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Tensoren und einfachste Tensoroperationen

In: Grundƶüge der Tensorrechnung in Analytischer Darstellung

Author

Listed:
  • Adalbert Duschek

    (Technischen Hochschule Wien)

  • August Hochrainer

Abstract

Zusammenfassung Wir definieren also die Tensoren durch das Verhalten ihrer Koordinaten bei Ausführung einer Bewegung des Koordinatensystems, die durch (10, 01) % 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! \[{x_i} = {a_{ij}}{\bar x_j} + {b_i}\]$$ mit (10, 02) % 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! \[{a_{ij}}{a_{ih}} = {\delta _{jh}}\]$$ oder (10, 03) % 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! \[{a_{hi}}{a_{ji}} = {\delta _{hj}}\]$$ gegeben ist. (10, 01) kann als das Transformationsgesetz der Koordinaten des Ortsvektors, d. h. der Punktkoordinaten angesehen werden. Ist (10, 03) % 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! \[{A_i} = {y_i} - {x_i}\]$$ ein Vektor (Tensor I. Stufe), wobei x i und y i Anfangs- und Endpunkr sind, so folgt in den neuen Koordinaten % 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 (10, 04) % 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! \[{\bar A_i} = {a_{ij}}{A_j}\]$$ die Koordinaten des Vektors A i im neuen System, so daß (10, 05) % 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! \[{\bar A_i} = {a_{ji}}{A_j}\]$$ .

Suggested Citation

  • Adalbert Duschek & August Hochrainer, 1946. "Tensoren und einfachste Tensoroperationen," Springer Books, in: Grundƶüge der Tensorrechnung in Analytischer Darstellung, chapter 0, pages 56-66, Springer.
    • Handle: RePEc:spr:sprchp:978-3-7091-3476-4_11
    DOI: 10.1007/978-3-7091-3476-4_11
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