Diese Formelsammlung fasst Formeln und Definitionen der Tensoralgebra für Tensoren zweiter Stufe in der Kontinuumsmechanik zusammen. Es wird der dreidimensionale Raum zugrunde gelegt.
Allgemeines
Notation
- Operatoren wie
werden nicht kursiv geschrieben. - Buchstaben die als Indizes benutzt werden:
.
Ausnahme:
Die imaginäre Einheit
und die #Vektorinvariante
werden in Abgrenzung zu den Indizes nicht kursiv geschrieben. ![{\displaystyle p,q,r,s\in \{1,2,\ldots ,9\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67f04da0c7100d378aeabd1cbe2382af0ebce5c2)
![{\displaystyle u,v\in \{1,2,\ldots ,6\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79a7cef55472b9bf89eeb0b01c1030246a868d3f)
- Alle anderen Buchstaben stehen für reelle Zahlen oder komplexe Zahlen.
- Vektoren:
- Alle hier verwendeten Vektoren sind geometrische Vektoren im dreidimensionalen euklidischen Vektorraum
. - Vektoren werden mit Kleinbuchstaben bezeichnet.
Ausnahme #Dualer axialer Vektor ![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1013e8695183599633093784579c69129379c7c6)
- Einheitsvektoren mit Länge eins werden wie in ê mit einem Hut versehen. Die Standardbasis von
ist ê1,2,3. - Vektoren mit unbestimmter Länge werden wie in
mit einem Pfeil versehen. - Dreiergruppen von Vektoren wie in
oder
bezeichnen eine rechtshändige Basis von
. - Gleichnamige Basisvektoren mit unterem und oberem Index sind dual zueinander, z. B.
ist dual zu
.
- Tensoren zweiter Stufe werden wie in A mit fetten Großbuchstaben notiert. Die Menge aller Tensoren wird mit
bezeichnet. Tensoren höherer Stufe werden mit einer hochgestellten Zahl wie in
geschrieben. Tensoren vierter Stufe sind Elemente der Menge
. - Es gilt die Einstein'sche Summenkonvention ohne Beachtung der Indexstellung.
- Kommt in einer Formel in einem Produkt ein Index doppelt vor wie in
wird über diesen Index summiert:
. - Kommen mehrere Indizes doppelt vor wie in
wird über diese summiert:
. - Ein Index, der nur einfach vorkommt wie
in
, ist ein freier Index. Die Formel gilt dann für alle Werte der freien Indizes:
.
Glossar
Reservierte und besondere Symbole
Zeichen für Operatoren
Tensorfunktionen
Formelzeichen | Abschnitt in der Formelsammlung | Wikipedia-Artikel |
![{\displaystyle \mathrm {Sp,tr,I} _{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/400136f12d8d237337689f33d1ed6f5c6e27f08c) | #Spur | Spur (Mathematik), Hauptinvariante |
![{\displaystyle \mathrm {I} _{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86557baa2d3aa8319e59051fa3a5ce040b70d304) | #Zweite Hauptinvariante | Hauptinvariante |
| #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
Mengen
Formelzeichen | Elemente |
![{\displaystyle \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc) | Reelle Zahlen |
![{\displaystyle \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7) | Komplexe Zahlen |
![{\displaystyle \mathbb {V} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4ef4769e1c31b86792140f48d99ba83ebd2358) | Vektoren |
| Tensoren zweiter Stufe |
| #Tensoren vierter Stufe |
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5848b08018d21399dca43f661ded59381f6c60e5)
Für Summen gilt dann z. B.
![{\displaystyle v_{i}\delta _{ij}=v_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d02bdff673155c5412b6519aed8bc501b48b8d)
![{\displaystyle A_{ij}\delta _{ij}=A_{ii}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd1010266d2918ffe3478d6a2c0285e4af196fc8)
Dies gilt für die anderen Indexgruppen entsprechend.
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/841e6669653447e20d762bf2271c3827e238a95c)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b064740c16cb38fbec98678262bf3e587b9f57b)
![{\displaystyle \epsilon _{ijk}\epsilon _{klm}=\delta _{il}\delta _{jm}-\delta _{im}\delta _{jl}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0688d9c2f79e627159c59369faa411f918900f7f)
![{\displaystyle \epsilon _{ijk}\epsilon _{jkl}=2\delta _{il}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c19fdd496aac1b1bb4902d8935bca57006d76827)
![{\displaystyle \epsilon _{ijk}\epsilon _{ijk}=6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9674b4c35ceefbba4d9f8079f1f06325c77dde78)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0e04ccc8a0f45ca671ecbeb85b39c049c092279)
![{\displaystyle \epsilon _{ijk}{\hat {e}}_{k}={\hat {e}}_{i}\times {\hat {e}}_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab36c5f1a8a650bcff461c1c0c227efa8b1de71)
Spaltenvektoren und Matrizen
Siehe auch: Matrix (Mathematik)
Die hier verwendeten Vektoren sind Spaltenvektoren
![{\displaystyle {\vec {a}}=a_{i}{\hat {e}}_{i}={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fad13b14e2dc1df4a5b31561ec27660d8425f98)
Drei Vektoren
können spaltenweise in einer 3×3-Matrix
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a3da5a0e09f7936aa6fc5ff9652c0722619dd04)
Die Determinante der Matrix
![{\displaystyle |M|={\begin{vmatrix}{\vec {a}}&{\vec {b}}&{\vec {c}}\end{vmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba72d5d0ba1b0047c356fd1a2308da9ba8ae638)
ist
- ungleich null, wenn die Spaltenvektoren linear unabhängig sind und
- größer null, wenn die Spaltenvektoren zusätzlich ein Rechtssystem bilden.
Also gewährleistet
, dass die Vektoren
eine rechtshändige Basis bilden.
Die Spaltenvektoren bilden eine Orthonormalbasis, wenn
![{\displaystyle M^{\top }M={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2f2464dc349b1b9ebc9bffdb17adb1b8d0bbca)
worin
die transponierte Matrix ist. Bei der hier vorausgesetzten Rechtshändigkeit gilt dann zusätzlich
.
Vektoralgebra
Basis und Duale Basis
Basisvektoren
Duale Basisvektoren
Beziehungen zwischen den Basisvektoren
![{\displaystyle {\vec {g}}_{i}\cdot {\vec {g}}^{j}=\delta _{i}^{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47009eb787798be9030ec026d51ef08d9d5a71b8)
![{\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})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65fff5af26f0053da15a4e1292370c47ced5817e)
![{\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})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b05de46b692d765650ecc0cb22853eaa2cfd4fd2)
mit dem 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b0c4ea34866af3fad8c50b58bb861db59d9076)
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 {\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6af88e1004942cad510dab4603291cf4363091e2)
In der Standardbasis wie in jeder Orthonormalbasis sind die Basisvektoren
zu sich selbst dual:
![{\displaystyle {\hat {e}}_{i}={\hat {e}}^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c530bc53f800d589738e259ba0470a63dbb299e4)
Berechnung von Vektorkomponenten
![{\displaystyle {\vec {v}}=v_{i}{\hat {e}}_{i}\quad \rightarrow \;v_{i}={\vec {v}}\cdot {\hat {e}}_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d859a5bc470b668d706a026a67facfa2e6834f7)
![{\displaystyle {\vec {v}}=v^{i}{\vec {g}}_{i}\quad \rightarrow \;v^{i}={\vec {v}}\cdot {\vec {g}}^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5576ced1b1b8dfa6fd8c556ad34b841b3682059)
![{\displaystyle {\vec {v}}=v_{i}{\vec {g}}^{i}\quad \rightarrow \;v_{i}={\vec {v}}\cdot {\vec {g}}_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc469bce8644a75ffb7943ce724733f9c37d3150)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e28f285951f2b4103d14c7c1fab2efb3b19187a)
Wechsel der Basis bei Vektoren
Wechsel von
Basis
mit dualer Basis
nach
Basis
mit dualer Basis
:
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c39e1852ee543a889e29c86281bd17d55c6d4ef0)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33e398a545671b2635ae605a0bdf77c8598714b7)
Dyadisches Produkt
Die grundlegenden Eigenschaften des dyadischen Produkts „⊗“ sind:
Abbildung
![{\displaystyle {\vec {a}}\otimes {\vec {g}}=\mathbf {T} \in {\mathcal {L}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5427c433d73df3b025d5388250d8b43d9e71074)
Multiplikation mit 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26509d6816ff665626baae4233ce193b5ba872c1)
Distributivität:
![{\displaystyle (x+y){\vec {a}}\otimes {\vec {g}}=x{\vec {a}}\otimes {\vec {g}}+y{\vec {a}}\otimes {\vec {g}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86d9ac5ee0125009e73bc269fca78e8110f62629)
![{\displaystyle ({\vec {a}}+{\vec {b}})\otimes {\vec {g}}={\vec {a}}\otimes {\vec {g}}+{\vec {b}}\otimes {\vec {g}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5097c5885f69ffc1886127464c34d164bd72686b)
![{\displaystyle {\vec {a}}\otimes ({\vec {g}}+{\vec {h}})={\vec {a}}\otimes {\vec {g}}+{\vec {a}}\otimes {\vec {h}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d9a5fe15d698b26a7b471064b22ce96d336841)
Skalarprodukt:
![{\displaystyle ({\vec {a}}\otimes {\vec {g}}):({\vec {b}}\otimes {\vec {h}})=({\vec {a}}\cdot {\vec {b}})({\vec {g}}\cdot {\vec {h}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d86e8bc139d192bbe5be110bc0616a3ace556a50)
Weitere Eigenschaften von Dyaden siehe #Dyade und den folgenden Abschnitt.
Tensoren als Elemente eines Vektorraumes
Durch die Eigenschaften des dyadischen Produktes wird
zu einem euklidischen Vektorraum und entsprechend kann jeder Tensor komponentenweise bezüglich einer Basis von
dargestellt werden:
mit Komponenten
.
Die Dyaden
und
bilden Basissysteme von
.
