer axialer Vektor:

${\displaystyle {\begin{aligned}&\mathbf {A} _{\times }:={\stackrel {A}{\overrightarrow {\mathbf {A} }}}:=-{\frac {1}{2}}\mathbf {1} \times \mathbf {A} ^{\top }=-{\frac {1}{2}}\mathbf {1} \cdot \!\!\times \mathbf {A} =-{\frac {1}{2}}{\vec {\mathrm {i} }}(\mathbf {A} )\\&\rightarrow \quad \mathbf {A} \cdot {\vec {v}}={\stackrel {A}{\overrightarrow {\mathbf {A} }}}\times {\vec {v}}\quad \forall {\vec {v}}\in \mathbb {V} \end{aligned}}}$

mit #Vektorinvariante ${\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} )}$. Der zum Eigenwert null gehörende #Eigenvektor ist proportional zum dualen axialen Vektor ${\displaystyle \mathbf {A} _{\times }}$ denn

${\displaystyle \mathbf {A\cdot A} _{\times }=\mathbf {A} _{\times }\times \mathbf {A} _{\times }={\vec {0}}}$

${\displaystyle \mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\quad \rightarrow \;\mathbf {A} _{\times }=-{\frac {1}{2}}A_{ij}{\hat {e}}_{i}\times {\hat {e}}_{j}={\begin{pmatrix}-A_{23}\\A_{13}\\-A_{12}\end{pmatrix}}}$

${\displaystyle \mathbf {A} =A_{ij}({\vec {a}}_{i}\otimes {\vec {b}}_{j}-{\vec {b}}_{j}\otimes {\vec {a}}_{i})\quad \rightarrow \;\mathbf {A} _{\times }=-A_{ij}{\vec {a}}_{i}\times {\vec {b}}_{j}}$

Axialer Tensor oder Kreuzproduktmatrix

Kreuzproduktmatrix ${\displaystyle [{\vec {u}}]_{\times }}$ eines Vektors ${\displaystyle {\vec {u}}}$:

${\displaystyle {\begin{aligned}{\vec {u}}=u_{i}{\hat {e}}_{i}=&{\begin{pmatrix}u_{1}\\u_{2}\\u_{3}\end{pmatrix}}\\\rightarrow \;[{\vec {u}}]_{\times }=&{\vec {u}}\times \mathbf {1} ={\vec {u}}\times {\hat {e}}_{i}\otimes {\hat {e}}_{i}=-{\stackrel {3}{\mathbf {E} }}\cdot {\vec {u}}={\begin{pmatrix}0&-u_{3}&u_{2}\\u_{3}&0&-u_{1}\\-u_{2}&u_{1}&0\end{pmatrix}}\in {\mathcal {L}}\end{aligned}}}$

Kofaktor: ${\displaystyle \mathrm {cof} ({\vec {u}}\times \mathbf {1} )=\mathrm {adj} ({\vec {u}}\times \mathbf {1} )={\vec {u}}\otimes {\vec {u}}}$

#Invarianten:

${\displaystyle \mathrm {Sp} =0}$

${\displaystyle \mathrm {I} _{2}={\vec {u}}\cdot {\vec {u}}=u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}$

${\displaystyle \mathrm {det} =0}$

${\displaystyle \|{\vec {u}}\times \mathbf {1} \|={\sqrt {2{\vec {u}}\cdot {\vec {u}}}}={\sqrt {2}}{\sqrt {u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}}}$

${\displaystyle {\stackrel {A}{\overrightarrow {{\vec {u}}\times \mathbf {1} }}}={\vec {u}}}$

#Eigensystem:

${\displaystyle {\begin{aligned}\lambda _{1}=&0\,,&{\vec {v}}_{1}=&{\vec {u}}\\\lambda _{2,3}=&\mp \mathrm {i} |{\vec {u}}|\,,&{\vec {v}}_{2,3}&\simeq &{\frac {u_{1}}{|{\vec {u}}|}}{\begin{pmatrix}u_{1}\\u_{2}\\u_{3}\end{pmatrix}}\pm \mathrm {i} {\begin{pmatrix}\pm \mathrm {i} |{\vec {u}}|\\-u_{3}\\u_{2}\end{pmatrix}}\end{aligned}}}$

Eigenschaften:

${\displaystyle {\vec {u}}\times {\vec {v}}=({\vec {u}}\times \mathbf {1} )\cdot {\vec {v}}={\vec {u}}\cdot ({\vec {v}}\times \mathbf {1} )}$

${\displaystyle {\vec {u}}\times \mathbf {1} =\mathbf {1} \times {\vec {u}}}$

${\displaystyle ({\vec {u}}\times \mathbf {1} )^{\top }=-{\vec {u}}\times \mathbf {1} }$

${\displaystyle {\vec {u}}=-{\frac {1}{2}}\mathbf {1} \cdot \!\!\times ({\vec {u}}\times \mathbf {1} )=-{\frac {1}{2}}(\mathbf {1} \times {\vec {u}})\times \mathbf {1} ={\frac {1}{2}}\mathbf {1} \times ({\vec {u}}\times \mathbf {1} )}$

${\displaystyle {\vec {u}}\times ({\vec {v}}\times \mathbf {1} )=({\vec {u}}\times \mathbf {1} )\cdot ({\vec {v}}\times \mathbf {1} )={\vec {v}}\otimes {\vec {u}}-({\vec {u}}\cdot {\vec {v}})\mathbf {1} }$

${\displaystyle {\vec {u}}\times ({\vec {v}}\times \mathbf {1} )\cdot {\vec {w}}={\vec {u}}\times ({\vec {v}}\times {\vec {w}})=({\vec {u}}\cdot {\vec {w}}){\vec {v}}-({\vec {u}}\cdot {\vec {v}}){\vec {w}}}$

