tensor calculus 02 - tensor calculus - tensor the word tensor was introduced in 1846 by william rowan hamilton. it was tensor algebra 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.

02 - tensor calculus 1 tensor calculus 2

tensor calculus - repetition vector algebra - notation • vector algebra • einstein‘s summation convention 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 • summation over any indices that appear twice in a term decomposition, vol-dev decomposition, orthogonal tensor • tensor analysis derivatives, gradient, divergence, laplace operator, integral transformations

tensor calculus 3 tensor calculus 4 vector algebra - notation vector algebra - euklidian vector space • euklidian vector space • kronecker symbol

• is defined through the following axioms

• permutation symbol • zero element and identity

• linear independence of if is the only (trivial) solution to

tensor calculus 5 tensor calculus 6

vector algebra - euklidian vector space vector algebra - euklidian vector space • euklidian vector space equipped with norm • euklidian vector space equipped with euklidian norm

• norm defined through the following axioms • representation of 3d vector

with coordinates (components) of relative to the basis

tensor calculus 7 tensor calculus 8 vector algebra - scalar product vector algebra - vector product • euklidian norm enables definition of scalar (inner) product • vector product

• properties of scalar product

• properties of vector product

• positive definiteness • orthogonality

tensor calculus 9 tensor calculus 10

vector algebra - scalar triple product tensor algebra - second order tensors

• scalar triple product • second order tensor

area volume with coordinates (components) of relative to • properties of scalar triple product the basis • transpose of second order tensor

• linear independency

tensor calculus 11 tensor calculus 12 tensor algebra - second order tensors tensor algebra - third order tensors

• second order unit tensor in terms of kronecker symbol • third order tensor

with coordinates (components) of relative to the basis with coordinates (components) of relative • matrix representation of coordinates to the basis

• third order permutation tensor in terms of permutation symbol • identity

tensor calculus 13 tensor calculus 14

tensor algebra - fourth order tensors tensor algebra - fourth order tensors • fourth order tensor • symmetric fourth order unit tensor

• screw-symmetric fourth order unit tensor with coordinates (components) of relative to the basis • volumetric fourth order unit tensor • fourth order unit tensor

• transpose of fourth order unit tensor • deviatoric fourth order unit tensor

tensor calculus 15 tensor calculus 16 tensor algebra - scalar product tensor algebra - scalar product • scalar (inner) product • scalar (inner) product

of second order tensor and vector of two second order tensors and • zero and identity • positive definiteness • zero and identity • properties of scalar product • properties of scalar product

tensor calculus 17 tensor calculus 18

tensor algebra - scalar product tensor algebra - dyadic product • scalar (inner) product • dyadic (outer) product

of two vectors introduces second order tensor of two second order tensors • properties of dyadic product (tensor notation) • scalar (inner) product

of fourth order tensors and second order tensor • zero and identity

tensor calculus 19 tensor calculus 20 tensor algebra - dyadic product • dyadic (outer) product

of two vectors introduces second order tensor • properties of dyadic product (index notation)

tensor calculus 21