1, A.J.Mcconnell, applications of tensor analysis, dover publications,Inc , NEW York

1. 1, A.J.Mcconnell, applications of tensor analysis, dover publications,Inc , NEW York 2, Garl E.pearson, handbook of applied mathematics, Van Nostrand Reinhold ; 2nd edition 3, I.s. sokolnikoff, tensor analysis, Walter de Gruyter ; 2Rev Ed edition 4. C.WREDE, introduction to vectors and tensor analysis, Dover Publications ; New Ed edition 5.WILHELM Flugge, tensor analysis and continuum mechanics , Springer ; 1 edition 6.Eutioquio C.young , vector and tensor analysis, CRC ; 2 edition 7. 黃克智,薛明德,陸明萬, 張量分析, 北京清華大學出版社

2. Tensor analysis 1、Vector in Euclidean 3-D 2、Tensors in Euclidean 3-D 3、general curvilinear coordinates in Euclidean 3-D 4、tensorcalculus

3. 1-1 Orthonormal base vector： 1、Vector in euclidean 3-D Let (e1,e2,e3) be a right-handed set of three mutually perpendicular vector of unit magnitude

4. ei(i = 1,2,3)may be used to define the kronecker delta δij and the permutation symbol eijk by means of the equations and

5. Ex1. [prove]

6. Thus δij = 1 for i=j ; e123 =e312=e231=1;e132=e321=e213=-1; All other eijk = 0. in turn,a pair of eijk is related to a determinate of δij by (1-1-1) By setting r= i we recover the e – δrelation

7. Rotated set of orthonormal base vector is introduced. the corresponding new components of F may be computed in terms of the old ones by writing 1-2 Cartesian component of vectors transformation rule We have transformation rule Here

8. These direction cosines satisfy the useful relations (1-2-1) [prove]

9. 1-3 General base vectors： vector components with respect to a triad of base vector need not be resticted to the use of orthonomal vector .let ε1, ε2 , ε3 be any three noncoplanar vector that play the roles of general base vectors

10. (metric tensor) and (permutation tensor) From (1-1-1) the general vector identity can be established (*)

11. Note that the volume v of the parallelopiped having the base vectors as edges equals ; consequently the permutation tensor and the permutation symbols are related by Denote by the determinate of the matrix having as its element then, by (*), , so that

12. 1-4 General components of vectors ;transformation rules convariant component (free index) contravariant component (dummy index) Summation notation: the repeated index i, called “dummy index”, is to be summed from 1 to n. This notation is due to Einstein.

13. Figure. Convariant tensor and contravariant tensor in Euclidean 2-D The two kinds of components can be related with the help of the metric tensor . Substituting into yields [prove]

14. Use to denote the (i , j)th element of the inverse of the matrix [g] Where that the indices in the Kronecker delta have been placed in superscript and subscript position to conform to their placement on the left-hand side of the equation. Proof.

15. When general vector components enter , the summation convention invariably applies to summation over a repeated index that appears once as a superscript and once as a subscript The metric tensor can introduce an auxiliary set of base vectors by means of means of the definition

16. Consider , finally , the question of base vector , a direct calculation gives for the transformed covariant components. We also find easily that

17. 2、Tensors in Euclidean 3-D 2-1 Dyads, dyadics, and second-order tensors The mathematical object denote by AB, where A and B are given vectors, is called a dyad. The meaning of AB is simply that the operation A sum of dyads, of the form Is called a dyadic

18. Any dyadic can be expressed in terms of an arbitrary set of general base vectors εi ;since It follows that T can always be written in the form (2-1-1) (contravariant components of the tensor).