1, a.j.mcconnell, applications of tensor analysis, dover publications,inc, new york 2, garl...
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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 、 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
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