Operatoren
Transposition
Abbildung
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})^{\top }:={\vec {g}}\otimes {\vec {a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbe5191f87dd6c7fc88a19cc03180a0a9f6be335)
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02e7fc5f9bd7d381306adbba0c035045cf19f3a8)
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/273ceacec07b3c273fe3647e56535bb832b34b09)
![{\displaystyle \left(\mathbf {A} ^{\top }\right)^{\top }=\mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/56021b854992886d23b72aefbc9c846859f1a46e)
![{\displaystyle (\mathbf {A+B} )^{\top }=\mathbf {A} ^{\top }+\mathbf {B} ^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/442b6b9872c0922905613d7834c9839c2cca7948)
![{\displaystyle (\mathbf {A\cdot B} )^{\top }=\mathbf {B} ^{\top }\cdot \mathbf {A} ^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e03b4197a7444ae1b0a119e85dc161e29918f44e)
Abbildung
oder
Dyaden:
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})\cdot {\vec {h}}:=({\vec {g}}\cdot {\vec {h}}){\vec {a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ac51649d88a81ff1f05eeaed447c7ff8a9fb6a)
![{\displaystyle {\vec {b}}\cdot ({\vec {a}}\otimes {\vec {g}}):=({\vec {a}}\cdot {\vec {b}}){\vec {g}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94b9b6a97f04e8dd0641834fcb73ed2b376609b1)
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})\cdot {\vec {h}}={\vec {h}}\cdot ({\vec {a}}\otimes {\vec {g}})^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44ee4fbf908ee09af165e0f46411fe9ee9b24b79)
![{\displaystyle {\vec {b}}\cdot ({\vec {a}}\otimes {\vec {g}})=({\vec {a}}\otimes {\vec {g}})^{\top }\cdot {\vec {b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a3d0e136d463bd469a9ebbdf26971e31609b10d)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dba32fa3732decbe300cc3ba588de04160834f)
![{\displaystyle A^{ij}({\vec {a}}_{i}\otimes {\vec {g}}_{j})\cdot {\vec {v}}=A^{ij}({\vec {v}}\cdot {\vec {g}}_{j}){\vec {a}}_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5639e47cf629f1bd8bbd8b901aa4ed966127d91a)
![{\displaystyle {\vec {v}}\cdot A_{ij}({\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{ij}({\vec {v}}\cdot {\hat {e}}_{i}){\hat {e}}_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80af71281e1907529d1704a6b7daadfdd8a8bc07)
![{\displaystyle {\vec {v}}\cdot A^{ij}({\vec {a}}_{i}\otimes {\vec {g}}_{j})=A^{ij}({\vec {v}}\cdot {\hat {a}}_{i}){\vec {g}}_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b35543e4399d03fdacbc3b049c92ce89ae98dd34)
Symbolisch:
![{\displaystyle \mathbf {A} \cdot {\vec {v}}={\vec {v}}\cdot \mathbf {A} ^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14d607dc139d0890a9ac10eaaff277dd200b16e5)
![{\displaystyle {\vec {v}}\cdot \mathbf {A} =\mathbf {A} ^{\top }\cdot {\vec {v}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8302b30a25b1882b55c3628c1be1b447550dbfe2)
Tensorprodukt
Abbildung
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})\cdot ({\vec {h}}\otimes {\vec {u}}):=({\vec {g}}\cdot {\vec {h}}){\vec {a}}\otimes {\vec {u}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/917a4b6fa62350d2a9ac8314f6339376b85d1391)
![{\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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6f54a02fbcd09dcca83b05b6ced076503ef1eae)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/364d4e82902be07d973d06dccb2be4c18fae0849)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c0051de4f770fba53f6f5e48d92f7601e358f66)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66e640a3e8240a282386b851360312b31b3f1478)
Skalarprodukt von Tensoren
Abbildung
Definition über die #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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f65db693eba5a6b91b7fd1df9b459f666cacce5c)
![{\displaystyle \mathbf {A} :\mathbf {B} :=\mathrm {Sp} (\mathbf {A} ^{\top }\cdot \mathbf {B} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa86e050e129c4b241c9132bbfe578ab6b6cbef8)
Eigenschaften:
![{\displaystyle \mathbf {A} :\mathbf {B} =\mathbf {B} :\mathbf {A} =\mathbf {A} ^{\top }:\mathbf {B} ^{\top }=\mathbf {B} ^{\top }:\mathbf {A} ^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4fb3221987ccda37314b7f23080a5df62d8bd35)
![{\displaystyle \mathbf {A} ^{\top }:\mathbf {B} =\mathbf {A} :\mathbf {B} ^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4bcee264fd9c9bec41cfd6382c8ecb8545cef1b)
![{\displaystyle \mathbf {A} :(\mathbf {B\cdot C} )=(\mathbf {B} ^{\top }\cdot \mathbf {A} ):\mathbf {C} =(\mathbf {A\cdot C} ^{\top }):\mathbf {B} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/96de0ac685bcd75fefcc41c6e14c0afb2de5e33a)
![{\displaystyle (\mathbf {A\cdot B} ):\mathbf {C} =\mathbf {B} :(\mathbf {A} ^{\top }\cdot \mathbf {C} )=\mathbf {A} :(\mathbf {C\cdot B} ^{\top })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18ca0597ebf1f3bd7db258c6aa4ba2bfddb2f453)
![{\displaystyle ({\vec {u}}\otimes {\vec {v}}):\mathbf {A} ={\vec {u}}\cdot \mathbf {A} \cdot {\vec {v}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4745442a5feae540040f3731695e0a033f98c4f5)
Kreuzprodukt eines Vektors mit einem Tensor
Abbildung
oder
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/032de75550c81295294d15c585ecdd9f7e444e74)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a845cb1e3030dbbaef33818f320a9bb35d3c998)
![{\displaystyle {\vec {a}}\times {\vec {b}}\otimes {\vec {g}}=-[({\vec {b}}\otimes {\vec {g}})^{\top }\times {\vec {a}}]^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee045447b66e661f78e485fb51f163a478719f7)
![{\displaystyle {\vec {a}}\otimes {\vec {g}}\times {\vec {h}}=-[{\vec {h}}\times ({\vec {a}}\otimes {\vec {g}})^{\top }]^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2078c4ef7f0d54314cfe20bcae8529cfd95705ef)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6881749e3af8b59217ec00384615afe6c9ff633e)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bf3965df00146a1bef8edae93c55fcb9bcc1e64)
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 })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a42b2b303c9800ca18f2e6040bef2fbf982556f)
![{\displaystyle {\vec {b}}\cdot ({\vec {a}}\times \mathbf {A} ):=({\vec {b}}\times {\vec {a}})\cdot \mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe7b9a39e78ceed1a76a54490fd7d9577702b5f)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f642e89c825b7e8b0aeb51c5915c5840ac639f1b)
![{\displaystyle (\mathbf {A} \times {\vec {a}})\cdot {\vec {b}}=\mathbf {A} \cdot ({\vec {a}}\times {\vec {b}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/589de3e994dd4bdece04e0323979dd9bd936fb74)
![{\displaystyle {\vec {a}}\times \mathbf {A} =-\left(\mathbf {A} ^{\top }\times {\vec {a}}\right)^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/404b78b1d92b454ae02c6d873a7c4a072b0da99b)
![{\displaystyle \mathbf {A} \times {\vec {a}}=-\left({\vec {a}}\times \mathbf {A} ^{\top }\right)^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daceaa3113a4486f2931745eb2b6951f2355414e)
Symmetrische Tensoren:
Insbesondere Kugeltensoren:
Schiefsymmetrische Tensoren:
#Axialer Tensor oder Kreuzproduktmatrix mit 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b83fb4a7d1a8f56a89034d5203b6c56e6b942322)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18cb628b561f8cce919e87ad0b98ed72e77c8576)
![{\displaystyle {\vec {a}}\times ({\vec {b}}\times \mathbf {A} )={\vec {b}}\otimes {\vec {a}}\cdot \mathbf {A} -({\vec {a}}\cdot {\vec {b}})\mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e4c22d8f53e50bd850998a29457c6fc0c32a05d)
Meistens ist 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcd693fbebeab024fb48e8bf20b7deb1662d7e74)
![{\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} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/222ed06929dce1ea9902068605be1f2b853e9a3d)
Kreuzprodukt von Tensoren
Abbildung
![{\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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/335d7fc2acf50060fc1e795321ed56b659b7b561)
mit #Fundamentaltensor 3. Stufe
.
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})\times ({\vec {b}}\otimes {\vec {h}})=({\vec {g}}\cdot {\vec {h}}){\vec {a}}\times {\vec {b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/233ad0a8444f537c1d7d5b0bd8455bd162d50642)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ea9e6b7895db1d6c78cb573be56abd33d93cfc1)
Zusammenhang mit #Dualer axialer Vektor und #Vektorinvariante:
![{\displaystyle \mathbf {A\times B} =-2{\stackrel {A}{\overrightarrow {\mathbf {A\cdot B} ^{\top }}}}={\vec {\mathrm {i} }}(\mathbf {A\cdot B} ^{\top })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11b630c12f5c47ca51a7710c0240a59a765d8845)
Mit #Einheitstensor:
![{\displaystyle \mathbf {1\times A} =2{\stackrel {A}{\overrightarrow {\mathbf {A} }}}=-{\vec {\mathrm {i} }}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/828f5306802cc895950d12fb2454bc2fdf6441a2)
Mehrfachprodukte:
![{\displaystyle (\mathbf {A\cdot B} )\times \mathbf {C} =\mathbf {A} \times (\mathbf {C\cdot B} ^{\top })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c1237f9a9cfb960301f585db18e9891180bf244)
![{\displaystyle \mathbf {A} \times (\mathbf {B\cdot C} )=(\mathbf {A\cdot C} ^{\top })\times \mathbf {B} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/72f098cae58e2a6df536a9968fedbd302e59078f)
Zusammenhang mit dem #Skalarkreuzprodukt von Tensoren:
![{\displaystyle \mathbf {A\times B} =\mathbf {A} \cdot \!\!\times (\mathbf {B} ^{\top })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/347a827c426d4b311dfd796692adf565b265257b)
Skalarkreuzprodukt von Tensoren
Abbildung
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f98c76eb08301296550e60f21741b517a45211)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce5db296cfa14b66cc0845927afbe48b3943a73)
Das Skalarkreuzprodukt mit dem #Einheitstensor vertauscht das dyadische Produkt durch das Kreuzprodukt:
![{\displaystyle \mathbf {1} \cdot \!\!\times ({\vec {a}}\otimes {\vec {b}})={\vec {a}}\times {\vec {b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6178be1d55ed3904ada6f1eaa3af5c21e9240c5a)
Allgemein:
![{\displaystyle \mathbf {A} \cdot \!\!\times \mathbf {B} =-(\mathbf {B} ^{\top })\cdot \!\!\times (\mathbf {A} ^{\top })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5756c8cfd377861b125664f606885f033a934df8)
![{\displaystyle \mathbf {A} \cdot \!\!\times (\mathbf {B\cdot C} )=(\mathbf {A\cdot B} )\cdot \!\!\times \mathbf {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f74ac8237b4a886cb6b70105b9820e20e811c5fb)
![{\displaystyle (\mathbf {A\cdot B} )\cdot \!\!\times \mathbf {C} =\mathbf {A} \cdot \!\!\times (\mathbf {B\cdot C} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/643700a3987814a8e0713f28859b6ee280d866e5)
Zusammenhang mit dem #Kreuzprodukt von Tensoren:
![{\displaystyle \mathbf {S} \cdot \!\!\times \mathbf {T} =\mathbf {S\times (T^{\top })} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b854beb9724f92fe58863b8c7ab6a1073b2bf870)
Zusammenhang mit #Vektorinvariante und #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} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1684e9bb89b50ecc9dbf1fc2d48cd68cd11ab3bb)
Doppeltes Kreuzprodukt von Tensoren
Siehe auch #Äußeres Tensorprodukt #
Abbildung
![{\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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da7769a1e8ab842607efc29f4f3325098b326248)
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b04a1d5ecd7d95b8604e343287771adabfb7093)
![{\displaystyle \mathbf {A} \times \!\!\times \mathbf {B} =\mathbf {A} ^{\top }\#\mathbf {B} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a2e892c4f4891aff3337422d664446c19114cc8)
Äußeres Tensorprodukt
Abbildung
![{\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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22be4e628119feb6a4c78f308ab959920ac2e8a8)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c11f40d299b532695652c3b803691b895f466af5)
