02 - tensor calculus - tensor algebra tensor calculus

Transcription

1 tensor algebra 02-1

2 tensor the word tensor was introduced in 1846 by william rowan hamilton. it was used in its current meaning by woldemar voigt in was developed around 1890 by gregorio ricci-curbastro 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 introduction of einsteins's theory of general relativity around tensors are used also in other fields such as continuum mechanics. 2

3 - 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 3

4 vector algebra - notation einstein s summation convention summation over any indices that appear twice in a term 4

5 kronecker symbol vector algebra - notation permutation symbol 5

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

7 vector algebra - euklidian vector space euklidian vector space equipped with norm norm defined through the following axioms 7

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

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

10 vector product vector algebra - vector product properties of vector product 10

11 vector algebra - scalar triple product scalar triple product area volume properties of scalar triple product linear independency 11

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

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 13

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

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 15

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 16

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

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

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

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

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