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 e ijk by means of the equations

and

ijji ee

ijkkji eeee

))(( 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

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

ktkskr

jtjsjr

itisir

tkskrk

tjsjrj

tisiri

tsrkjirstijk

eeeeee

eeeeee

eeeeee

eeeeeeee

))((

ksjtktjsistijkee

(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)3,2,1( iei

3,2,1i

We have transformation rule

Here

jiijj eeFeFF

iijj FlF

jiij eel

ii eFF

1-2 Cartesian component of vectors transformation rule

These direction cosines satisfy the useful relations

ijpjpijpip llll (1-2-1)

[prove]

ijjkikmkjmik

mjmkikji

lleell

elelee

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

ijji g

321

321

321

jjj

jjj

iii

ijkkji

(metric tensor)

(permutation tensor)

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

tkskrk

tjsjrj

tisiri

tsrkji

))((

can be established

ktkskr

jtjsjr

itisir

ggg

ggg

ggg

(*)

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

321

ijkijk ve

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

22123)( vg

ijkijk eg

333231

232221

131211

ggg

ggg

ggg

g

then, by (*), , so that

1-4 General components of vectors ;transformation rules

ii FF

iiFF

(convariant component)

(contravariant component)

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

iFF ijg

jiji FgF

[prove]

ijj

ijj

ii gFFFF

ijpj

ipgg

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.

jiji FgF

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

ii

ijji

jj

ii FFgFFFFFF )()(

for the transformed covariant components. We also find easily that

)( ji

ij FF

)()(ji

i

j

iij

FFF

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

jiji g

i

)()( jii

jii

j FFF

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

)()( VBAVAB

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

EFCDABT

2 、 Tensors in Euclidean 3-D

Any dyadic can be expressed in terms of an arbitrary set of general base vectors ε i ;since

It follows that

T can always be written in the form

(2-1-1)

,,, ii

ii

ii CCBBAA

jiji

jiji DCBAT

jiijTT

(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

)( VTVT jiij

ijijVT )(

pipi g q

jqj g

qppqTT

Where are nine quantities

ijjqippq TggT

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

pjip

ji

iqjq

ij

TgT

TgT

.

.

That appear in the representation

jij

ij

iij TTT . .

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

qppq

qpq

j

p

iij TTT ))((

))((

))((

qjpi

ij

pq

qjpiij

pq

TT

TT

qq

jj

pp

ii

)(

)(

iijT

2.2 Transformation rule

2.3 Cartesian components of second-order Tensors

jiij eeTT

qjpipqij llTT

Cartesian components

2.4 Tensors operations

(scalar)

(scalar)

(tensor) .

ii

ijij

ijij

ikjk

ij

TTg

ST

PST

Quotient laws

jkjkij

ijijij

iijiij

tensorsallfortensoraisST

tensorsallforscalaraisST

YXvectorsallforscalaraisYXT

S

S

,

2.5 The metric Tensor ijg

YXYXYXg iiji

ij

iig

Substituting pp

ii )(

pp

pp

iig )(

jiijgg

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

kjiijkTT

kji

ijkT ..

kji

ijkT .

tsml

kjistijk

lmTT ....

N indices N base vectros

))(())()((

))((t

g

s

fme

lk

dc

j

b

i

afgabc

destijk

lm TT

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

ii

ii

kjikjiE )(

kjiijk

(2-4-1)

ijk

i

ijkijk eg

1

3 、 general curvilinear coordinates in Euclidean 3-D

321 ,, 1 2

Suppose that general coordinates are( ); this means that the position vectors x of a point is a known function of , and ,

then the choice that is usually made for the base vectors is

For consistency with the right-handedness of the εi , the coordinates

must be numbered in such a way that

3

i

i x

0321

xxx

3-1 coordinate system and general in Euclidean 3-D

1

2

3

1x2

x3

x

1x

2x

3x

0321

xxx

With in terms of the Cartesian coordinates and the Cartesian base vectors , where , we have