Mit der Formel für das 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f30aa56e3900ae28a4a8e1574d73d05da236c6c3)
Grundlegende Eigenschaften:
![{\displaystyle \mathbf {A} \#\mathbf {B} =\mathbf {B} \#\mathbf {A} =(\mathbf {A} ^{\top }\#\mathbf {B} ^{\top })^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d43d9f9dcd105a14f3b447842d4b6057f9908e4a)
![{\displaystyle (\mathbf {A+B} )\#\mathbf {C} =\mathbf {A} \#\mathbf {C} +\mathbf {B} \#\mathbf {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/71bbfc8d3fe54f553e23525810179ad2a34f749b)
![{\displaystyle \mathbf {A} \#(\mathbf {B+C} )=\mathbf {A} \#\mathbf {B} +\mathbf {A} \#\mathbf {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf4d9ccf73925b338d019fb220fdd310e7f6f3f4)
Kreuzprodukt und #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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6825d2c59b6e106a2ee831b7230a82ea0bde770d)
![{\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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de03ebca727a702723430504eb843b0627eb3b71)
#Hauptinvarianten:
![{\displaystyle {\frac {1}{2}}(\mathbf {A\#1} ):\mathbf {1} =\mathrm {Sp} (\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ff54309a258885335bd6008985e7ad6ce4d796)
![{\displaystyle {\frac {1}{2}}(\mathbf {A\#A} ):\mathbf {1} =\mathrm {I} _{2}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b2057aa026c0bfcd57187e9ee93cdeefb70c31c)
![{\displaystyle {\frac {1}{6}}(\mathbf {A\#A} ):\mathbf {A} =\det(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4fc9414e4b7da1b4df713612b24fdc00266c648)
Weitere Eigenschaften:
![{\displaystyle \mathbf {1} \#\mathbf {1} =2\,\mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/29cf57576b5af9c6cad701bddd33cbbfa7f3cace)
![{\displaystyle \mathbf {A} \#\mathbf {1} =\mathrm {Sp} (\mathbf {A} )\mathbf {1} -\mathbf {A} ^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a344b6362bd5eff3991fc46289053cb54b47b732)
![{\displaystyle (\mathbf {A} \#\mathbf {B} ):\mathbf {C} =(\mathbf {B} \#\mathbf {C} ):\mathbf {A} =(\mathbf {C} \#\mathbf {A} ):\mathbf {B} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/17057bab60b50eeed54f52e8caaaa88dac27c495)
![{\displaystyle \mathrm {Sp} (\mathbf {A} \#\mathbf {B} )=\mathrm {Sp} (\mathbf {A} )\mathrm {Sp} (\mathbf {B} )-\mathrm {Sp} (\mathbf {A\cdot B} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c549c82e01dccde8ebad90d29d6631aec323fb37)
![{\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} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/488b922c2c1f40ded9183c674ea9bf3015f6dfa9)
Aber meistens:
![{\displaystyle (\mathbf {A} \#\mathbf {B} )\#\mathbf {C} \neq \mathbf {A} \#(\mathbf {B} \#\mathbf {C} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dca4ce5a68d7839685fc16088fa5ffded44dcb6)
.
Produkte von Tensoren, Dyaden und Vektoren
![{\displaystyle \mathbf {A} \cdot ({\vec {a}}\otimes {\vec {g}})=(\mathbf {A} \cdot {\vec {a}})\otimes {\vec {g}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f42a8e6fcdbed9914151efb9750fd1b1bdcfd68)
![{\displaystyle {\vec {a}}\otimes (\mathbf {A} \cdot {\vec {g}})=({\vec {a}}\otimes {\vec {g}})\cdot \mathbf {A} ^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b02a972e52e48ecde1a28b6e6a9b7ace5d883d0)
![{\displaystyle {\vec {a}}\cdot \mathbf {A} \cdot {\vec {g}}=\mathbf {A} :({\vec {a}}\otimes {\vec {g}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/302f52352cfbbbb370f333b32a6c195e78f79cec)
Spatprodukt und #Determinante eines 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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbc7c49b3eb8ff7c9ab37bd7703895306b7aade1)
Kreuzprodukt und #Kofaktor:
![{\displaystyle (\mathbf {A} \cdot {\vec {a}})\times (\mathbf {A} \cdot {\vec {b}})=\mathrm {cof} (\mathbf {A} )\cdot ({\vec {a}}\times {\vec {b}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e15bea83d2d0dcd6f7b896d7bf8e3dcc6f41f41a)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f3c6625c0cf3bcd406a93c4b713a03a35700f4a)
#Axialer Tensor oder Kreuzproduktmatrix, #Kreuzprodukt von Tensoren, #Skalarkreuzprodukt von Tensoren, #Dualer axialer Vektor und #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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45efb3282ee07d634d3faba73bf58c4a027b2e3b)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/534b10662a6cbd07c638d0d70cd4d7a2c72334c7)
![{\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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c16651ab44a5ce7b209e51ef8b489785062edb)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15e229f9035c9ed63d9f53d09f159e9710a47953)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f1f9d4ad8252ec0d01471bb2475e6656dae418a)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93f500735c5bad26e01a14cb16e2f9a52a090080)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0dec8e420096aae94a6b412c919ed2ff90717f)
Die Komponenten
ergeben sich durch Vor- und Nachmultiplikation mit dem #Einheitstensor
:
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/260787ee390018e377236cd5533a2d3bbf3c56b0)
Allgemein:
![{\displaystyle \mathbf {A} =A_{ij}{\vec {a}}^{i}\otimes {\vec {g}}^{j}=A_{ij}^{\ast }{\vec {b}}^{i}\otimes {\vec {h}}^{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4676d57dd71dfa713effd01e00cc8c33a54a27bb)
Basiswechsel mit
:
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22bffba7cf126ad08a7e17c762a5258ecafe5742)
Definition für einen Tensor A:
![{\displaystyle \langle {\vec {u}},{\vec {v}}\rangle :={\vec {u}}\cdot \mathbf {A} \cdot {\vec {v}}=\mathbf {A} :({\vec {u}}\otimes {\vec {v}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f9b14f0a1c035c10fc27741b7baaaf788a64f33)
Zwei Tensoren A und B sind 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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8e497a1cd140fcade1fae2c0635230fe8643e9)
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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/150101c86c39898d119ff013bbd0d1bd2e8d65a0)
#Invarianten:
Wenn λ1,2,3 die #Eigenwerte des Tensors A sind, dann hat cof(A) die Eigenwerte λ1λ2, λ2λ3, λ3λ1.
#Hauptinvarianten:
![{\displaystyle \mathrm {I} _{1}(\mathrm {cof} (\mathbf {A} ))=\mathrm {I} _{2}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be693872303df48e606b8d8a478f64df7b7b141e)
![{\displaystyle \mathrm {I} _{2}(\mathrm {cof} (\mathbf {A} ))=\mathrm {I} _{1}(\mathbf {A} )\mathrm {det} (\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ddd583dfc9910a14e346ab55f58981c58fcb412)
![{\displaystyle \mathrm {det} (\mathrm {cof} (\mathbf {A} ))=\mathrm {det} ^{2}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52ea4ae46b56d131c86ac5aeccfea46092a27f44)
#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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/326a83aa99ebcb967adabff0ff339a1ac71290eb)
Weitere Eigenschaften:
![{\displaystyle \mathrm {cof} (x\mathbf {A} )=x^{2}\mathrm {cof} (\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aca83688a49724e04cc9a16382af8a06d19831b1)
![{\displaystyle \mathrm {det} (\mathbf {A} )\neq 0\quad \rightarrow \quad \mathrm {cof} (\mathbf {A} )=\det(\mathbf {A} )\mathbf {A} ^{\top -1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/580ee6d9bb8f81632ac5f99de750f9c7de817041)
![{\displaystyle \mathbf {A} ^{\top }\cdot \mathrm {cof} (\mathbf {A} )=\mathrm {cof} (\mathbf {A} )\cdot \mathbf {A} ^{\top }=\mathrm {det} (\mathbf {A} )\mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c2ec5baaa1822796c07d1710cc48212225224d4)
![{\displaystyle \mathrm {cof} (\mathbf {A\cdot B} )=\mathrm {cof} (\mathbf {A} )\cdot \mathrm {cof} (\mathbf {B} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a794ff78c714a9c171923bc048a12e481a9a21d)
![{\displaystyle \mathrm {cof} (\mathbf {A} ^{\top })=\mathrm {cof} (\mathbf {A} )^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2372db1703f5b85f9671e2247502c46bfca666cc)
![{\displaystyle \mathrm {cof} \left(\mathrm {cof} (\mathbf {A} )\right)=\mathrm {det} (\mathbf {A} )\mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac1261a8d2c8193e8d0c7b19b2e982edf64df95a)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d3d3b4a1f37e9ddf3458a35833a3373d748b57f)
Kofaktor und #Äußeres Tensorprodukt:
![{\displaystyle \mathrm {cof} (\mathbf {A} )={\frac {1}{2}}\mathbf {A} \#\mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6466e7934deef9d664c654a78c8497a6e8716e9f)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c945fb4c609cff70fdeb417b9e883ea782d71c8)
Kreuzprodukt und Kofaktor:
![{\displaystyle (\mathbf {A} \cdot {\vec {a}})\times (\mathbf {A} \cdot {\vec {b}})=\mathrm {cof} (\mathbf {A} )\cdot ({\vec {a}}\times {\vec {b}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e15bea83d2d0dcd6f7b896d7bf8e3dcc6f41f41a)
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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7220f7ade02878b7f344ae1d0c3906fb67ce5720)
#Hauptinvarianten:
![{\displaystyle \mathrm {I} _{1}(\mathrm {adj} (\mathbf {A} ))=\mathrm {I} _{2}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2770bdd5ca90204da86d2565389a6c8bb037e906)
![{\displaystyle \mathrm {I} _{2}(\mathrm {adj} (\mathbf {A} ))=\mathrm {I} _{1}(\mathbf {A} )\mathrm {det} (\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e10ecfdac072a24b0c419f3f20d9da09189ab9f)
![{\displaystyle \mathrm {det} (\mathrm {adj} (\mathbf {A} ))=\mathrm {det} ^{2}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e69b56f20b00ac8a7df7fe7cdbce1c61c1be6c6a)
#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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4727c60fbbd3b7d579f7b75ff6cffd82842c6391)
Weitere Eigenschaften:
![{\displaystyle \mathrm {adj} (x\mathbf {A} )=x^{2}\mathrm {adj} (\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50f157a9f522feb79ffa802f607b61b6f6021e57)
![{\displaystyle \mathrm {det} (\mathbf {A} )\neq 0\quad \rightarrow \quad \mathrm {adj} (\mathbf {A} )=\det(\mathbf {A} )\mathbf {A} ^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e21fd25955218b5944b2c3f4e9737666867795ea)
![{\displaystyle \mathbf {A} \cdot \mathrm {adj} (\mathbf {A} )=\mathrm {adj} (\mathbf {A} )\cdot \mathbf {A} =\mathrm {det} (\mathbf {A} )\mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f629dd4cd52fc8dbf694049988525a100c79d64b)
![{\displaystyle \mathrm {adj} (\mathbf {A\cdot B} )=\mathrm {adj} (\mathbf {B} )\cdot \mathrm {adj} (\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d725c33f11e192c2333e402f70dccdcf860a7a66)
![{\displaystyle \mathrm {adj} (\mathbf {A} ^{\top })=\mathrm {adj} (\mathbf {A} )^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a123d2a89ae4a23bedc5b86c694eff01219846e)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1e3843c17b351e6bae6a77a2937d02ff1570e80)
![{\displaystyle \mathrm {adj} \left(\mathrm {adj} (\mathbf {A} )\right)=\mathrm {det} (\mathbf {A} )\mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f86a53e3fb7c418d1b2e47625fa42e5d1aeac19)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/262b071891edb78620f8139980c43b58ff7b5f7c)
Inverse
Definition
![{\displaystyle \mathbf {A} ^{-1}:\quad \mathbf {A} ^{-1}\cdot \mathbf {A} =\mathbf {A\cdot A} ^{-1}=\mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/742d4d0033c51965fa36c1bc0a4ceb718bb80644)
Die Inverse ist nur definiert, wenn
Zusammenhang mit dem adjungierten Tensor
:
![{\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\mathrm {adj} (\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53f5d91c2943d9ce68acbe7d957abe3dcfccfe7c)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/155a11762ee05ad2018013fbae471fc210829642)
Werden die Spalten von A mit Vektoren bezeichnet, also
, 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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db0af006ab11f16b170b67de4c71b4dc45dcb8be)
Satz von 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} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/083e5768d964d003c38ebe7a7789993e59018886)
worin
die drei #Hauptinvarianten sind.
