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

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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

Tensor analysis

1、Vector in Euclidean 3-D

2、Tensors in Euclidean 3-D

3、general curvilinear coordinates in Euclidean 3-D

4、tensorcalculus

1-1 Orthonormal base vector：

Let (e1,e2,e3) be a right-handed

set of three mutually perpendicular

vector of unit magnitude

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

Ex1.

[prove]

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

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

These direction cosines satisfy the useful relations

(1-2-1)

[prove]

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

(metric tensor)

and

(permutation tensor)

From (1-1-1) the general vector identity

can be established

(*)

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

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.

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]

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.

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

Consider , finally , the question of base vector , a direct calculation gives

for the transformed covariant components. We also find easily that

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

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).

We re-emphasize the basic meaning of T by noting that ,for all vector V

By introduce the contravariant base vectors єi we can define other kinds of components of the tensor T. thus, substituting

Into (2-1-1), we get

Where are nine quantities

are called the covariantcomponent of the tensor. Similarly ,we can define two, generally ,kinds of mixed components

That appear in the representation

2.2 Transformation rule

Suppose new base vectors are introduced； what are the new components of T？ substitution

Into (2-1-1) gives

Is the desired transformation rule. Many different, but equivalent, relations are easily derived；for example

2.3 Cartesian components of second-order Tensors

Cartesian components

2.4 Tensors operations

Quotient laws

2.5 The metric Tensor

Substituting

2.6 Nth _ order Tensors

A third-order tensor, or triadic, is the sum of triads ,as follows：

It is easily established that any third-order tensor can written

As well as in the alternative form

N indices

N base vectros

2.7 The permutation tensor

Choose a particular set of base vectors ，and define the third –order tensor

(2-4-1)

Since it follow that E has the same from as (2-4-1) with respect to all sets of base vector , so is indeed a tensor