iiexX ixie

zx

rx

rx

3

2

1

sin

cos

3321

3

21321

2

21321

1

])sin()cos[(

)cos()sin(])sin()cos[(

)(sin)(cos])sin()cos[(

ez

zeerer

z

X

ererzeererX

eer

zeerer

r

X

And so

As an example , consider the cylindrical coordinate

z

r

3

2

1

33

22

11 exexexX

33

212

211

)cos()sin(

)(sin)(cos

e

erer

ee

100

010

001

ijg

We have already seen that is a tensor ; it will now be shown why it is

called the metric tensor The definition , together with ,give

Note that

So that an element of arc length satisfies

i

i x

jiij

xxg

jiij

ji ddgddxx

dxdxds

2)(

ijji gi

id

xdx

3-2 metric tensor and jacobian

ijg

ds

The jacobian of the transformation relating Cartesian coordinates and curvilinear coordinates is determinant of the array and the element of volume having the vectors

As edges is

)(),).(( 33

22

11

dx

dx

dx

321321

321 )( dddddJddV

j

ix

Note that is the same as iid

xdx

iiddx )(

gJ

3-3 Transformation rule for change of coordinates

Suppose a new set general coordinates is introduced, with the understanding that the relations between and are known, at least in principle. The rule for changing to new tensor components is

i

i i

))()((.... krj

q

i

ppqr

ijk TT

ikjk

ijjiijj

ijj

ijiggandgxgg

,),(

))()((

))()((

..

....

k

n

nuru

q

m

tm

jt

p

l

sl

ispqr

kuru

tq

jt

sp

ispqr

ijk

xxg

xxg

xxgT

xxg

xxg

xxgTT

))()((.... k

r

q

j

p

i

pqr

ijk TT

321 ,,

4-1 gradient of a scalar

If f is a scalar function, then

But grad ; hence

An alternative way to conclude that is a vector is to note

that is a scalar for all recall that is a vector ,and

invoke the appropriate quotient law.

ijji

i

ji

xe

x

fx

x

ff

)(

jj exfff

iif

f

)(

if

jjd

fdf

jd

4 、 tensor calculus

is the convariant component of i

f

thi f

ii

ff

jd

4-2 Derivative of a vector ; christoffel` symbol; covariant derivative

Consider the partial derivative of a vector F. with F = ,

we have

write

the contravriant component of the derivative with respect to of the base vector. Note that

jii

ij

i

jF

FF

jF

jik

kkijj

i

i

j

jiji x

2

iiF

jthk

christoffel system of the second kind

kji

kij

We can now write

iiki

kjjF

FF

)(

Introduce the notation

iijj

FF

,This means that ----- called the convariant derivative of --- is definded as the contravariant component of the vector comparing (4-2-1) and (4-2-2) then gives us the formula

ijF,

iFthi j

F

ikj

kj

iij F

FF

,

iji

jj

jdFd

FdF

)( ,

Although is not necessarily a tensor, is one , forj

iF

ijF,

(4-2-1)

(4-2-2)

The covariant derivative of writing as , is defined as the convariant component of ; hence

jiF ,

kjkiji FgF ,,

iFthi

iiF

)(, j

k

ki

j

ij

jijF

FF

F

kii

k

kijl

lij

kjik

j

k

i

)(

ikijj

k

kijkj

iji F

FF

,

A direct calculation of is more instructive; with F= ,we have

Now , whence

And therefore

consequently

And while this be the same as (4-2-3) it shows the explicit addition to needed to provide the covariant derivative of

jF

jiF ,

(4-2-3)

iFjiF

Other notations are common for convariant derivatiives; they are, in approximate order of popularity

ijijjiji FFDFF ; ;

Although is not a third-order tensor, the superscript can nevertheless be lowered by means of the operation

kij

ji

pj

ik

kpkijkp gpijg

],[

and the resultant quantity , denoted by , is the Christoffel symbol of the first kind. The following relations are easily verified

],[ pij

jiki

jkj

ik

xxkij

2

],[

],[],[)( ijkjikg

k

jijk

ijikk

ij

}{2

1],[

ki

jk

jikp

ijkp

gggkijg

(4-2-4)

],[

]},[],{[2

1

]},[],[],[],[],[],{[2

1

}{2

1

kij

kjikij

jikijkkjijkikijikj

gggki

jk

jik

[Prove] ：

4-3 covariant derivatives of Nth –order Tensors

Let us work out the formula for the covariant derivatives of . write ijkA..

kji

ijkAA ..