Inverse des transponierten Tensors:
![{\displaystyle (\mathbf {A} ^{\top })^{-1}=(\mathbf {A} ^{-1})^{\top }=\mathbf {A} ^{\top -1}=\mathbf {A} ^{-\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f83aa0c88e1c80ac758282c83e166293b3eac979)
Inverse eines Tensorprodukts:
![{\displaystyle (\mathbf {A\cdot B} )^{-1}=\mathbf {B} ^{-1}\cdot \mathbf {A} ^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f949f4974909487e45c6e76f5d4d9caedbdc0386)
![{\displaystyle (x\mathbf {A} )^{-1}={\frac {1}{x}}\mathbf {A} ^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c96437b656e8c29e5a0d370aa42e987ec8042cd1)
#Äußeres Tensorprodukt und Inverse einer 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ca2aaecb1a055d07a1bbd87925f375293aea518)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de02616589e90272e41ee437b2517a849e8c48da)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ca799d52963fee1ca4bd96035b510f468daabfb)
![{\displaystyle ({\vec {a}}_{i}\otimes {\vec {g}}_{i})^{-1}={\vec {g}}^{i}\otimes {\vec {a}}^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e240bf3746ba2aec15dc19fd6cd9a4d5361c734)
Eigensystem
Eigenwertproblem
![{\displaystyle \mathbf {A} \cdot {\hat {v}}=\lambda {\hat {v}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4707ee1085c41eb8f86e17a99f4b46e67c30f0a6)
mit Eigenwert
und Eigenvektor
. Die Eigenvektoren werden auf die Länge eins normiert.
Jeder Tensor hat drei Eigenwerte und drei dazugehörige Eigenvektoren. Mindestens ein Eigenwert und Eigenvektor sind reell. Die beiden anderen Eigenwerte und -vektoren können reell oder komplex sein.
Eigenwerte
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77f1b126e6a1f960afc2ca49181d2ba003dab7ab)
Lösung siehe Cardanische Formeln. Die Koeffizienten sind die #Hauptinvarianten :
![{\displaystyle \mathrm {I} _{1}(\mathbf {A} ):=\mathrm {Sp} (\mathbf {A} )=\lambda _{1}+\lambda _{2}+\lambda _{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/373762e8e90d5edf3cc5ac4e548328f27b6bb11d)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b24a31fd8023d3f6a5aff015a032dd9c9ae91bd)
![{\displaystyle \mathrm {I} _{3}(\mathbf {A} ):=\mathrm {det} (\mathbf {A} )=\lambda _{1}\lambda _{2}\lambda _{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52aea0b4be8aae04bd04a086fc3a706bbd4ecd1d)
Eigenvektoren
Eigenvektoren
sind nur bis auf einen Faktor ≠ 0 bestimmt. Der Nullvektor ist kein Eigenvektor.
Bestimmungsgleichung:
Tensor
:
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eee01965a09c39a4743d118fb3b24660a8c8dd6)
Bestimmung mit gegebenem/angenommenem
:
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46f20a269701c5720da8499aa9b47b78a8e163c1)
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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b711d5efc0b238b8030de258eab3cef242b6ddf4)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8611ce6469191cf6b68e1beb244c9a278c5049a1)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/519cac3e078a64413897bc4146cf59f56d2ce6a0)
Die Formeln bleiben richtig, wenn die Indizes {1,2,3} zyklisch vertauscht werden.
Symmetrischen Tensoren: Für das Betragsquadrat der Komponenten
der auf Betrag 1 normierten Eigenvektoren
des (komplexen) Tensors
gilt mit dessen Eigenwerten
und den Eigenwerten
der Hauptuntermatrizen von
:[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 )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01f7a426de10d6ecb30a5e2f88269d8f9b376218)
Eigensystem symmetrischer Tensoren
Sei
symmetrisch.
Symmetrische Tensoren haben reelle Eigenwerte und paarweise zueinander senkrechte oder orthogonalisierbare Eigenvektoren, die also eine Orthonormalbasis aufbauen. Die Eigenvektoren werden so nummeriert, dass sie ein Rechtssystem bilden.
Hauptachsentransformation mit Eigenwerten
und Eigenvektoren
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dccc0601da555e7ac431ab08f78f718e767b3ae9)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65fa7457719a38386e903ffdbc2214b7b9322764)
Eigensystem schiefsymmetrischer Tensoren
Sei
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
ist. Siehe auch #Axialer Tensor oder Kreuzproduktmatrix.
Eigensystem allgemeiner auch unsymmetrischer Tensoren
Sei
und
eine Basis und
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/898d2b7f467ae73cee102c9fa59212ea4ea5541d)
hat die Eigenwerte
![{\displaystyle \lambda _{1}=a,\;\lambda _{2}=b,\;\lambda _{3}=c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d729afb21c4aad10e82372eeafa292ea8e888e3)
und Eigenvektoren
![{\displaystyle {\vec {v}}_{1}={\vec {a}}_{1},\;{\vec {v}}_{2}={\vec {a}}_{2},\;{\vec {v}}_{3}={\vec {a}}_{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9bd402f4f7a67de98adee9df536ea46cdabc9ba)
Der #transponierte Tensor hat dieselben Eigenwerte zu den dualen Eigenvektoren
![{\displaystyle {\vec {v}}_{1}={\vec {a}}^{1},\;{\vec {v}}_{2}={\vec {a}}^{2},\;{\vec {v}}_{3}={\vec {a}}^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/214a32d5417f3d74bc4fa8a203b1c3cc61b4c1c9)
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e52d39de491290e8cee9e9acb0cb65955a975deb)
hat die Eigenwerte
![{\displaystyle \lambda _{1}=c,\;\lambda _{2}=a+\mathrm {i} \,b,\;\lambda _{3}=a-\mathrm {i} \,b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afa900638441873c6fa1c2c3ce9ceb35056d4b93)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd2abe5bc66855c6f40d8e1a592359e66b265fc)
Der #transponierte Tensor hat dieselben Eigenwerte zu 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/886c531912bfbecafbe089c63daaf86af9f87e42)
Invarianten
Eigenwerte des Tensors
Die #Eigenwerte
sind Invarianten.
Hauptinvarianten
Charakteristisches Polynom
Die Hauptinvarianten des Tensors A sind 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} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ba9676801f9872ec181e4dc4256fb200ef065dd)
Spezialfall:
![{\displaystyle \mathrm {det} ({\vec {b}}\otimes {\vec {c}}+a\mathbf {1} )=a^{2}(a+{\vec {b}}\cdot {\vec {c}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bb8c27e1ccd3968b680194f1fd7a8849802279b)
Satz von 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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/60768c09efb826fc98e17fdd823e6b88bb76b9b4)
Spur
Siehe auch: Spur (Mathematik)
Abbildung
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f65cba55f9b503b6502cf36ee7f6c26a1e9f0a63)
mit #Eigenwerten λ1,2,3 von A.
![{\displaystyle \mathrm {Sp} ({\vec {a}}\otimes {\vec {g}})=\mathrm {Sp} ({\vec {g}}\otimes {\vec {a}}):={\vec {a}}\cdot {\vec {g}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/482bc8c64b203628840406df3a490afb2d3b7b9e)
Linearität:
![{\displaystyle \mathrm {Sp} (\mathbf {A} )=\mathrm {Sp} (\mathbf {A} ^{\top })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bba5b69c269c5f6d079da53b2517998031b11097)
![{\displaystyle \mathrm {Sp} (\mathbf {A\cdot B} )=\mathrm {Sp} (\mathbf {B\cdot A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d99058c6657da4c51c570271706387d03f11c396)
![{\displaystyle \mathrm {Sp} (\mathbf {A^{\top }\cdot B} )=\mathrm {Sp} (\mathbf {A\cdot B^{\top }} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd40c028368e6261c41141ce0e05f159f433f88)
![{\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} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/942bb461b3f465527a8fd9fde1c23ddcc3e5fd76)
In Komponenten:
![{\displaystyle \mathrm {Sp} \left(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\right)=A_{ii}=A_{11}+A_{22}+A_{33}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ac5aec4ecec7c6d17833193b62d9b55444cd06)
![{\displaystyle \mathrm {Sp} \left(A^{ij}{\vec {a}}_{i}\otimes {\vec {b}}_{j}\right)=A^{ij}{\vec {a}}_{i}\cdot {\vec {b}}_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b855d7310a3bc8ebfa9a20a8bf3b023645381cb)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83cde640aa7896cb142873127d2db7c55ca687d1)
Zweite Hauptinvariante
Abbildung
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84ca8ed8f53afa2ebd6301bd0baf8ada5847c9db)
mit #Eigenwerten λ1,2,3 von A.