Potenzen von ${\displaystyle [{\vec {u}}]_{\times }={\vec {u}}\times \mathbf {1} }$

${\displaystyle [{\vec {u}}]_{\times }^{2}=[{\vec {u}}]_{\times }\cdot [{\vec {u}}]_{\times }={\vec {u}}\otimes {\vec {u}}-({\vec {u}}\cdot {\vec {u}})\mathbf {1} }$

${\displaystyle [{\vec {u}}]_{\times }^{3}=-({\vec {u}}\cdot {\vec {u}})[{\vec {u}}]_{\times }}$

Deviatorische Tensoren

Siehe auch: Deviator

Definition

${\displaystyle \mathbf {A}$ :\quad \mathrm {Sp} (\mathbf {A} )=0}

Kofaktor: ${\displaystyle \mathrm {cof} (\mathbf {A} )=\left(\mathbf {A} ^{2}\right)^{\top }+\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} =\left(\mathbf {A} ^{2}\right)^{\top }-{\frac {1}{2}}\mathrm {Sp} (\mathbf {A} ^{2})\mathbf {1} }$

#Hauptinvarianten:

${\displaystyle \mathrm {Sp} (\mathbf {A} ):=0}$

${\displaystyle \mathrm {I} _{2}(\mathbf {A} )=-{\frac {1}{2}}\mathrm {Sp} (\mathbf {A} ^{2})}$

${\displaystyle \mathrm {det} (\mathbf {A} )={\frac {1}{3}}\mathrm {Sp} (\mathbf {A} ^{3})}$

Bezüglich d​er Standardbasis:

${\displaystyle \mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}={\begin{pmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&-A_{11}-A_{22}\end{pmatrix}}}$

${\displaystyle \mathrm {Sp} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=0}$

${\displaystyle \mathrm {I} _{2}(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=-A_{11}^{2}-A_{22}^{2}-A_{11}A_{22}-A_{12}A_{21}-A_{13}A_{31}-A_{23}A_{32}}$

${\displaystyle {\begin{aligned}\mathrm {det} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=&-A_{11}(A_{11}A_{22}+A_{22}^{2}+A_{23}A_{32})\\&+A_{12}(A_{23}A_{31}+A_{21}A_{11}+A_{21}A_{22})\\&+A_{13}(A_{21}A_{32}-A_{22}A_{31})\end{aligned}}}$

${\displaystyle \parallel A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\parallel ={\sqrt {\begin{array}{r}2A_{11}^{2}+2A_{22}^{2}+2A_{11}A_{22}+A_{12}^{2}+A_{21}^{2}+\ldots \\\ldots +A_{13}^{2}+A_{31}^{2}+A_{23}^{2}+A_{32}^{2}\end{array}}}}$

Kugeltensoren

Siehe auch: Kugeltensor

Definition

${\displaystyle \mathbf {A}$ :\quad \mathbf {A} =a\mathbf {1} ={\begin{pmatrix}a&0&0\\0&a&0\\0&0&a\end{pmatrix}}}

Kofaktor: ${\displaystyle \mathrm {cof} (\mathbf {A} )=\mathrm {adj} (\mathbf {A} )=a^{2}\mathbf {1} }$

${\displaystyle \mathrm {Sp} (\mathbf {A} )=3a}$

${\displaystyle \mathrm {I} _{2}(\mathbf {A} )=3a^{2}}$

${\displaystyle \mathrm {det} (\mathbf {A} )=a^{3}}$

${\displaystyle \parallel \mathbf {A} \parallel ={\sqrt {3}}|a|}$

Dekompositionen eines Tensors

Gegeben ein beliebiger Tensor ${\displaystyle \mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}}$

Symmetrischer Anteil

${\displaystyle \mathbf {A} ^{\mathrm {S} }=\mathrm {sym} (\mathbf {A} ):={\frac {1}{2}}(\mathbf {A} +\mathbf {A} ^{\top })}$

${\displaystyle \mathbf {A} ^{\mathrm {S} }={\frac {1}{2}}{\begin{pmatrix}2A_{11}&A_{12}+A_{21}&A_{13}+A_{31}\\A_{12}+A_{21}&2A_{22}&A_{23}+A_{32}\\A_{13}+A_{31}&A_{23}+A_{32}&2A_{33}\end{pmatrix}}}$

${\displaystyle \mathrm {Sp} (\mathbf {A} ^{\mathrm {S} })=\mathrm {Sp} (\mathbf {A} )=A_{11}+A_{22}+A_{33}}$

${\displaystyle {\begin{aligned}\mathrm {I} _{2}(\mathbf {A} ^{\mathrm {S} })=&{\frac {1}{2}}\mathrm {I} _{2}(\mathbf {A} )+{\frac {1}{4}}\mathrm {Sp} ^{2}(\mathbf {A} )-{\frac {1}{4}}\mathbf {A:A} \\=&A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33}\\&-{\frac {1}{4}}\left[(A_{12}+A_{21})^{2}+(A_{13}+A_{31})^{2}+(A_{23}+A_{32})^{2}\right]\end{aligned}}}$

${\displaystyle {\begin{aligned}\mathrm {det} (\mathbf {A} ^{\mathrm {S} })=&{\frac {1}{4}}\mathrm {det} (\mathbf {A} )+{\frac {1}{4}}\mathbf {A}$ :\mathrm {adj} (\mathbf {A} )\\=&A_{11}A_{22}A_{33}+{\frac {1}{4}}(A_{12}+A_{21})(A_{23}+A_{32})(A_{13}+A_{31})\\&-{\frac {1}{4}}\left[A_{11}(A_{23}+A_{32})^{2}+A_{22}(A_{13}+A_{31})^{2}+A_{33}(A_{12}+A_{21})^{2}\right]\end{aligned}}}

