02 - tensor calculus - tensor algebra

1 102 - tensor calculus02 - tensor calculus - tensor algebra2tensor calculustensor the word tensor was introducedin 1846 by william rowan hamilton. it wasused in its current meaning by woldemarvoigt in 1899. tensor calculus was deve-loped around 1890 by gregorio ricci-curba-stro under the title absolute differentialcalculus. in the 20th century, the subjectcame to be known as tensor analysis, andachieved broader acceptance with the intro-duction of einsteins's theory of generalrelativity around 1915. tensors are usedalso in other fields such as calculus3tensor calculustensor calculus - repetition tensor analysis vector algebranotation, euklidian vector space, scalar product, vectorproduct, scalar triple productnotation, scalar products, dyadic product, invariants, trace,determinant, inverse, spectral decomposition, sym-skewdecomposition, vol-dev decomposition, orthogonal tensorderivatives, gradient, divergence, laplace operator.

2 Integraltransformations tensor algebra4tensor calculusvector algebra - notation summation over any indices that appear twice in a term einstein s summation convention5tensor calculusvector algebra - notation permutation symbol kronecker symbol6tensor calculusvector algebra - euklidian vector space zero element and identity euklidian vector space is defined through the following axioms linear independence ofifis the only (trivial) solution to7tensor calculusvector algebra - euklidian vector space euklidian vector space equipped with norm norm defined through the following axioms8tensor calculusvector algebra - euklidian vector space euklidian vector space equipped with representation of 3d vectoreuklidian normwith coordinates (components) of relative tothe basis9tensor calculusvector algebra - scalar product euklidian norm enables definition of scalar (inner)

3 Product properties of scalar product positive definiteness orthogonality10tensor calculusvector algebra - vector product vector product properties of vector product11tensor calculusvector algebra - scalar triple product scalar triple product properties of scalar triple productareavolume linear independency12tensor calculustensor algebra - second order tensors second order tensor transpose of second order tensorwith coordinates (components) of relative tothe basis13tensor calculustensor algebra - second order tensors second order unit tensor in terms of kronecker symbol matrix representation of coordinateswith coordinates (components) of relative to thebasis identity14tensor calculustensor algebra - third order tensors third order tensor third order permutation tensor in terms of permutationwith coordinates (components) of relativeto the basissymbol15tensor calculustensor algebra - fourth order tensors fourth order tensor fourth order unit tensorwith coordinates (components)

4 Ofrelative to the basis transpose of fourth order unit tensor16tensor calculustensor algebra - fourth order tensors symmetric fourth order unit tensor screw-symmetric fourth order unit tensor volumetric fourth order unit tensor deviatoric fourth order unit tensor17tensor calculustensor algebra - scalar product scalar (inner) product properties of scalar productof second order tensor and vector zero and identity positive definiteness18tensor calculustensor algebra - scalar product scalar (inner) product properties of scalar productof two second order tensors and zero and identity19tensor calculustensor algebra - scalar product scalar (inner) productof fourth order tensors and second order tensor zero and identity scalar (inner) productof two second order tensors20tensor calculustensor algebra - dyadic product dyadic (outer) product properties of dyadic product ( tensor notation)of two vectors introduces second order tensor21tensor calculustensor algebra - dyadic product dyadic (outer) product properties of dyadic product (index notation)of two vectors introduces second order tensor