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} )=\mathrm {Sp(cof} (\mathbf {A} ))=\mathrm {Sp(adj} (\mathbf {A} ))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05a8bbaeecbe7d864b107f16df70a25f3d920192)
![{\displaystyle \mathrm {I} _{2}(x\mathbf {A} )=x^{2}\mathrm {I} _{2}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/232583c87f9dfef8062db33c0f587501507cbb34)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} ^{\top })=\mathrm {I} _{2}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/148cc4fd9b8bd98203c7bc701185fe1ba177a862)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} \cdot \mathbf {B} )=\mathrm {I} _{2}(\mathbf {B} \cdot \mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c56bec2c8073bb065616abd79ccad2434da27a65)
![{\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} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e12c10a1f6d9c37cd20af9a43e2b4b81356a5eb2)
![{\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} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aac73f887f02f75d9987fbdaf6a0122f0d01932)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d38eb9aa20dd0dd4b407e78cb5e1efb937d57c2d)
![{\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})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be21f1963544d3e79b4f74e6ffbc3b1dcdcb63d)
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82613e4cdfd2ec892c41cf56ad617869cecf5609)
Determinante
Abbildung
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26799c2a1a0cc6969049c5aecf89b90401b8fbf2)
mit #Eigenwerten λ1,2,3 von A.
![{\displaystyle \mathrm {det} (\mathbf {A} ^{\top })=\mathrm {det} (\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/637e8bbf79277e76d6db29faae137c2b799b21a5)
Determinantenproduktsatz:
![{\displaystyle \mathrm {det} (\mathbf {A\cdot B} )=\mathrm {det} (\mathbf {B\cdot A} )=\mathrm {det} (\mathbf {A} )\mathrm {det} (\mathbf {B} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07f25ce0f0b268020f871dbf7ae21717231601e6)
![{\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} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b535813b47c1b8fa15ee302e76237510a688421)
![{\displaystyle \mathrm {det} (\mathbf {A} ^{-1})={\frac {1}{\mathrm {det} (\mathbf {A} )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4c15559d56f4cc1ad9618443f66f4d380d65ecb)
Multiplikation mit Skalaren
:
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7f0b8b26e7d32d73e3e7bae653a0aa2d096e3db)
![{\displaystyle \mathrm {det} (x\mathbf {A} )=x^{3}\mathrm {det} (\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90d9c103b3690d12fc17f21a99f030809d372b15)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b48cd08b2cca51f2b006df8eaef2dd3a16e6786)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9939430ae83474ad64ef71a38e28468b657b18a3)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/def5da3471bfb6dce0cbe5655f2d11bdfbd5891a)
Zusammenhang mit 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f27a10ba68da17b63b286a7d278dae4011f46157)
Zusammenhang mit 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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83831d75d34abf6528c1cf9dc375f75b0446ddf0)
Zusammenhang mit #Äußeres Tensorprodukt:
![{\displaystyle \det(\mathbf {A} )={\frac {1}{6}}(\mathbf {A} \#\mathbf {A} ):\mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7fb5389e3036ad2a132cfc929adaa5c4265228)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e426aed55baf7aefe3499c145a0626c81aae91d7)
Zusammenhang mit 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} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e96fa0e01ded3d32b4930c0091720d7f6a049ae)
Betrag
Abbildung
![{\displaystyle \parallel \mathbf {A} \parallel :={\sqrt {\mathbf {A} :\mathbf {A} }}={\sqrt {\mathrm {Sp} (\mathbf {A} ^{\top }\cdot \mathbf {A} )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fbb3cf59fead4a0255f26b3aef3d35b38f1c126)
![{\displaystyle \parallel {\vec {a}}\otimes {\vec {g}}\parallel =|{\vec {a}}|\,|{\vec {g}}|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efb5c1d178a38c3f1f32e1e937e58a5367950bc0)
![{\displaystyle \parallel A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\parallel ={\sqrt {A_{ij}A_{ij}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3fda954f48dfdf6ffaf25265ff2ec27654cd56)
![{\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})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1487989dbee06a6ac4c73491d74d9145ecb9765)
![{\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})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/768930ca7aaab22bbf2a942ea58e2ce838df0d90)
Falls
:
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d6c959ae0e8ae5f5bc7b8053a3a46a3c2876c45)
Falls
:
![{\displaystyle \quad \parallel \mathbf {A} \parallel ={\sqrt {2\mathrm {I} _{2}(\mathbf {A} )}}={\sqrt {-\mathrm {Sp} (\mathbf {A} ^{2})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/772de10a9a235de191a5ebd5dd5ce251f69f922f)
Dualer axialer Vektor
Für #Schiefsymmetrische Tensoren
gibt es einen dualen axialen Vektor
für den gilt:
für alle ![{\displaystyle {\vec {v}}\in \mathbb {V} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bae916139ff0c226e1f5fc8a988605f3cc3ce3)
Der duale axiale Vektor ist proportional zur #Vektorinvariante:
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}:=-{\frac {1}{2}}{\vec {\mathrm {i} }}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f232702ab948d5b8a1550f27a2b3a5a819febb7)
Berechnung mit #Fundamentaltensor 3. Stufe
, #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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c75300d4d7a5760b032d5e01af9687b40d8afdf9)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23df918f1b1464f14fb68a37e5c3aa55a5c77036)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81925d8529e942f3dbeb7e4c20965ca082c2ca52)
#Symmetrische Tensoren und #Kugeltensoren haben keinen dualen axialen Vektor:
Ein #Symmetrischer Anteil oder #Kugelanteil trägt nichts zum dualen axialen Vektor bei:
Seien x eine beliebige Zahl,
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67a782a3499593a7433737b52e0c3e9cd7bdf695)
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}\times {\vec {v}}=\mathbf {A} ^{\mathrm {A} }\cdot {\vec {v}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f75728c1f8cf681eed7fd28555b8b64b72b4c22)
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} ^{\top }}}}\quad \;=-{\stackrel {A}{\overrightarrow {\mathbf {A} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7faae3f4316873ad7c8cbbed87b29a86158c215)
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A+B} }}}={\stackrel {A}{\overrightarrow {\mathbf {A} }}}+{\stackrel {A}{\overrightarrow {\mathbf {B} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3026e771163f6638bc261a085547960e66ca80a)
![{\displaystyle {\stackrel {A}{\overrightarrow {x\mathbf {A} }}}\quad \;=x\,{\stackrel {A}{\overrightarrow {\mathbf {A} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c92e46b60344ad4538734bb718263ee121dadb5e)
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} \#\mathbf {B} }}}\;=\mathbf {A} \cdot {\stackrel {A}{\overrightarrow {\mathbf {B} }}}+\mathbf {B} \cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9288d1e9a02dfd23df97cec53c658c6f700cbb)
![{\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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/464b300f93557753ac15716874b49fb55b190eeb)
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathrm {cof} (\mathbf {A} )}}}=\mathbf {A} \cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82754241273639d0005be619dea0386f9f022047)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebfeb4a8913dff67dfb9a1d9f318e28ab3cdb2b)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0313d67bb9b35121cf91c7eedc741a0da93a1a8b)
![{\displaystyle {\stackrel {A}{\overrightarrow {{\vec {v}}\times \mathbf {1} }}}\;\;={\vec {v}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c912a682e63aa4b1a37f02f62a2f1ca2d6357aa)
![{\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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daa8b2ecc1b1bf8e2b448079c88e3b485c9bb055)
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {B\cdot A\cdot B} ^{\top }}}}\;\;=\mathrm {cof} (\mathbf {B} )\cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30bf82109dce9879b7f153059cd4a964e5ed13cf)
Darin ist „#“ ein #Äußeres Tensorprodukt, cof(·) ist 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} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d29e1b6d9c3222ba4523aa329b88529a4094d5)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aba8cb2f2f906439a3b142583c5c52f8a94d5a4)
![{\displaystyle {\vec {\mathrm {i} }}(A^{ij}({\vec {a}}_{i}\otimes {\vec {b}}_{j}))=A^{ij}{\vec {a}}_{i}\times {\vec {b}}_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/484fb07e3a363b8c1ec8f60705f1770742488856)
Zusammenhang mit dem #Skalarkreuzprodukt von Tensoren:
![{\displaystyle \mathbf {A} \cdot \!\!\times \mathbf {B} ={\vec {\mathrm {i} }}(\mathbf {A} \cdot \mathbf {B} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1da900ea7c6343a5dfe89b9b0ab1e3f075fb6c3)
#Symmetrische Tensoren haben keine Vektorinvariante:
Die Eigenschaften des dualen axialen Vektors sind hierher übertragbar. Seien x eine beliebige Zahl,
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/209bf741254bbd04b23a4beafd25c513d64866af)
![{\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} )\times {\vec {v}}\;=-2\mathbf {A} ^{\mathrm {A} }\cdot {\vec {v}}=(\mathbf {A^{\top }-A} )\cdot {\vec {v}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09e3ec250a602d3306b26829ade384203131b328)
![{\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} ^{\top })\quad \;=-{\vec {\mathrm {i} }}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cab7837568aa7c5509b309bcb31011aac0c3f2fc)
![{\displaystyle {\vec {\mathrm {i} }}(\mathbf {A+B} )={\vec {\mathrm {i} }}(\mathbf {A} )+{\vec {\mathrm {i} }}(\mathbf {B} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8fce22bc2412d4c0eb827428dd7ad6f492c6ac3)
![{\displaystyle {\vec {\mathrm {i} }}(x\mathbf {A} )\quad \;=x\,{\vec {\mathrm {i} }}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9adf4ec372f65c24b7c6585fd7e83dbeec8e759d)
![{\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} \#\mathbf {B} )\;=\mathbf {A} \cdot {\vec {\mathrm {i} }}(\mathbf {B} )+\mathbf {B} \cdot {\vec {\mathrm {i} }}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aec4743160cce3379a8340ea38ee55707bff8c24)
![{\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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6774c00fb5ceb2e8c67147ad82cfa6d31082bd4)
![{\displaystyle {\vec {\mathrm {i} }}(\mathrm {cof} (\mathbf {A} ))=\mathbf {A} \cdot {\vec {\mathrm {i} }}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb500ca5bfd5e5e7a2f8864e622784da630648a7)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cb0823c5ae06470bc0b939ace532ce021b26d0f)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12a75f769ca5f67a6c6f5c41455611178aefe02c)
![{\displaystyle {\vec {\mathrm {i} }}({\vec {v}}\times \mathbf {1} )\;\;=-2{\vec {v}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/462195dbba5bfd4abad2214ab2a7ed4e3b1bc46d)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7274d48f0b790b75b0186819fcacd3f7d69bde98)
![{\displaystyle {\vec {\mathrm {i} }}(\mathbf {B\cdot A\cdot B} ^{\top })=\mathrm {cof} (\mathbf {B} )\cdot {\vec {\mathrm {i} }}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3876fa3e5edfae4a7216e3006997d386e2418f1)
Darin ist „#“ ein #Äußeres Tensorprodukt, cof(·) ist der #Kofaktor.