${\displaystyle {\begin{aligned}\parallel (\mathbf {A} ^{\mathrm {S} })\parallel =&{\sqrt {\mathbf {A:A} ^{\mathrm {S} }}}\\=&{\sqrt {A_{11}^{2}+A_{22}^{2}+A_{33}^{2}+{\frac {1}{2}}[(A_{12}+A_{21})^{2}+(A_{13}+A_{31})^{2}+(A_{23}+A_{32})^{2}]}}\end{aligned}}}$

Schiefsymmetrischer Anteil

${\displaystyle \mathbf {A} ^{\mathrm {A} }=\mathrm {skw} (\mathbf {A} ):={\frac {1}{2}}(\mathbf {A} -\mathbf {A} ^{\top })}$

${\displaystyle \mathbf {A} ^{\mathrm {A} }={\frac {1}{2}}{\begin{pmatrix}0&A_{12}-A_{21}&A_{13}-A_{31}\\A_{21}-A_{12}&0&A_{23}-A_{32}\\A_{31}-A_{13}&A_{32}-A_{23}&0\end{pmatrix}}}$

${\displaystyle \mathrm {Sp} (\mathbf {A} ^{\mathrm {A} })=0}$

${\displaystyle {\begin{aligned}\mathrm {I} _{2}(\mathbf {A} ^{\mathrm {A} })=&{\frac {1}{4}}\left[\mathbf {A:A} -\mathrm {Sp} (\mathbf {A} ^{2})\right]\\=&{\frac {1}{4}}\left[(A_{12}-A_{21})^{2}+(A_{13}-A_{31})^{2}+(A_{23}-A_{32})^{2}\right]\end{aligned}}}$

${\displaystyle \mathrm {det} (\mathbf {A} ^{\mathrm {A} })=0}$

${\displaystyle \parallel \mathbf {A} ^{\mathrm {A} }\parallel ={\sqrt {\mathbf {A:A} ^{\mathrm {A} }}}={\sqrt {\frac {1}{2}}}{\sqrt {(A_{12}-A_{21})^{2}+(A_{13}-A_{31})^{2}+(A_{32}-A_{23})^{2}}}}$

Deviator

Siehe auch: Deviator

${\displaystyle \mathbf {A} ^{\mathrm {D} }=\mathrm {dev} (\mathbf {A} ):=\mathbf {A} -{\frac {1}{3}}\mathrm {Sp} (\mathbf {A} )\mathbf {1} }$

${\displaystyle \mathbf {A} ^{\mathrm {D} }={\begin{pmatrix}{\frac {2A_{11}-A_{22}-A_{33}}{3}}&A_{12}&A_{13}\\A_{21}&{\frac {2A_{22}-A_{11}-A_{33}}{3}}&A_{23}\\A_{31}&A_{32}&{\frac {2A_{33}-A_{11}-A_{22}}{3}}\end{pmatrix}}}$

${\displaystyle \mathrm {Sp} (\mathbf {A} ^{\mathrm {D} })=0}$

${\displaystyle {\begin{aligned}\mathrm {I} _{2}(\mathbf {A} ^{\mathrm {D} })=&\mathrm {I} _{2}(\mathbf {A} )-{\frac {1}{3}}\mathrm {Sp} ^{2}(\mathbf {A} )={\frac {1}{6}}\mathrm {Sp} ^{2}(\mathbf {A} )-{\frac {1}{2}}\mathrm {Sp} (\mathbf {A} ^{2})\\=&{\frac {1}{3}}(A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33}-A_{11}^{2}-A_{22}^{2}-A_{33}^{2})\\&-A_{12}A_{21}-A_{13}A_{31}-A_{23}A_{32}\end{aligned}}}$

${\displaystyle {\begin{aligned}\mathrm {det} (\mathbf {A} ^{\mathrm {D} })=&\mathrm {det} (\mathbf {A} )+{\frac {2}{27}}\mathrm {Sp} ^{3}(\mathbf {A} )-{\frac {1}{3}}\mathrm {Sp} (\mathbf {A} )\mathrm {I} _{2}(\mathbf {A} )\\=&{\frac {1}{27}}{\Big [}12A_{11}A_{22}A_{33}+2(A_{11}^{3}+A_{22}^{3}+A_{33}^{3})\ldots \\&\qquad \ldots -3A_{11}^{2}(A_{22}+A_{33})-3A_{22}^{2}(A_{11}+A_{33})-3A_{33}^{2}(A_{11}+A_{22}){\Big ]}\\&-{\frac {1}{3}}{\Big [}(2A_{11}-A_{22}-A_{33})A_{23}A_{32}+(2A_{22}-A_{11}-A_{33})A_{13}A_{31}+\ldots \\&\qquad \ldots +(2A_{33}-A_{11}-A_{22})A_{12}A_{21}{\Big ]}\\&+A_{13}A_{32}A_{21}+A_{12}A_{23}A_{31}\end{aligned}}}$

${\displaystyle \parallel \mathbf {A} ^{\mathrm {D} }\parallel ={\sqrt {\begin{array}{r}{\frac {2}{3}}(A_{11}^{2}+A_{22}^{2}+A_{33}^{2}-A_{11}A_{22}-A_{11}A_{33}-A_{22}A_{33})+\ldots \\\ldots +A_{12}^{2}+A_{21}^{2}+A_{13}^{2}+A_{31}^{2}+A_{23}^{2}+A_{32}^{2}\end{array}}}}$

Kugelanteil

Siehe auch: Kugeltensor

${\displaystyle \mathbf {A} ^{\mathrm {K} }=\mathrm {sph} (\mathbf {A} ):={\frac {1}{3}}\mathrm {Sp} (\mathbf {A} )\mathbf {1} }$

${\displaystyle \mathbf {A} ^{\mathrm {K} }={\frac {1}{3}}(A_{11}+A_{22}+A_{33}){\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}$

