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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

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slide1

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

slide2

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 vector in euclidean 3 d
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

slide4
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

slide5

Ex1.

[prove]

slide6

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

slide7

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

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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

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(metric tensor)

and

(permutation tensor)

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

can be established

(*)

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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

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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.

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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]

slide14

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.

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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

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

for the transformed covariant components. We also find easily that

2 tensors in euclidean 3 d
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

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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).

slide19

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

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

That appear in the representation

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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

slide25
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

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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

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