Spezielle Tensoren
Dyade
Definition
![{\displaystyle \mathbf {A} :={\vec {a}}\otimes {\vec {b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e1d3d7606c01434206d043cbf5e4d767b4ccbb)
Kofaktor:
#Invarianten:
![{\displaystyle \mathrm {Sp} (\mathbf {A} )={\vec {a}}\cdot {\vec {b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4727a6d4dd930f15213311d501bfb42d6d9c1e19)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} )=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7ac432adb406235cd42d0706f7ed892a0bfadb)
![{\displaystyle \mathrm {det} (\mathbf {A} )=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/caf7f9929850f7d5d3a89267efb891c9b2832235)
![{\displaystyle \parallel \mathbf {A} \parallel =|{\vec {a}}|\,|{\vec {b}}|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f96df1161a68493ee2b9be417bffc6754434564f)
#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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e64653346435ef2f2d44b3d92279f770c4e367fc)
Dyadentripel
Gegeben ein beliebiger Tensor 2. Stufe A. Dieser kann immer als 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f99c7fb839a92a30616389cc5a7db3166876db66)
mit Spaltenvektoren
, Zeilenvektoren
und
.
#Hauptinvarianten (
):
![{\displaystyle \mathrm {I} _{1}(\mathbf {A} )=s_{i,i}=z_{i,i}={\vec {a}}_{i}\cdot {\vec {g}}_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ba61fc99cc0370f61a05c4f0f5a76776a9cb7c5)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70bf824dff001aaebe367d8bcc20502088bcc953)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25cfe954e51d6b49381962ad41e958442010532d)
#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})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/012b2938dcbd96bf6b8715c8631628a5fd74156a)
#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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2b2fd3392e3ae3270a996630e35549b9150d68)
#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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce043ca8133bea116278d2982a60f0c14bc666bc)
#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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13273b8bc47a24d8e527dcfb85ab7a5834852cfa)
#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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9359ce2ec160ca9a2e56821b87ecb3cbe317827)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f66685859bd022625bd60390102ab4bc8299f83a)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0ad37b2d6db45fddb62e84a6d2e7ea24bc18168)
mit
Allgemein:
![{\displaystyle \mathbf {1} =({\vec {a}}^{i}\cdot {\vec {g}}^{j}){\vec {a}}_{i}\otimes {\vec {g}}_{j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0989efc71f8b86115c6665fd5f059abe797d0f3c)
#Transposition und #Inverse:
![{\displaystyle \mathbf {1} =\mathbf {1} ^{\top }=\mathbf {1} ^{-1}=\mathbf {1} ^{\top -1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b53e15a98b0b6e735b157d743a8a517af0209c0)
Kofaktor:
Vektortransformation
![{\displaystyle \mathbf {1} \cdot {\vec {v}}={\vec {v}}\cdot \mathbf {1} ={\vec {v}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfad47ec2103b8f515aede800ac6348b3581b58)
Tensorprodukt
![{\displaystyle \mathbf {A\cdot 1} =\mathbf {1\cdot A} =\mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c06ad3382f017cf7ebcf2ccd3e499a67e47438)
Skalarprodukt
![{\displaystyle \mathbf {A} :\mathbf {1} =\mathrm {Sp} (\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bb89d44289ff5145d0f7f23871fbd1ed65fb051)
#Invarianten:
![{\displaystyle \mathrm {Sp} (\mathbf {1} )=\mathbf {1} :\mathbf {1} =3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e7319bfa1ca69739fc8953ad8c3770d1b89c8b)
![{\displaystyle \mathrm {I} _{2}(\mathbf {1} )=3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74b5c751ddcc294cd9e5b430368f4aa29f33384f)
![{\displaystyle \mathrm {det} (\mathbf {1} )=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e60e3f4692e328ce814572ee90f1c933b6a9e5)
![{\displaystyle \parallel \mathbf {1} \parallel ={\sqrt {3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b9c61c5a7679f2d69aa24f51cba3f37e63d443)
#Eigenwerte:
![{\displaystyle \lambda _{1,2,3}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/359d574814a189ac0ec034749111d8bb6bf01b37)
Alle Vektoren sind #Eigenvektoren.
Unimodulare Tensoren
Definition
![{\displaystyle \mathbf {H} :\quad \mathrm {det} (\mathbf {H} )=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a28f5fa77eecd3d7de687970a0ef5ac157131897)
Kofaktor:
Determinantenproduktsatz:
![{\displaystyle \mathrm {det} (\mathbf {A\cdot H} )=\mathrm {det} (\mathbf {H\cdot A} )=\mathrm {det} (\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cfa17b0fee1f6d215913ccf86e9af3b92c0416c)
Orthogonale Tensoren
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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f59f3e182453d919b27a807e53a1ce12ebe99d5d)
Kofaktor:
#Invarianten (
ist der Drehwinkel):
![{\displaystyle \mathrm {Sp} (\mathbf {Q} )=\mathrm {det} (\mathbf {Q} )+2\cos(\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2739ac0e4821a4655b0e3a0ad7c5d5a5caa0def1)
![{\displaystyle \mathrm {I} _{2}(\mathbf {Q} )=\mathrm {det} (\mathbf {Q} )\cdot \mathrm {Sp} (\mathbf {Q} )=1+2\,\mathrm {det} (\mathbf {Q} )\cos(\alpha )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cdc37efb4737b52a3fc9065c6b51079c5a25b5c)
![{\displaystyle \mathrm {det} (\mathbf {Q} )=\pm 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12fc19d6cd42709391d57bed1bfbc8e36d68b7a4)
![{\displaystyle \parallel \mathbf {Q} \parallel ={\sqrt {3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f987906ea27f8943025b870a8e2604bbbd8678e)
Eigentlich orthogonaler Tensor
, entspricht einer Drehung.
Uneigentlich orthogonaler Tensor
, 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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4670b76716a241c5c47e1b5ccc37eb0a29925d26)
Kreuzprodukt und #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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99eae1be8a453b0a8b673324c9207b9a5b428310)
![{\displaystyle \mathrm {cof} (\mathbf {Q} )=\mathrm {det} (\mathbf {Q} )\mathbf {Q} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/efe64318ff5d83a030f24cc3c3b37f6abe492a27)
Gegeben ein Einheitsvektor
und Drehwinkel α. Dann sind die folgenden Tensoren R zueinander gleich, orthogonal und drehen um die Achse
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ab023d4cefdfa46bc9fcdd121190ba5dd2822a)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ac17e0c83ee7bf288eb17ae997e6ee8f2ba66a0)
mit
.
Euler-Rodrigues-Formel:
also
:
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b5a2d803244247a0eb3b713a736ca3a2ffe77dc)
Formulierung mit Drehvektor:
Drehvektor | | Orthogonaler Tensor |
![{\displaystyle {\vec {\alpha }}=\alpha {\vec {n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc616f2f13266cd68c2da1905e8d9a72fa1cf6d) | → | |
![{\displaystyle {\vec {\alpha }}=\tan(\alpha ){\vec {n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b215fdac8e84e1ca5491c6ba2007091d8d9d717a) | → | |
| → | |
![{\displaystyle {\vec {\alpha }}=\sin(\alpha )\;{\vec {n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab4b7eede36638772e41a90545413377e934c98) | → | |
![{\displaystyle {\vec {\alpha }}=\sin \left({\frac {\alpha }{2}}\right)\;{\vec {n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/367e2d80adaddca65d7e98f8e91eb064966aa857) | → | |
![{\displaystyle {\vec {\alpha }}=\cos(\alpha )\;{\vec {n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cd7fb4827e84b6e7b8f30e5fb2383a772c670b1) | → | |
| → | |
Darin ist
Beispiel für Drehspiegelung:
![{\displaystyle \mathbf {Q} =-\mathbf {1} +\sin(\alpha ){\hat {n}}\times \mathbf {1} -(1+\cos(\alpha ))({\hat {n}}\times \mathbf {1} )^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72ed00db445d6b8ca9f5d18bcfdb3c6f8abfa358)
Drehung von Vektorraumbasis
mit Drehachse
:
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd0b43379de312d7c7334e1ae1348a6cdd4a9132)
![{\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} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96b162a20c44078d79a707f919038e784ce660af)
mit #Dualer axialer Vektor
und #Vektorinvariante
.
Gegeben Orthonormalbasis
, Drehwinkel
und
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afc0175881ebe5c1c223bd244a1b827132373c3b)
: Drehung,
: Drehspiegelung um ![{\displaystyle {\hat {v}}_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2071430f7617504c8d2c2ff069990e65bad7fd)
Wenn
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffbb7d6c7ceafb50a24cde805b190e83a63043ac)
Drehwinkel:
![{\displaystyle \cos(\alpha )={\frac {1}{2}}(\mathrm {Sp} (\mathbf {Q} )-\mathrm {det} (\mathbf {Q} ))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84ca2761578688a301ad57bee275213354852d33)
Drehachse
ist #Vektorinvariante:
![{\displaystyle {\hat {n}}\simeq {\vec {\mathrm {i} }}(\mathbf {Q} )=\mathbf {1} \cdot \!\!\times \mathbf {Q} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bda59c3e5c683771de3d3245c6fd340b5b701219)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df0db8310abd138a9c41592760c8251ec7be00b2)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b4fd2f3d96657fa50aaf356a2d0c790b1c2a51b)
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}}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3700d5dbbb658c47d95969d061a56e9230a9a05)
Kofaktor:
Notwendige Bedingungen für positive Definitheit:
![{\displaystyle \mathrm {det} (\mathbf {A} )>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63e6b5fcc238ac04ccaa440dd9c9579a28dfa0da)
![{\displaystyle \mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\quad \rightarrow \quad A_{11},\,A_{22},\,A_{33}>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3fe6d976192bd558c5ee247495150ccd0e43047)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/164ab9a9219e712b4ea8da472d95f866841f2e92)
Notwendige und hinreichende Bedingung für positive Definitheit: Alle #Eigenwerte von A sind größer als null.
Immer positiv definit falls det(A) ≠ 0:
- A·A⊤ und A⊤·A
Symmetrische Tensoren
Definition
![{\displaystyle \mathbf {A} :\quad \mathbf {A} =\mathbf {A} ^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6923eb92a0eac2b623563f0a47cb7e17b27b08c)
Kofaktor:
#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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d6c959ae0e8ae5f5bc7b8053a3a46a3c2876c45)
Bei Symmetrischen Tensoren verschwinden ihr #Dualer axialer Vektor und ihre #Vektorinvariante:
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} ^{\mathrm {S} }}}}={\vec {\mathrm {i} }}(\mathbf {A} ^{\mathrm {S} })={\vec {0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/602b86308ab16b9cc72f366be5e1682333472e52)
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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdcb1d8b8e00b884096d50c9fd0b79009c01d5be)
Alle #Eigenwerte λ1,2,3 sind reell. Alle #Eigenvektoren
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/accfe34edeae455099dece715ecb9a1a1992eb81)
Bezüglich der 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12d4144f83efdd79134261efc547b3f50c9a3939)
#Invarianten:
![{\displaystyle \mathrm {Sp} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{11}+A_{22}+A_{33}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/268e21edd787cc4026cbc121fae30499464b8473)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4771e7f7d95f09bc3a22cf18f346fd95bde96dc4)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/812a7333b8da50866cad310f2ae82f129cb724fc)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89f79accc67007b3d5bed0fbfa4502ee838723cb)
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}}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe124c1d9a80a27a402a54987f095f9098c7718d)
Kofaktor:
Mit den #Eigenwerten
, den #Eigenvektoren
und einer reellwertigen Funktion
eines reellen Argumentes
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e437d59e39698f6c84135a366cbdc0e044b92f5)
den Funktionswert des 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c50221154a0bb753dfd2af8e972a2380631cc275)
Ist f eine mehrdeutige Funktion, wie die Wurzel (Mathematik), mit n alternativen Werten, dann steht f(A) mehrdeutig für n3 alternative Tensoren.