${\displaystyle \mathrm {Sp} (\mathbf {A} ^{\mathrm {K} })=\mathrm {Sp} (\mathbf {A} )=A_{11}+A_{22}+A_{33}}$

${\displaystyle \mathrm {I} _{2}(\mathbf {A} ^{\mathrm {K} })={\frac {1}{3}}\mathrm {Sp} ^{2}(\mathbf {A} )={\frac {1}{3}}(A_{11}+A_{22}+A_{33})^{2}}$

${\displaystyle \mathrm {det} (\mathbf {A} ^{\mathrm {K} })={\frac {1}{27}}\mathrm {Sp} ^{3}(\mathbf {A} )={\frac {1}{27}}(A_{11}+A_{22}+A_{33})^{3}}$

${\displaystyle \parallel \mathbf {A} ^{\mathrm {K} }\parallel ={\frac {1}{\sqrt {3}}}|\mathrm {Sp} (\mathbf {A} )|={\frac {1}{\sqrt {3}}}|A_{11}+A_{22}+A_{33}|}$

Beziehung zwischen den Anteilen des Tensors

${\displaystyle \mathbf {A} =\mathbf {A} ^{\mathrm {S} }+\mathbf {A} ^{\mathrm {A} }=\mathbf {A} ^{\mathrm {D} }+\mathbf {A} ^{\mathrm {K} }}$

Symmetrische u​nd schiefsymmetrische Tensoren s​ind orthogonal zueinander:

${\displaystyle \mathbf {A} ^{\mathrm {S} }:\mathbf {B} ^{\mathrm {A} }=0}$

Deviatoren u​nd Kugeltensoren s​ind orthogonal zueinander:

${\displaystyle \mathbf {A} ^{\mathrm {D} }:\mathbf {B} ^{\mathrm {K} }=0}$

Polarzerlegung

Siehe auch: Polarzerlegung

Für j​eden Tensor F m​it #Determinante ≠ 0 g​ibt es #Orthogonale Tensoren Q u​nd #Symmetrische u​nd positiv definite Tensoren U i​n eindeutiger Weise, sodass

F = Q·U

Im Fall d​es Deformationsgradienten i​st U d​er rechte Strecktensor, s​iehe #Symmetrische u​nd positiv definite Tensoren. Der Anteil U berechnet s​ich wie d​ort angegeben aus

${\displaystyle \mathbf {U} =+{\sqrt {\mathbf {F^{\top }\cdot F} }}}$

Dann i​st U·U = F^⊤·F und

${\displaystyle \mathbf {Q} =\mathbf {F\cdot U} ^{-1}}$

Bei det(F)=0 ergeben s​ich U s​owie Q a​us der Singulärwertzerlegung v​on F u​nd U i​st nur n​och symmetrisch positiv semidefinit.

Projektionen

Punkt auf Gerade

Gegeben sei die Gerade durch den Punkt ${\displaystyle {\vec {x}}}$ mit Richtungsvektor ${\displaystyle {\vec {g}}}$ und ein beliebiger anderer Punkt ${\displaystyle {\vec {p}}}$.

Dann ist

${\displaystyle {\begin{aligned}{\vec {p}}=&{\vec {x}}+{\vec {a}}+{\vec {b}}\quad {\textsf {mit}}\quad {\vec {a}}\|{\vec {g}}\quad {\text{und}}\quad {\vec {b}}\bot {\vec {g}}\\\mathbf {G} =&{\frac {{\vec {g}}\otimes {\vec {g}}}{{\vec {g}}\cdot {\vec {g}}}}\quad \rightarrow \quad \mathbf {G} \cdot {\vec {g}}={\vec {g}}\,,\quad (\mathbf {1} -\mathbf {G} )\cdot {\vec {g}}={\vec {0}}\\&{\vec {n}}\cdot {\vec {g}}=0\quad \rightarrow \quad \mathbf {G} \cdot {\vec {n}}={\vec {0}}\,,\quad (\mathbf {1} -\mathbf {G} )\cdot {\vec {n}}={\vec {n}}\\{\vec {a}}=&\mathbf {G} \cdot ({\vec {p}}-{\vec {x}})={\frac {{\vec {g}}\cdot ({\vec {p}}-{\vec {x}})}{{\vec {g}}\cdot {\vec {g}}}}{\vec {g}}\\{\vec {b}}=&\left(\mathbf {1} -\mathbf {G} \right)\cdot ({\vec {p}}-{\vec {x}})={\vec {p}}-{\vec {x}}-{\vec {a}}\end{aligned}}}$

Der Punkt ${\displaystyle {\vec {x}}+{\vec {a}}}$ ist die senkrechte Projektion von ${\displaystyle {\vec {p}}}$ auf die Gerade. Der Tensor G extrahiert den Anteil eines Vektors in Richtung von ${\displaystyle {\vec {g}}}$ und 1-G den Anteil senkrecht dazu.

Punkt oder Gerade auf Ebene

Gegeben sei die Ebene durch den Punkt ${\displaystyle {\vec {x}}}$ und zwei die Ebene aufspannende Vektoren ${\displaystyle {\vec {u}}}$ und ${\displaystyle {\vec {v}}\not \!\|{\vec {u}}}$ sowie ein beliebiger anderer Punkt ${\displaystyle {\vec {p}}}$. Dann verschwindet die Normale

