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 3,1.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.黄克智，薛明德，陸明萬，張量分析，北京清華大學出版社
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
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 Vector in euclidean 3-D 1-1 Orthonormal base vector: Let (e1,e2,e3)be a right-handed ,e2 set of three mutually perpendicular vector of unit magnitude
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
e(i=1,2,3)may be used to define the kronecker delta and the permutation symbol eik by means of the equations ☒无法显示该图片 erej=y and eiejxek=ejk
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 =
Ex1. A.D A-E A.F (A×B·C(D×E·F)=B.D B.E B.F C.D C.E C.F [prove] (A×B·C)=(C.A×B)= B B2 B = (D×E·F)=(F·D×E)= D 骨 会 E马E 会 D A·E A·F
(A BC)(D E F) = C D C E C F B D B E B F A D A E A F [prove] ( ) ( ) ( ) ( ) 3 3 3 2 2 2 1 1 1 1 2 3 1 2 3 1 2 3 3 3 3 2 2 2 1 1 1 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 C D C E C F B D B E B F A D A E A F D E F D E F D E F C C C B B B A A A D E F D E F D E F F F F E E E D D D E E E D D D F F F D E F F D E C C C B B B A A A B B B A A A C C C A B C C A B = = = = = = = = Ex1
Thusδ，=1 for i=j;e123=e312=e231=1e132=e321=e213=-1; All other ek=0.in turn,a pair of ek is related to a determinate of by erer eres erer Bir Bis Bn exers=(e;xej"erxerxes"e)=ej"er ejes ej"erBir 8js 8n (1-1-1) ek'er ekes eker ir Bis Snt By setting r=i we recover the e-6 relation eke=dDa-δnds
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)
1-2 Cartesian component of vectors transformation rule rei 元=1,2,3 ☒无法显示该图片， Rotated set of orthonormal base vector ei(i=1.2.3)is introduced.the corresponding new components F,of F may be computed in terms of the old ones by writing -F.ej=Feiej We have transformation rule F =lF Here i eiei
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 Fi ( 1,2,3) i ei = i =1,2,3 We have transformation rule Here j j j i i F F = = F e e e j ij i F l F = j ij i l e e F i i = F e 1-2 Cartesian component of vectors transformation rule
These direction cosines satisfy the useful relations p'p==δi (1-2-1) [prove] e,e=kk·mm lik jmkSm=lik jk=y
These direction cosines satisfy the useful relations ip jp pi pj ij l 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 =E2,E3 be any three noncoplanar vector that play the roles of general base vectors
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
6:×8j=8 (metric tensor) and eil 22 B *e*k=张=i1无i2无3 (permutation tensor) ejl jj From(1-1-1)the general vector identity can be established e,×E,8,×8 e,×e, gir 8is 8u (E,×8,×EkE,×8,×8,)=8,×8,8,×e, 8,×ε， = gis 8n (倒 8 Eis gk
and ij ε = g i j ×ε ε ε ε i1 i2 i3 ε ε ε ε ε ε ε i j k ijk j1 j2 j3 ε ε ε j1 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 kr ks kt jr js jt ir is it g g g g g g g g g = (* )