Insbesondere mit dem Deformationsgradient F:
Rechter Strecktensor
![{\displaystyle \mathbf {U} =+{\sqrt {\mathbf {F} ^{\top }\cdot \mathbf {F} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a258b88407ad255cf9e195742afabfd022a1c395)
Linker Strecktensor
![{\displaystyle \mathbf {v} =+{\sqrt {\mathbf {F\cdot F} ^{\top }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20b8f07cffe62f287ea656d12b807c3a4585331c)
Henky-Dehnung
![{\displaystyle \mathbf {E} _{H}:=\ln(\mathbf {U} )={\frac {1}{2}}\ln(\mathbf {F} ^{\top }\cdot \mathbf {F} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25c163e76301c5b1f48ac3f46d189920a02edda1)
Voigt-Notation symmetrischer Tensoren zweiter Stufe
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a7cb219fe337438b9bd4778c2640471b99ada8b)
bilden eine Basis im Vektorraum
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc740ed14994011c46d49f1f888732a8541d4cf9)
Diese Vektoren dürfen addiert, subtrahiert und mit einem 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7119303ad76a47dd3ddde609ef295c2ea84800e9)
berücksichtigt werden. Siehe auch #Voigt'sche Notation von Tensoren vierter Stufe.
Schiefsymmetrische Tensoren
Definition
![{\displaystyle \mathbf {A} :\quad \mathbf {A} =-\mathbf {A} ^{\top }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee8f96268af1c4f864e08e786337a9183f00b0a5)
Kofaktor:
#Invarianten:
![{\displaystyle \mathrm {Sp} (\mathbf {A} )=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2983711d45a9666298f36bf79d5cf2114a2f720e)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} )=-{\frac {1}{2}}\mathrm {Sp} (\mathbf {A} ^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e96e5155f9fce8f1d2753d3f45cf7bd445a4dc1)
![{\displaystyle \mathrm {det} (\mathbf {A} )=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/caf7f9929850f7d5d3a89267efb891c9b2832235)
![{\displaystyle \quad \parallel \mathbf {A} \parallel ={\sqrt {2\mathrm {I} _{2}(\mathbf {A} )}}={\sqrt {-\mathrm {Sp} (\mathbf {A} ^{2})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/772de10a9a235de191a5ebd5dd5ce251f69f922f)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f81b4977ebd7e12eec181d3e2b6c2f865916fa46)
#Invarianten:
![{\displaystyle \mathrm {Sp} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fca4deee63e7836b931da028e0e1192834b042a0)
![{\displaystyle \mathrm {I} _{2}(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{12}^{2}+A_{13}^{2}+A_{23}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2343bb73670a8cb9d0110590967a71e126317b60)
![{\displaystyle \mathrm {det} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e30c7c22909c466b70f95f6239efe45323a972c)
![{\displaystyle \parallel A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\parallel ={\sqrt {2}}{\sqrt {A_{12}^{2}+A_{13}^{2}+A_{23}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff99161653d6e9629c7fc538bf1e3c6f1d09d83)
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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd60b2dff040b7409e0f98f96e58bd8dfa98faaa)
![{\displaystyle {\vec {v}}\cdot \mathbf {A} \cdot {\vec {v}}=0\quad \forall {\vec {v}}\in \mathbb {V} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/454cedc15675e4c6ccfdcfb50d626822ecc2a59b)
Ein Eigenwert ist null, zwei imaginär konjugiert komplex, siehe #Axialer Tensor oder 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/016dc2398bc5cc4bd513e68eb7ee1d4e43796c39)
mit #Vektorinvariante
. Der zum Eigenwert null gehörende #Eigenvektor ist proportional zum dualen axialen Vektor
denn
![{\displaystyle \mathbf {A\cdot A} _{\times }=\mathbf {A} _{\times }\times \mathbf {A} _{\times }={\vec {0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6df82828759e192cbf65e16eef4d79f2775c74a)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19f3d5185d745644c9fc614a0824557529f67e76)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0431ea3ded6b2c37ef20bb5ed8c67dcbd23ce25)
Axialer Tensor oder Kreuzproduktmatrix
Kreuzproduktmatrix
eines Vektors
:
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f72e329dc94d224e41bbd6db6d0974d64d19fc4a)
Kofaktor:
#Invarianten:
![{\displaystyle \mathrm {Sp} =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dde01f883cf7aa4c112e74831ed7ad0807b2f52)
![{\displaystyle \mathrm {I} _{2}={\vec {u}}\cdot {\vec {u}}=u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f921c5b9d20c53a13904e1d1ec470ddc40f8fddd)
![{\displaystyle \mathrm {det} =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d173d7b4109a4a11359e21ab6b52241d1a4dfc)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a76e3cfec164d9b40a281c5789c7983560b1f13)
![{\displaystyle {\stackrel {A}{\overrightarrow {{\vec {u}}\times \mathbf {1} }}}={\vec {u}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74d4eee561aa10b7e2812b78c24323cf34d91ea1)
#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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3445f05b880054378d3b94f07446ad6eb3d5cebe)
Eigenschaften:
![{\displaystyle {\vec {u}}\times {\vec {v}}=({\vec {u}}\times \mathbf {1} )\cdot {\vec {v}}={\vec {u}}\cdot ({\vec {v}}\times \mathbf {1} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e34935c452dd8f8fe0b0d842dd6fca3e302d33c2)
![{\displaystyle {\vec {u}}\times \mathbf {1} =\mathbf {1} \times {\vec {u}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a755ce8b05dcfe5cdaf9ee25f8a583eec64b8c7)
![{\displaystyle ({\vec {u}}\times \mathbf {1} )^{\top }=-{\vec {u}}\times \mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d330a938bb9f51d3a19b55a11fa62484318f593)
![{\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} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2f5beea7a46aaccf6798d6658e06b388e562c4)
![{\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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/47801d8aca493975baf2a98820007b6edfc62472)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fabaa729c06b9d84925a54df16fdb1682dfbd098)
Potenzen von
![{\displaystyle [{\vec {u}}]_{\times }^{2}=[{\vec {u}}]_{\times }\cdot [{\vec {u}}]_{\times }={\vec {u}}\otimes {\vec {u}}-({\vec {u}}\cdot {\vec {u}})\mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f14903b24e3e44d457e73394ce7b6aed137b866e)
![{\displaystyle [{\vec {u}}]_{\times }^{3}=-({\vec {u}}\cdot {\vec {u}})[{\vec {u}}]_{\times }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f102b85bacccd189ffa7d45e231516f4d368f6dc)
Deviatorische Tensoren
Definition
![{\displaystyle \mathbf {A} :\quad \mathrm {Sp} (\mathbf {A} )=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6791dfbd7668bf0181cf427074aec76f62b432)
Kofaktor:
#Hauptinvarianten:
![{\displaystyle \mathrm {Sp} (\mathbf {A} ):=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3eae3b2f87c47da00c0cd75d35088ec8fb51e8)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} )=-{\frac {1}{2}}\mathrm {Sp} (\mathbf {A} ^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e96e5155f9fce8f1d2753d3f45cf7bd445a4dc1)
![{\displaystyle \mathrm {det} (\mathbf {A} )={\frac {1}{3}}\mathrm {Sp} (\mathbf {A} ^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f137312137f2dae5829910cd1a64c387ce0f43)
Bezüglich der 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36c814ab757c0f9924a49d0ca13116837b1d2907)
![{\displaystyle \mathrm {Sp} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fca4deee63e7836b931da028e0e1192834b042a0)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf51494d11c7857e60f392ada08c5d0223c2e358)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a917481318b4566c1450a8e2ce9536308bc1f9e0)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6134e573aac2de1eeb2080533b1b3302844c3217)
Kugeltensoren
Definition
![{\displaystyle \mathbf {A} :\quad \mathbf {A} =a\mathbf {1} ={\begin{pmatrix}a&0&0\\0&a&0\\0&0&a\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f3846a5a0dbc9f8c26fde9add551de180f28867)
Kofaktor:
![{\displaystyle \mathrm {Sp} (\mathbf {A} )=3a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/989a752b4b91beb98a8b0ce9d2315a1d9d3e9ea1)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} )=3a^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93c25505673a67c0192b63c88c80d4985fa077dd)
![{\displaystyle \mathrm {det} (\mathbf {A} )=a^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1533052abe47834b389753f2a6cf9f73855f38b5)
![{\displaystyle \parallel \mathbf {A} \parallel ={\sqrt {3}}|a|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73895b5465a98b47203420cdd805d1d66fea8706)
Dekompositionen eines Tensors
Gegeben ein beliebiger Tensor
Symmetrischer Anteil
![{\displaystyle \mathbf {A} ^{\mathrm {S} }=\mathrm {sym} (\mathbf {A} ):={\frac {1}{2}}(\mathbf {A} +\mathbf {A} ^{\top })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16c4bf4257a274ef72a6443caaa0ae29729f62d9)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00073c4588774ed2f921974fb4d740868b8db4a7)
![{\displaystyle \mathrm {Sp} (\mathbf {A} ^{\mathrm {S} })=\mathrm {Sp} (\mathbf {A} )=A_{11}+A_{22}+A_{33}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/595314f909262b02fe6cd3ff598c3bdc9a7d8b08)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4b87ed4f2e19999cbd0b913d9248e7805e5a1f)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce01ffbfe3bb963d107415f25429ea5d98f9924c)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73143912adce0a407a74247ec14a1a5335893902)
Schiefsymmetrischer Anteil
![{\displaystyle \mathbf {A} ^{\mathrm {A} }=\mathrm {skw} (\mathbf {A} ):={\frac {1}{2}}(\mathbf {A} -\mathbf {A} ^{\top })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc1bd64655361d645341c2e0db6d191376e067dc)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cf093da909c6e841f70be77a331f618eb2de2b0)
![{\displaystyle \mathrm {Sp} (\mathbf {A} ^{\mathrm {A} })=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a629d531239d1edd5cce732c08d6226c566d448)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b309ec8c1571cf5cc87371062eae8507bc44c5)
![{\displaystyle \mathrm {det} (\mathbf {A} ^{\mathrm {A} })=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0274b69fcdcba2fecc8d06cf404acbacbffdf8a1)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d6d1b1e6c8cb11876acba79c0be7f63e478ecba)
Deviator
![{\displaystyle \mathbf {A} ^{\mathrm {D} }=\mathrm {dev} (\mathbf {A} ):=\mathbf {A} -{\frac {1}{3}}\mathrm {Sp} (\mathbf {A} )\mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/86770d0c1bbadcc7acd4a0254dce579b15cb40f1)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e89d6daec43fb3b230fbe90187c643f512d5563)
![{\displaystyle \mathrm {Sp} (\mathbf {A} ^{\mathrm {D} })=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d96c9898b5e21a95c411132364d0f802b3d0209)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b9e5b8b70297174331c5ec17aae882f77a8e297)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90d9773006d1fa1d5dd4ddd744274787a9619f78)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/006dac2e91673e08517ca63f9ff6db2abddb1308)
Kugelanteil
![{\displaystyle \mathbf {A} ^{\mathrm {K} }=\mathrm {sph} (\mathbf {A} ):={\frac {1}{3}}\mathrm {Sp} (\mathbf {A} )\mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/740f8af76d431d3fda9eebcbea3a299398d6d2b7)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/413673accf90e24a0f0b68883845c4717df8dab5)
![{\displaystyle \mathrm {Sp} (\mathbf {A} ^{\mathrm {K} })=\mathrm {Sp} (\mathbf {A} )=A_{11}+A_{22}+A_{33}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e64396664dcffee237f31ecc98a2b3e9e4a08965)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bce72e19460286e8e49774569276ed428f2cc79)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2432bdfd006365a920627f25e8db43b491eb77b)
![{\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}|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ec5d37fe4190154d04639a33c99ca760399bc9b)
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} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae9b900aa940ffe957b5b9c61522449710ae8d7)
Symmetrische und schiefsymmetrische Tensoren sind orthogonal zueinander:
![{\displaystyle \mathbf {A} ^{\mathrm {S} }:\mathbf {B} ^{\mathrm {A} }=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef73c2b0c4ea8f0aae5e112f5029cd652d4294f5)
Deviatoren und Kugeltensoren sind orthogonal zueinander:
![{\displaystyle \mathbf {A} ^{\mathrm {D} }:\mathbf {B} ^{\mathrm {K} }=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb93cfdf0a890a71a5ba8cab9436c45b7d8230f1)
Polarzerlegung
Für jeden Tensor F mit #Determinante ≠ 0 gibt es #Orthogonale Tensoren Q und #Symmetrische und positiv definite Tensoren U in eindeutiger Weise, sodass
- F = Q·U
Im Fall des Deformationsgradienten ist U der rechte Strecktensor, siehe #Symmetrische und positiv definite Tensoren. Der Anteil U berechnet sich wie dort angegeben aus
![{\displaystyle \mathbf {U} =+{\sqrt {\mathbf {F^{\top }\cdot F} }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed2c3312861f4a0b9010bcc017c129338c6dd9c)
Dann ist U·U = F⊤·F und
![{\displaystyle \mathbf {Q} =\mathbf {F\cdot U} ^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4993f7eaf4c78f5bfcc11ab5cbd22e19d63c579b)
Bei det(F)=0 ergeben sich U sowie Q aus der Singulärwertzerlegung von F und U ist nur noch symmetrisch positiv semidefinit.
