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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
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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
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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
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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
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))(( 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.
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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)
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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
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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
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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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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
(*)
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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
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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
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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
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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.
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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
( ) ( )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
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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
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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
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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)
, , ,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).
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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
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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
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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
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2.3 Cartesian components of second-order Tensors
ij i jTT e e
ij pq pi qjT T l l
Cartesian components
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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
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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
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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
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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