${\displaystyle {\hat {n}}={\frac {{\vec {u}}\times {\vec {v}}}{|{\vec {u}}\times {\vec {v}}|}}}$

nicht. Dann ist

${\displaystyle {\begin{aligned}{\vec {p}}=&{\vec {x}}+{\vec {a}}+{\vec {b}}\quad {\textsf {mit}}\quad {\vec {a}}\bot {\hat {n}}\quad {\text{und}}\quad {\vec {b}}\|{\hat {n}}\\\mathbf {P} =&{\frac {({\vec {v}}\cdot {\vec {v}}){\vec {u}}\otimes {\vec {u}}-({\vec {u}}\cdot {\vec {v}})({\vec {u}}\otimes {\vec {v}}+{\vec {v}}\otimes {\vec {u}})+({\vec {u}}\cdot {\vec {u}}){\vec {v}}\otimes {\vec {v}}}{({\vec {u}}\cdot {\vec {u}})({\vec {v}}\cdot {\vec {v}})-({\vec {u}}\cdot {\vec {v}})^{2}}}=\mathbf {1} -{\hat {n}}\otimes {\hat {n}}\\&\rightarrow \mathbf {P} \cdot {\vec {u}}={\vec {u}}\,,\quad \mathbf {P} \cdot {\vec {v}}={\vec {v}}\,,\quad \mathbf {P} \cdot {\hat {n}}={\vec {0}}\,,\quad (\mathbf {1} -\mathbf {P} )\cdot {\hat {n}}={\hat {n}}\\&\rightarrow \mathbf {P} \cdot (x{\vec {u}}+y{\vec {v}})=x{\vec {u}}+y{\vec {v}}\quad {\text{und}}\quad (\mathbf {1} -\mathbf {P} )\cdot (x{\vec {u}}+y{\vec {v}})={\vec {0}}\quad \forall x,y\in \mathbb {R} \\{\vec {a}}=&\mathbf {P} \cdot ({\vec {p}}-{\vec {x}})\\{\vec {b}}=&(\mathbf {1} -\mathbf {P} )\cdot ({\vec {p}}-{\vec {x}})={\vec {p}}-{\vec {x}}-{\vec {a}}\end{aligned}}}$

Der Punkt ${\displaystyle {\vec {x}}+{\vec {a}}}$ ist die senkrechte Projektion von ${\displaystyle {\vec {p}}}$ auf die Ebene.[2] Der Tensor P extrahiert den Anteil eines Vektors in der Ebene und 1-P den Anteil senkrecht dazu.

Die Projektion der Geraden, die durch die Punkte ${\displaystyle {\vec {x}}}$ und ${\displaystyle {\vec {p}}}$ verläuft, liegt in der Ebene in Richtung des Vektors ${\displaystyle {\vec {a}}}$.

Falls ${\displaystyle |{\vec {u}}|=|{\vec {v}}|=1}$ und ${\displaystyle {\vec {u}}\bot {\vec {v}}}$ folgt:

${\displaystyle {\hat {n}}={\vec {u}}\times {\vec {v}}\quad {\text{mit}}\quad |{\hat {n}}|=1}$

${\displaystyle \mathbf {P} ={\vec {u}}\otimes {\vec {u}}+{\vec {v}}\otimes {\vec {v}}=\mathbf {1} -{\hat {n}}\otimes {\hat {n}}}$

${\displaystyle {\vec {a}}=({\vec {u}}\otimes {\vec {u}}+{\vec {v}}\otimes {\vec {v}})\cdot ({\vec {p}}-{\vec {x}})=(\mathbf {1} -{\hat {n}}\otimes {\hat {n}})\cdot ({\vec {p}}-{\vec {x}})}$

${\displaystyle {\vec {b}}=(\mathbf {1} -{\vec {u}}\otimes {\vec {u}}-{\vec {v}}\otimes {\vec {v}})\cdot ({\vec {p}}-{\vec {x}})=({\hat {n}}\otimes {\hat {n}})\cdot ({\vec {p}}-{\vec {x}})}$

Fundamentaltensor 3. Stufe

Siehe auch: Epsilon-Tensor

Definition:

${\displaystyle {\begin{aligned}{\stackrel {3}{\mathbf {E} }}:=&\epsilon _{ijk}\,{\hat {e}}_{i}\otimes {\hat {e}}_{j}\otimes {\hat {e}}_{k}\\=&({\hat {e}}_{j}\times {\hat {e}}_{k})\otimes {\hat {e}}_{j}\otimes {\hat {e}}_{k}\\=&{\hat {e}}_{i}\otimes ({\hat {e}}_{k}\times {\hat {e}}_{i})\otimes {\hat {e}}_{k}\\=&{\hat {e}}_{i}\otimes {\hat {e}}_{j}\otimes ({\hat {e}}_{i}\times {\hat {e}}_{j})\end{aligned}}}$

Kreuzprodukt v​on Vektoren:

${\displaystyle {\vec {u}}\times {\vec {v}}={\stackrel {3}{\mathbf {E} }}:({\vec {u}}\otimes {\vec {v}})={\vec {v}}\cdot {\stackrel {3}{\mathbf {E} }}\cdot {\vec {u}}=-{\vec {u}}\cdot {\stackrel {3}{\mathbf {E} }}\cdot {\vec {v}}=-{\stackrel {3}{\mathbf {E} }}:({\vec {v}}\otimes {\vec {u}})=-{\vec {v}}\times {\vec {u}}}$

${\displaystyle {\vec {e}}_{i}\times {\vec {e}}_{j}=\epsilon _{ijk}\,{\hat {e}}_{k}}$

#Kreuzprodukt v​on Tensoren, #Skalarkreuzprodukt v​on Tensoren:

${\displaystyle {\begin{aligned}{\stackrel {3}{\mathbf {E} }}:\mathbf {A} =&\mathbf {A}$ :{\stackrel {3}{\mathbf {E} }}=-{\stackrel {3}{\mathbf {E} }}:(\mathbf {A} ^{\top })=-(\mathbf {A} ^{\top }):{\stackrel {3}{\mathbf {E} }}\\=&\mathbf {1} \times \mathbf {A} ^{\top }=\mathbf {1} \cdot \!\!\times \mathbf {A} \end{aligned}}}

#Dualer axialer Vektor u​nd #Vektorinvariante:

${\displaystyle {\stackrel {3}{\mathbf {E} }}:\mathbf {A} =-2{\stackrel {A}{\overrightarrow {\mathbf {A} }}}={\vec {\mathrm {i} }}(\mathbf {A} )}$

#Kreuzprodukt v​on Tensoren:

${\displaystyle \mathbf {A} \times \mathbf {B} ={\stackrel {3}{\mathbf {E} }}:(\mathbf {A\cdot B} ^{\top })}$

${\displaystyle (A_{ik}{\vec {e}}_{i}\otimes {\vec {e}}_{k})\times (B_{jl}{\vec {e}}_{j}\otimes {\vec {e}}_{l})=A_{ik}B_{jk}{\vec {e}}_{i}\times {\vec {e}}_{j}=\epsilon _{ijk}A_{jl}B_{kl}{\vec {e}}_{i}}$

#Skalarkreuzprodukt v​on Tensoren:

${\displaystyle \mathbf {A} \cdot \!\!\times \mathbf {B} ={\stackrel {3}{\mathbf {E} }}:(\mathbf {A\cdot B} )}$

${\displaystyle (A_{ik}{\vec {e}}_{i}\otimes {\vec {e}}_{k})\cdot \!\!\times (B_{lj}{\vec {e}}_{l}\otimes {\vec {e}}_{j})=A_{ik}B_{kj}{\vec {e}}_{i}\times {\vec {e}}_{j}=\epsilon _{ijk}A_{jl}B_{lk}{\vec {e}}_{i}}$

#Axialer Tensor o​der Kreuzproduktmatrix:

${\displaystyle {\stackrel {3}{\mathbf {E} }}\cdot {\vec {u}}={\vec {u}}\cdot {\stackrel {3}{\mathbf {E} }}=-{\vec {u}}\times \mathbf {1} =-\mathbf {1} \times {\vec {u}}}$

Tensoren vierter Stufe

Tensoren zweiter Stufe sind ebenfalls Elemente eines Vektorraums ${\displaystyle {\mathcal {L}}}$ wie im Abschnitt #Tensoren als Elemente eines Vektorraumes dargestellt. Daher kann man Tensoren vierter Stufe definieren, indem man in dem Kapitel formal die Tensoren zweiter Stufe durch Tensoren vierter Stufe und die Vektoren durch Tensoren zweiter Stufe ersetzt, z. B.:

${\displaystyle {\stackrel {4}{\mathbf {A} }}=A_{pq}(\mathbf {A} _{p}\otimes \mathbf {G} _{q})}$

mit Komponenten ${\displaystyle A_{pq}}$ und die Tensoren ${\displaystyle \mathbf {A} _{1},\mathbf {A} _{2},\ldots ,\mathbf {A} _{9}\in {\mathcal {L}}}$ sowie ${\displaystyle \mathbf {G} _{1},\mathbf {G} _{2},\ldots ,\mathbf {G} _{9}\in {\mathcal {L}}}$ bilden eine Basis von ${\displaystyle {\mathcal {L}}}$.

Standardbasis in ${\displaystyle {\mathcal {L}}}$:

${\displaystyle \mathbf {E} _{1}={\vec {e}}_{1}\otimes {\vec {e}}_{1},\mathbf {E} _{2}={\vec {e}}_{1}\otimes {\vec {e}}_{2},\mathbf {E} _{3}={\vec {e}}_{1}\otimes {\vec {e}}_{3},\mathbf {E} _{4}={\vec {e}}_{2}\otimes {\vec {e}}_{1},\ldots ,\mathbf {E} _{9}={\vec {e}}_{3}\otimes {\vec {e}}_{3}}$

Tensortransformation:

${\displaystyle {\stackrel {4}{\mathbf {A} }}:\mathbf {H} =A_{pq}(\mathbf {A} _{p}\otimes \mathbf {G} _{q}):\mathbf {H}$ :=A_{pq}(\mathbf {G} _{q}:\mathbf {H} )\mathbf {A} _{p}}

Tensorprodukt:

${\displaystyle [A_{pq}(\mathbf {A} _{p}\otimes \mathbf {G} _{q})]:[B_{rs}(\mathbf {H} _{r}\otimes \mathbf {U} _{s})]:=A_{pq}(\mathbf {G} _{q}:\mathbf {H} _{r})B_{rs}\mathbf {A} _{p}\otimes \mathbf {U} _{s}}$

Übliche Schreibweisen für Tensoren vierter Stufe:

${\displaystyle {\stackrel {4}{\mathbf {A} }}=\mathbb {A} =A_{ijkl}\;{\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l}}$

Transpositionen

Transposition:

${\displaystyle (\mathbf {A} \otimes \mathbf {B} )^{\top }=\mathbf {B} \otimes \mathbf {A} }$

${\displaystyle (A_{ijkl}\;{\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})^{\top }:=A_{ijkl}\;{\vec {e}}_{k}\otimes {\vec {e}}_{l}\otimes {\vec {e}}_{i}\otimes {\vec {e}}_{j}}$

Spezielle Transposition ${\displaystyle {\stackrel {4}{\mathbf {A} }}{}^{\stackrel {mn}{\top }}}$ vertauscht ${\displaystyle m}$-tes mit ${\displaystyle n}$-tem Basissystem.