Projektionen
Punkt auf Gerade
Gegeben sei die Gerade durch den Punkt
mit Richtungsvektor
und ein beliebiger anderer Punkt
.
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abc8667212d279485a1b354f4e4d310e417416e7)
Der Punkt
ist die senkrechte Projektion von
auf die Gerade. Der Tensor G extrahiert den Anteil eines Vektors in Richtung von
und 1-G den Anteil senkrecht dazu.
Punkt oder Gerade auf Ebene
Gegeben sei die Ebene durch den Punkt
und zwei die Ebene aufspannende Vektoren
und
sowie ein beliebiger anderer Punkt
. Dann verschwindet die Normale
![{\displaystyle {\hat {n}}={\frac {{\vec {u}}\times {\vec {v}}}{|{\vec {u}}\times {\vec {v}}|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afa0f5aeec49d5180f070836465984f97ed4a785)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a78d531f06c25409c16ffa18c8eb327144b7512)
Der Punkt
ist die senkrechte Projektion von
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
und
verläuft, liegt in der Ebene in Richtung des Vektors
.
Falls
und
folgt:
![{\displaystyle {\hat {n}}={\vec {u}}\times {\vec {v}}\quad {\text{mit}}\quad |{\hat {n}}|=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2270c6e87983bd30ec892e730db8fc387c91837a)
![{\displaystyle \mathbf {P} ={\vec {u}}\otimes {\vec {u}}+{\vec {v}}\otimes {\vec {v}}=\mathbf {1} -{\hat {n}}\otimes {\hat {n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f502117290303656624ed54d03cf871feb700ecf)
![{\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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e82347b42982c405da0f6db7f3c29f4f1091561e)
![{\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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec6c1d41435b9c3f160ecea7d5bc81f8fc97cc70)
Fundamentaltensor 3. Stufe
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa5bc30e4c8a96178459911d40f5cbc35f393c43)
Kreuzprodukt von 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d07d32c070e13c0c40141655bcf03a3a0199463)
![{\displaystyle {\vec {e}}_{i}\times {\vec {e}}_{j}=\epsilon _{ijk}\,{\hat {e}}_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5fe8b5401313a634a88a3cdffdcfb4e7bdaa19)
#Kreuzprodukt von Tensoren, #Skalarkreuzprodukt von 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b29641349946980ccdbbb1f513724fccabc4c926)
#Dualer axialer Vektor und #Vektorinvariante:
![{\displaystyle {\stackrel {3}{\mathbf {E} }}:\mathbf {A} =-2{\stackrel {A}{\overrightarrow {\mathbf {A} }}}={\vec {\mathrm {i} }}(\mathbf {A} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21867dbb8bdc099cd524304e7576d7feda150e9c)
#Kreuzprodukt von Tensoren:
![{\displaystyle \mathbf {A} \times \mathbf {B} ={\stackrel {3}{\mathbf {E} }}:(\mathbf {A\cdot B} ^{\top })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ad467cf91df2fa099756da6968502b2cec0a46c)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06309c28b27b0185f52f1d2944e71b2465722539)
#Skalarkreuzprodukt von Tensoren:
![{\displaystyle \mathbf {A} \cdot \!\!\times \mathbf {B} ={\stackrel {3}{\mathbf {E} }}:(\mathbf {A\cdot B} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcfe4cf9ac708650706c735b60818bb0147bf483)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/735599596ee497a9f553a295253c419b4bf09342)
#Axialer Tensor oder 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f769fbc1a684a8fe9e0d813e4a9e8ec739963857)
Tensoren vierter Stufe
Tensoren zweiter Stufe sind ebenfalls Elemente eines Vektorraums
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78bfab4c9cd1ab364808efd09fe7a6c3aa8eadd7)
mit Komponenten
und die Tensoren
sowie
bilden eine Basis von
.
Standardbasis in
:
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/117f6c41da69b5ba6f1fc84c683a89cee60f8911)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf613e69a34dc6f79f8319b884723a31d075aaf)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9525703bddeaed75db48bef45fecca155895224)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43def320e42740a292d08b922a30797f5192a90f)
Transpositionen
Transposition:
![{\displaystyle (\mathbf {A} \otimes \mathbf {B} )^{\top }=\mathbf {B} \otimes \mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb8935008c4e612fbe704e9288d68846fc945d2)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a66a43d19ff28a11e5b1bfa785b31dfc831eab)
Spezielle Transposition
vertauscht
-tes mit
-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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35dcca595ebba7dba683e9ed5e6608607173d36e)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4367be00c130189346f9e9293dee215bb1d5e9d)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/240bfba43675239b91cd1dada64713efc4934b52)
Symmetrische Tensoren vierter Stufe
Definition:
Dann gilt:
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79ad5e3655a8470eb4225423225370b4c41c6ff6)
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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fbbcd6b3e02bbb2da78373e0de0a091a3a255ae)
![{\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} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32e3e18d92bea0d8389a7ed9537f996d5b9dd737)
![{\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} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef304bc67a9c3bc7720540183414f19d270acdb)
![{\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} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a281c9c8dc8fd6c59da3cf576d56245bcc9b224)
![{\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} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e292baf2ceebd1f6fac49664b162e1ad245d0f45)
Diese fünf Tensoren sind sämtlich symmetrisch.
Mit beliebigen Tensoren zweiter Stufe A, B und 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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5dba566aea8229c318ee0055a754c161da2e1b)
![{\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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a23f8ef554f15711426f2bd8d7839311fdf4ce5)
![{\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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/687a9a91fc3fbe970263bdf7f2900dc6be3c7b95)
![{\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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fddf3de3db0b0a636018fb6f6e2f0aa38d16b2c3)
In dem in diesen Formeln im Tensor vierter Stufe B durch B⊤ und die Transpositionen
durch
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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/63651d08a16816ddda62b8d3319ed14623fc4c87)
![{\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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f492e3e69d7dbfc2b308a200ae98c60da1694da4)
![{\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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cafe32d0c2317e37e5a1f0958291cd4f184c0286)
![{\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 }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a910e654b40a690884679e968ac46711aec734)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42c1659462f023c4921ce99acfd8662d10e33776)
Hooke'sches Gesetz
Mit den Spannungen
und den Dehnungen
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} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f048f6a911f6f904ddc6504f3feec91bbb81b0)
mit den Lamé-Konstanten
und
. Dieser Elastizitätstensor ist symmetrisch.
Invertierungsformel mit
,
und
:
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eed170c6bd3a24b8fe41c2180396cb1e5b9464a)
mit der Querdehnzahl
und dem Elastizitätsmodul
.
Voigt'sche Notation von Tensoren vierter Stufe
Aus der Basis
des Vektorraums
der symmetrischen Tensoren zweiter Stufe, siehe #Voigt-Notation symmetrischer Tensoren zweiter Stufe, kann eine Basis des Vektorraums
der linearen Abbildungen von symmetrischen Tensoren auf symmetrische Tensoren konstruiert werden. Die 36 Komponenten der Tensoren vierter Stufe aus
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ac88d5af2bf0c5e3aa3e3e8de900667d37581fe)
Die Vektoren und Matrizen in Voigt'scher Notation können addiert, subtrahiert und mit einem Skalar multipliziert werden. Beim Matrizenprodukt in Voigt'scher Notation muss eine Diagonalmatrix
![{\displaystyle I=\mathrm {diag} (1,1,1,2,2,2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffc37b88f739e07366c653e193bd43140c4b86a0)
mit den Einträgen
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1023b8d8a1144dd451c63a647f0d892e8604a151)
![{\displaystyle \left[{\stackrel {4}{\mathbf {A} }}:\mathbf {T} \right]=\left[{\stackrel {4}{\mathbf {A} }}\right]I[\mathbf {T} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ca5d512aecfed8938663251e15908632637d87)
![{\displaystyle \left[{\stackrel {4}{\mathbf {A} }}:{\stackrel {4}{\mathbf {B} }}\right]=\left[{\stackrel {4}{\mathbf {A} }}\right]I\left[{\stackrel {4}{\mathbf {B} }}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d5f433e8e0b5eb5758cd39f4a74d7abcf21981e)
Darin steht [x] für die Voigt-Notation von x.
Einzelnachweise
- ↑ 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).
- ↑ 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.