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02 - tensor calculus

02 - tensor calculus - tensor algebra. 02 - tensor calculus. tensor calculus.

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02 - tensor calculus

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  1. 02 - tensor calculus - tensor algebra 02 - tensor calculus

  2. tensor calculus tensor the word tensor was introduced in 1846 by william rowan hamilton. it was used in its current meaning by woldemar voigt in 1899. tensor calculus was deve-loped around 1890 by gregorio ricci-curba-stro under the title absolute differential calculus. in the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the intro-duction of einsteins's theory of general relativity around 1915. tensors are used also in other fields such as continuum mechanics. tensor calculus

  3. tensor calculus - repetition • vector algebra notation, euklidian vector space, scalar product, vector product, scalar triple product • tensor algebra notation, scalar products, dyadic product, invariants, trace, determinant, inverse, spectral decomposition, sym-skew decomposition, vol-dev decomposition, orthogonal tensor • tensor analysis derivatives, gradient, divergence, laplace operator, integral transformations tensor calculus

  4. vector algebra - notation • einstein‘s summation convention • summation over any indices that appear twice in a term tensor calculus

  5. vector algebra - notation • kronecker symbol • permutation symbol tensor calculus

  6. vector algebra - euklidian vector space • euklidian vector space • is defined through the following axioms • zero element and identity • linear independence of if is the only (trivial) solution to tensor calculus

  7. vector algebra - euklidian vector space • euklidian vector space equipped with norm • norm defined through the following axioms tensor calculus

  8. vector algebra - euklidian vector space • euklidian vector space equipped with euklidian norm • representation of 3d vector with coordinates (components) of relative to the basis tensor calculus

  9. vector algebra - scalar product • euklidian norm enables definition of scalar (inner) product • properties of scalar product • positive definiteness • orthogonality tensor calculus

  10. vector algebra - vector product • vector product • properties of vector product tensor calculus

  11. vector algebra - scalar triple product • scalar triple product area volume • properties of scalar triple product • linear independency tensor calculus

  12. tensor algebra - second order tensors • second order tensor with coordinates (components) of relative to the basis • transpose of second order tensor tensor calculus

  13. tensor algebra - second order tensors • second order unit tensor in terms of kronecker symbol with coordinates (components) of relative to the basis • matrix representation of coordinates • identity tensor calculus

  14. tensor algebra - third order tensors • third order tensor with coordinates (components) of relative to the basis • third order permutation tensor in terms of permutation symbol tensor calculus

  15. tensor algebra - fourth order tensors • fourth order tensor with coordinates (components) of relative to the basis • fourth order unit tensor • transpose of fourth order unit tensor tensor calculus

  16. tensor algebra - fourth order tensors • symmetric fourth order unit tensor • screw-symmetric fourth order unit tensor • volumetric fourth order unit tensor • deviatoric fourth order unit tensor tensor calculus

  17. tensor algebra - scalar product • scalar (inner) product of second order tensor and vector • zero and identity • positive definiteness • properties of scalar product tensor calculus

  18. tensor algebra - scalar product • scalar (inner) product of two second order tensors and • zero and identity • properties of scalar product tensor calculus

  19. tensor algebra - scalar product • scalar (inner) product of two second order tensors • scalar (inner) product of fourth order tensors and second order tensor • zero and identity tensor calculus

  20. tensor algebra - dyadic product • dyadic (outer) product of two vectors introduces second order tensor • properties of dyadic product (tensor notation) tensor calculus

  21. tensor algebra - dyadic product • dyadic (outer) product of two vectors introduces second order tensor • properties of dyadic product (index notation) tensor calculus

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