Beispielsweise:

${\displaystyle {\stackrel {4}{\mathbf {A} }}{}^{\stackrel {13}{\top }}:=A_{ijkl}\;{\vec {e}}_{k}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{i}\otimes {\vec {e}}_{l}}$

${\displaystyle {\stackrel {4}{\mathbf {A} }}{}^{\stackrel {24}{\top }}:=A_{ijkl}\;{\vec {e}}_{i}\otimes {\vec {e}}_{l}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{j}}$

${\displaystyle {\stackrel {4}{\mathbf {A} }}\,^{\top }=\left({\stackrel {4}{\mathbf {A} }}{}^{\stackrel {13}{\top }}\right){}^{\stackrel {24}{\top }}=A_{ijkl}\;{\vec {e}}_{k}\otimes {\vec {e}}_{l}\otimes {\vec {e}}_{i}\otimes {\vec {e}}_{j}}$

Symmetrische Tensoren vierter Stufe

Definition: ${\displaystyle {\stackrel {4}{\mathbf {A} }}={\stackrel {4}{\mathbf {A} }}{}^{\top }}$

Dann gilt: ${\displaystyle {\stackrel {4}{\mathbf {A} }}:\mathbf {B} =\mathbf {B}$ :{\stackrel {4}{\mathbf {A} }}}

Einheitstensor vierter Stufe

${\displaystyle {\begin{aligned}{\stackrel {4}{\mathbf {1} }}:=&\mathbf {E} _{p}\otimes \mathbf {E} _{p}={\stackrel {4}{\mathbf {1} }}{}^{\top }=(\mathbf {1} \otimes \mathbf {1} )\,^{\stackrel {23}{\top }}\\=&{\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{i}\otimes {\vec {e}}_{j}=\delta _{ik}\delta _{jl}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\end{aligned}}}$

Spezielle Tensoren vierter Stufe

Für beliebige Tensoren zweiter Stufe A gilt:

${\displaystyle {\stackrel {4}{\mathbf {C} }}=\mathbf {E} _{p}^{\top }\otimes \mathbf {E} _{p}=\delta _{il}\delta _{jk}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {A} =\mathbf {A} ^{\top }}$

${\displaystyle {\stackrel {4}{\mathbf {C} }}={\frac {1}{3}}\mathbf {1} \otimes \mathbf {1} ={\frac {1}{3}}\delta _{ij}\delta _{kl}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {A} =\mathbf {A} ^{\mathrm {K} }}$

${\displaystyle {\stackrel {4}{\mathbf {C} }}={\stackrel {4}{\mathbf {1} }}-{\frac {1}{3}}\mathbf {1} \otimes \mathbf {1} =(\delta _{ik}\delta _{jl}-{\frac {1}{3}}\delta _{ij}\delta _{kl})({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {A} =\mathbf {A} ^{\mathrm {D} }}$

${\displaystyle {\stackrel {4}{\mathbf {C} }}={\frac {1}{2}}\left({\stackrel {4}{\mathbf {1} }}+\mathbf {E} _{p}^{\top }\otimes \mathbf {E} _{p}\right)={\frac {1}{2}}(\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk})({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {A} =\mathbf {A} ^{\mathrm {S} }}$

${\displaystyle {\stackrel {4}{\mathbf {C} }}={\frac {1}{2}}\left({\stackrel {4}{\mathbf {1} }}-\mathbf {E} _{p}^{\top }\otimes \mathbf {E} _{p}\right)={\frac {1}{2}}(\delta _{ik}\delta _{jl}-\delta _{il}\delta _{jk})({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {A} =\mathbf {A} ^{\mathrm {A} }}$

Diese fünf Tensoren s​ind sämtlich symmetrisch.

Mit beliebigen Tensoren zweiter Stufe A, B u​nd G gilt:

${\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} \otimes \mathbf {B} ^{\top })^{\stackrel {23}{\top }}=A_{ik}B_{lj}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A\cdot G\cdot B} }$

${\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} ^{\top }\otimes \mathbf {B} ^{\top })^{\stackrel {23}{\top }}=A_{ki}B_{lj}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A} ^{\top }\cdot \mathbf {G\cdot B} }$

${\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} \otimes \mathbf {B} )^{\stackrel {23}{\top }}=A_{ik}B_{jl}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A\cdot G\cdot B} ^{\top }}$

${\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} ^{\top }\otimes \mathbf {B} )^{\stackrel {23}{\top }}=A_{ki}B_{jl}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A} ^{\top }\cdot \mathbf {G\cdot B} ^{\top }}$

In dem in diesen Formeln im Tensor vierter Stufe B durch B^⊤ und die Transpositionen ${\displaystyle {\stackrel {23}{\top }}}$ durch ${\displaystyle {\stackrel {24}{\top }}}$ ersetzt werden, entstehen die Ergebnisse mit transponiertem G:

${\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} \otimes \mathbf {B} )^{\stackrel {24}{\top }}=A_{il}B_{kj}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A\cdot G} ^{\top }\cdot \mathbf {B} }$

${\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} ^{\top }\otimes \mathbf {B} )^{\stackrel {24}{\top }}=A_{li}B_{kj}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A} ^{\top }\cdot \mathbf {G} ^{\top }\cdot \mathbf {B} }$

${\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} \otimes \mathbf {B} ^{\top })^{\stackrel {24}{\top }}=A_{il}B_{jk}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A\cdot G} ^{\top }\cdot \mathbf {B} ^{\top }}$

${\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} ^{\top }\otimes \mathbf {B} ^{\top })^{\stackrel {24}{\top }}=A_{li}B_{jk}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A} ^{\top }\cdot \mathbf {G} ^{\top }\cdot \mathbf {B} ^{\top }}$

Invertierungsformel

${\displaystyle \left(a{\stackrel {4}{\mathbf {1} }}+\mathbf {B} \otimes \mathbf {C} \right)^{-1}={\frac {1}{a}}\left({\stackrel {4}{\mathbf {1} }}-{\frac {1}{a+\mathbf {B}$ :\mathbf {C} }}\mathbf {B} \otimes \mathbf {C} \right)}

Hooke'sches Gesetz

Siehe auch: Hooke'sches Gesetz

Mit den Spannungen ${\displaystyle \mathbf {T} }$ und den Dehnungen ${\displaystyle \mathbf {E} }$ im Hooke'schen Gesetz gilt:

${\displaystyle {\stackrel {4}{\mathbf {C} }}:=2\mu {\stackrel {4}{\mathbf {1} }}+\lambda \mathbf {1} \otimes \mathbf {1} \quad \rightarrow \quad {\stackrel {4}{\mathbf {C} }}:\mathbf {E} =\mathbf {T} }$

mit den Lamé-Konstanten ${\displaystyle \lambda }$ und ${\displaystyle \mu }$. Dieser Elastizitätstensor ist symmetrisch.

