Wir definieren also die Tensoren durch das Verhalten ihrer Koordinaten bei Ausführung einer Bewegung des Koordinatensystems, die durch
EquationSource% MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!
% MathType!MTEF!2!1!+-
% feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa
% aaleaacaWGPbaabeaakiabg2da9iaadggadaWgaaWcbaGaamyAaiaa
% dQgaaeqaaOGabmiEayaaraWaaSbaaSqaaiaadQgaaeqaaOGaey4kaS
% IaamOyamaaBaaaleaacaWGPbaabeaaaaa!412F!
\[{x_i} = {a_{ij}}{\bar x_j} + {b_i}\]$$
((10, 01))
mit
EquationSource% MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!
% MathType!MTEF!2!1!+-
% feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa
% aaleaacaWGPbGaamOAaaqabaGccaWGHbWaaSbaaSqaaiaadMgacaWG
% Obaabeaakiabg2da9iabes7aKnaaBaaaleaacaWGQbGaamiAaaqaba
% aaaa!4095!
\[{a_{ij}}{a_{ih}} = {\delta _{jh}}\]$$
((10, 02))
oder
EquationSource% MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!
% MathType!MTEF!2!1!+-
% feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa
% aaleaacaWGObGaamyAaaqabaGccaWGHbWaaSbaaSqaaiaadQgacaWG
% Pbaabeaakiabg2da9iabes7aKnaaBaaaleaacaWGObGaamOAaaqaba
% aaaa!4095!
\[{a_{hi}}{a_{ji}} = {\delta _{hj}}\]$$
((10, 03))
gegeben ist. (10, 01) kann als das Transformationsgesetz der Koordinaten des Ortsvektors, d. h. der Punktkoordinaten angesehen werden. Ist
EquationSource% MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!
% MathType!MTEF!2!1!+-
% feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa
% aaleaacaWGPbaabeaakiabg2da9iaadMhadaWgaaWcbaGaamyAaaqa
% baGccqGHsislcaWG4bWaaSbaaSqaaiaadMgaaeqaaaaa!3E08!
\[{A_i} = {y_i} - {x_i}\]$$
((10, 03))
ein Vektor (Tensor I. Stufe), wobei x
i
und y
i
Anfangs- und Endpunkr sind, so folgt in den neuen Koordinaten
EquationSource% MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!
% MathType!MTEF!2!1!+-
% feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa
% aaleaacaWGPbaabeaakiabg2da9iaadMhadaWgaaWcbaGaamyAaaqa
% baGccqGHsislcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0Jaai
% ikaiaadggadaWgaaWcbaGaamyAaiaadQgaaeqaaOGabmyEayaaraWa
% aSbaaSqaaiaadQgaaeqaaOGaey4kaSIaamOyamaaBaaaleaacaWGPb
% aabeaakiaacMcacqGHsislcaGGOaGaamyyamaaBaaaleaacaWGPbGa
% amOAaaqabaGcceWG4bGbaebadaWgaaWcbaGaamOAaaqabaGccqGHRa
% WkcaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaaiykaiabg2da9iaadgga
% daWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiikaiqadMhagaqeamaaBa
% aaleaacaWGQbaabeaakiabgkHiTiqadIhagaqeamaaBaaaleaacaWG
% QbaabeaakiaacMcaaaa!5DB2!
\[{A_i} = {y_i} - {x_i} = ({a_{ij}}{\bar y_j} + {b_i}) - ({a_{ij}}{\bar x_j} + {b_i}) = {a_{ij}}({\bar y_j} - {\bar x_j})\]$$
nun sind aber
EquationSource% MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!
% MathType!MTEF!2!1!+-
% feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaara
% WaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaamyyamaaBaaaleaacaWG
% PbGaamOAaaqabaGccaWGbbWaaSbaaSqaaiaadQgaaeqaaaaa!3DD4!
\[{\bar A_i} = {a_{ij}}{A_j}\]$$
((10, 04))
die Koordinaten des Vektors A
i
im neuen System, so daß
EquationSource% MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!
% MathType!MTEF!2!1!+-
% feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaara
% WaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaamyyamaaBaaaleaacaWG
% QbGaamyAaaqabaGccaWGbbWaaSbaaSqaaiaadQgaaeqaaaaa!3DD4!
\[{\bar A_i} = {a_{ji}}{A_j}\]$$
((10, 05))
.