By definition

)( ..

,..

kji

ijkp

kji

ijpkp

A

AA

This leads directly to the formula

rpk

ijr

jrp

irk

irp

rjkp

ijkij

pk AAAA

A

........

,..

4-4 divergence of a vector A useful formula for

will be developed for general coordinate systems. We have

But, by determinant theory

Hence

And therefore

iip

pi

iii FFFdivFF

)(,

gg

2

1

]g

-gg

[g2

1

s][ip,

pisis

s

ip

i

ps

iisis

isiip g

pisis

p

ggg

g

ppiip

g

g

g

g

)(1

2

1

)(1

gFg

F ii

4-5 Riemann-Christoffel Tensor

Since the order of differentiation in repeated partial differentiation of Cartesian tensor is irrelevant, it follows that the indices in repeated covariant differentiation of general tensors in Euclidean 3-D may also be interchanged at will. Thus, identities like

jiij ,, and

jikijk ff ,, Eq (4-5-1) is easily verified directly, since

pijpjiij

,

2

,

(4-5-1)

However, the assertion of (4-5-1) in Euclidean 3-D leads to some nontrivial information. By direct calculation it can be shown that

ppkijjikijk fRff .,,

rki

prj

rkj

prij

pki

i

pkjp

kijR

.

With help of (4-2-4) it can be shown that , the Riemann-christoffel tensor, is given by

pkilR

][][2

12222

mkj

rpi

mki

rpjrmip

kj

jk

pi

jpki

ik

pjpkil g

ggggR

But since the left-hand side of vanishes for all vectors , it follows that kf

0pkijR

Although (4-5-2) represents 81equations, most of them are either identities or redundant, since . Only six distinct nontrivial conditions are specified by (4-5-2), and they may be written as

(4-5-2)

031312331232312311223 RRRRRRpkij

ijpkkpijpkjipkij RRRR

(4-5-3)

[Note]

prki

prj

rkj

prjj

pki

i

pkj

jikijk fff )(,,

[prove]

prki

prj

rkj

prij

pki

i

pkj

pi

pkj

rrpj

pjk

pjkpj

pjk

rrpi

pkjjikijk

krpji

pjir

rpj

pkjj

ppkii

ppkji

pjk

jik

jik

kppijp

kpijr

rpi

pkji

ppkjj

ppkipj

pki

ijk

ijk

kppkp

jirrpjj

ppkjp

pkjj

kijik

kprkr

ijrkpi

ppkjp

pkii

ki

pijpi

pkji

ikijk

f

ffffff

fffff

f

ff

fff

ff

f

ff

ff

ff

f

ff

ff

ff

ff

f

)(

)()()(

)()()(

,,

,2

,

2

,

,

,,

Since is antisymmetrical in i and j as well as in p and k, no information is lost if (4-5-2) is multiplied by . Consequently , a set of six equations equivalent to (4-5-3) is given neatly by

pkijRtijspk

0stSWhere is the symmetrical, second-order tensor stS

pkijtijspkst RS

4

1

The tensor is related simply to the Ricci tensor ijS

ppijijij SgSR

pijpij RR .

So that (4-5-3) is also equivalent to the assertion 0ijR

The familiar divergence theorem relating integrals over a volume V and its boundary surface S can obvious be written in tensor notation as

Where Ni is the unit outward normal vector to S . Similar stokes ， theorem

for integrals over a surface S and its boundary line C is just

Where tk is the unit tangent vector to C , and the usual handedness rules apply for direction of Ni and ti

s

ii

v

ii dsNfdvf ,

c k

k

s

ijkijk dstfdsNf ,

4-6 Integral Relations