Invertierungsformel mit ${\displaystyle a=2\mu }$, ${\displaystyle \mathbf {B} =\lambda \mathbf {1} }$ und ${\displaystyle \mathbf {C} =\mathbf {1} }$:

${\displaystyle {\begin{aligned}&{\stackrel {4}{\mathbf {S} }}:={\stackrel {4}{\mathbf {C} }}{}^{-1}={\frac {1}{2\mu }}\left({\stackrel {4}{\mathbf {1} }}-{\frac {\lambda }{2\mu +3\lambda }}\mathbf {1} \otimes \mathbf {1} \right)={\frac {1}{2\mu }}{\stackrel {4}{\mathbf {1} }}-{\frac {\nu }{E}}\mathbf {1} \otimes \mathbf {1} \\&\rightarrow \quad {\stackrel {4}{\mathbf {S} }}:\mathbf {T} =\mathbf {E} \end{aligned}}}$

mit der Querdehnzahl ${\displaystyle \nu }$ und dem Elastizitätsmodul ${\displaystyle E}$.

Voigt'sche Notation von Tensoren vierter Stufe

Siehe auch: Voigtsche Notation

Aus der Basis ${\displaystyle \mathbf {S} _{1},\ldots ,\mathbf {S} _{6}}$ des Vektorraums ${\displaystyle {\mathcal {S}}=\mathrm {sym} (\mathbb {V} ,\mathbb {V} )}$ der symmetrischen Tensoren zweiter Stufe, siehe #Voigt-Notation symmetrischer Tensoren zweiter Stufe, kann eine Basis des Vektorraums ${\displaystyle {\stackrel {4}{\mathcal {S}}}=\mathrm {Lin} ({\mathcal {S}},{\mathcal {S}})}$ der linearen Abbildungen von symmetrischen Tensoren auf symmetrische Tensoren konstruiert werden. Die 36 Komponenten der Tensoren vierter Stufe aus ${\displaystyle {\stackrel {4}{\mathcal {S}}}}$ können als Voigt'scher Notation in eine 6×6-Matrix einsortiert werden:

${\displaystyle {\stackrel {4}{\mathbf {A} }}=A_{uv}\mathbf {S} _{u}\otimes \mathbf {S} _{v}{\hat {=}}{\begin{bmatrix}A_{11}&A_{12}&A_{13}&A_{14}&A_{15}&A_{16}\\A_{21}&A_{22}&A_{23}&A_{24}&A_{25}&A_{26}\\A_{31}&A_{32}&A_{33}&A_{34}&A_{35}&A_{36}\\A_{41}&A_{42}&A_{43}&A_{44}&A_{45}&A_{46}\\A_{51}&A_{52}&A_{53}&A_{54}&A_{55}&A_{56}\\A_{61}&A_{62}&A_{63}&A_{64}&A_{65}&A_{66}\end{bmatrix}}}$

Die Vektoren u​nd Matrizen i​n Voigt'scher Notation können addiert, subtrahiert u​nd mit e​inem Skalar multipliziert werden. Beim Matrizenprodukt i​n Voigt'scher Notation m​uss eine Diagonalmatrix

${\displaystyle I=\mathrm {diag} (1,1,1,2,2,2)}$

mit den Einträgen ${\displaystyle I_{uv}=\mathbf {S} _{u}:\mathbf {S} _{v}}$ zwischengeschaltet werden:

${\displaystyle \mathbf {A}$ :\mathbf {B} =[\mathbf {A} ]^{\top }I[\mathbf {B} ]=A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3}+2A_{4}B_{4}+2A_{5}B_{5}+2A_{6}B_{6}}

${\displaystyle \left[{\stackrel {4}{\mathbf {A} }}:\mathbf {T} \right]=\left[{\stackrel {4}{\mathbf {A} }}\right]I[\mathbf {T} ]}$

${\displaystyle \left[{\stackrel {4}{\mathbf {A} }}:{\stackrel {4}{\mathbf {B} }}\right]=\left[{\stackrel {4}{\mathbf {A} }}\right]I\left[{\stackrel {4}{\mathbf {B} }}\right]}$

Darin s​teht [x] für d​ie Voigt-Notation v​on x.

Einzelnachweise

1. P. B. Denton, S. J. Parke, T. Tao, X. Zhang: Eigenvectors from Eigenvalues. (PDF) 10. August 2019, S. 1–3, abgerufen am 29. November 2019 (englisch).
2. J. Hanson: Rotations in three, four, and five dimensions. Bei: arxiv.org. S. 4f.

Literatur

• Holm Altenbach: Kontinuumsmechanik. Einführung in die materialunabhängigen und materialabhängigen Gleichungen. 2. Auflage. Springer Vieweg, Berlin u. a. 2012, ISBN 978-3-642-24118-5.
• Philippe Ciarlet: Mathematical Elasticity. Band 1: Three-Dimensional Elasticity. North-Holland, Amsterdam 1988, ISBN 0-444-70259-8.
• Wolfgang Ehlers: Ergänzung zu den Vorlesungen Technische Mechanik und Höhere Mechanik. Vektor- und Tensorrechnung, Eine Einführung. 2015 (uni-stuttgart.de [PDF; abgerufen am 3. September 2020]).
• Ralf Greve: Kontinuumsmechanik. Ein Grundkurs für Ingenieure und Physiker. Springer, Berlin u. a. 2003, ISBN 3-540-00760-1.