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 、 tensor calculus

1-1 Orthonormal base vector：

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

set of three mutually perpendicular

vector of unit magnitude

1 、 Vector in euclidean 3-D

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

i j ij e e

i j k ijk e e e e

))(( FEDCBA

FCECDC

FBEBDB

FAEADA

[prove]

)()(

)()(

333

222

111

321

321

321

333

222

111

321

321

321

321

321

321

321

321

321

321

321

321

FCECDC

FBEBDB

FAEADA

FED

FED

FED

CCC

BBB

AAA

FED

FED

FED

FFF

EEE

DDD

EEE

DDD

FFF

EDFFED

CCC

BBB

AAA

BBB

AAA

CCC

BACCBA

Ex1.

By setting r= i we recover the e – δ relation

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

( )( )i

i r i s i t ir is it

jk rst i j k r s t j r j s j t jr js jt

k r k s k t kr ks kt

e e e e e e

e e e e e e e e e e e e e e

e e e e e e

ijk ist js kt jt ks e e

(1-1-1)

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

iF( 1,2,3)ii e

3,2,1i

We have transformation rule

Here

j j ji iF F F e e e

j ij iF l F

jij il e e

i iF F e

1-2 Cartesian component of vectors transformation rule

These direction cosines satisfy the useful relations

ip jp pi pj ijl l l l (1-2-1)

[prove]

i j ik k jm m

ik jm k m ik jk ij

l l

l l l l

e e e e

e e

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

and

ijgε

i j× ε

ε ε εi1 i2 i3

ε ε ε ε ε ε εi j k ijk j1 j2 j3

ε ε εj1 j2 j3

× × = =

(metric tensor)

(permutation tensor)

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

i r i s i t

i j k r s t j r j s j t

k r k s k t

ε × ε ε × ε ε × ε

(ε × ε × ε )(ε × ε × ε ) = ε × ε ε × ε ε × ε

ε × ε ε × ε ε × ε

can be established

ktkskr

jtjsjr

itisir

ggg

ggg

ggg

(*)

then, by (*), , so that

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

1 2 3ε × ε × ε

ijk ijkv eε

Denote by the determinate of the matrix having as its elementg g ijg ( , )thi j

2 2123( ) g v ε

ijk ijkgε e

333231

232221

131211

ggg

ggg

ggg

g

1-4 General components of vectors ;transformation rules

i iF F ε

ii

FF ε

convariant component

contravariant component

Summation notation: the repeated index i, called “dummy index”, is to be summedfrom 1 to n. This notation is due to Einstein.

(free index)

(dummy index)

indexdummy xa

xa

xaxaxaS

ii

n

1iii

nn2211

The two kinds of components can be related with the help of the metric tensor . Substituting into yields ijg

ii FF iiFF

ijg

ji ij

F g F

[prove]

ijj jF F F g

i i j i F ε ε ε

2ε

1ε

2ε

1εFigure. Convariant tensor and contravariant tensor in Euclidean 2-D

ip ipj j

g g

Use to denote the (i , j)th element of the inverse of the matrix [g] ijg

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.

ijij

F g F

jij

jjij

jiiii

ii

FgFFFFF

FF

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

( ) ( )j ji i ii j ij i

F F F F g F F F F ε ε

The metric tensor can introduce an auxiliary set of base vectors by means of means of the definition

ijij

gε ε

for the transformed covariant components. We also find easily that

( )ij ji

F F ε ε

( ) ( )j j ji i

i iF F F ε ε ε ε

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

( ) ( )i ij i j i j

F F F ε ε ε ε

2-1 Dyads, dyadics, and second-order tensors

( ) ( ) AB V A B V

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

T AB + CD + EF

2 、 Tensors in Euclidean 3-D

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)

, , ,i i ii i i

A B C A ε B ε C ε

j ji ii j i j

A B C D T ε ε ε ε

iji j

TT ε ε

(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

iji j

T ( ) T V ε ε V

( )ijj i

T V ε

pi ip

gε εq

j jqgε ε

p qpqT ε ε

Where are nine quantities

ijpq ip jq

T g g T

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

.

.

iqij jq

j pji ip

T g T

T g T

That appear in the representation

. .j ji i

j i i jT T T ε ε ε ε

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

( )( )p q pqij

p q p qi jT T T ε ε ε ε ε ε ε ε

( )( )

( )( )

ijpq p qi j

p p jiq qij

T T

T T

ε ε ε ε

ε ε ε ε

( )

( )

ppi i

qqj j

ε ε ε ε

ε ε ε ε

iεij

T

2.2 Transformation rule

2.3 Cartesian components of second-order Tensors

ij i jTT e e

ij pq pi qjT T l l

Cartesian components

2.4 Tensors operations

(tensor)

(scalar)

(scalar)

ij ijk k

ijij

ij iij i

T S P

T S

g T T

Quotient laws

,

S

S

iji j i i

ijij ij

ijjk jk

T X Y is a scalar for all vectors X Y

T S is a scalar for all tensors

T S is a tensor for all tensors

2.5 The metric Tensor ijg

ji iij i

g X Y X Y X Y

ii

g ε ε

Substituting ( )p

pi i

( )p pi

p pig ε ε ε ε ε ε

jiij

g g ε ε

2.6 Nth _ order Tensors

GHIDEFABC

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

ijkT Ti j k

ε ε ε

..ij kk i j

T ε ε ε

.ji k

jk iT ε ε ε

....ijk st l mlm i j k s t

T T ε ε ε ε ε ε ε

N indices N base vectros

k s tel m f g

( )( )

( )( )( ) ( )( )

ijk i jfgst abclm de a b

dc

T T

ε ε ε ε

ε ε ε ε ε ε ε ε ε ε

2.7 The permutation tensor

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

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

iiii

ε ε ε ε

( ) ji ki j k

E ε ε ε ε ε ε

ijk i j k ε ε ε

(2-4-1)

ijk

iε

1ijk ijkeg