东南大学：《弹性力学》课程教学课件（英文讲稿）02 Mathematical Preliminaries

Kronecker Delta: Kronecker delta: a useful special symbol commonly usedin index notational schemes?Symmetric010l,if i=j (no sum)001..{o,ifij001:Replacement propertyij =Oji = 3,Qi= 1dijdj = ai, Oijai = ajOijk=aik,Ojkaik=aijOijaij = ai, OjOj = 311

Kronecker Delta • Kronecker delta: a useful special symbol commonly used in index notational schemes • Symmetric • Replacement property 11

Levi-Civita (Permutation) Symbol. Antisymmetric w.r.t. any pair of its indices(+1,if ijkisanevenpermutationof1,2,3-1,ifijkisanoddpermutationof1,2,3iik0.otherwise8123 = 8231 = 8312= 1,8321 = 8132 = 8213 =-1, 8112 = 8131 = 8222=.:.= 0.. Cross product of two vectorsaxb=(a,e,)x(b,e,)=ab, (e, xe,)=a,b,(se)=8ma,b,e. Scalar triple product of three vectorsaxb.c=Suab,ex*Cmem=ka,b,cm(exem)=Ska,b,cmOkm=Sua,b,Ck12

Levi-Civita (Permutation) Symbol • Antisymmetric w.r.t. any pair of its indices • Cross product of two vectors 12 a b e e e e e e        a b a b a b a b i i j j i j i j i j ijk k ijk i j k         • Scalar triple product of three vectors   ijk i j k m m ijk i j m k m ijk i j m km ijk i j k a b c e e e e              a b c a b c a b c a b c

Levi-Civita (Permutation) Symbol: Determinant of a matrix (easily verifiableai1a12a13detai=ai|=a21a22a23=Sikai,a2,a3k=Eikaiai2aka3a32a33: By the definition of Levi-Civita and note the followingSijkrstairajsak=Sjkaaj2ak3-Sjkanaj3ak2+Sjkai2aj3aklSjkai2ajiak3+Sjkai3ajiak2-Sijki3aj2akljkaiajmakn=Cimndetsraiaiak=6det13

13 • Determinant of a matrix (easily verifiable) 11 12 13 21 22 23 1 2 3 1 2 3 3 32 33 det ij ij ijk i j k ijk i j k a a a a a a a a a a a a a a a a a           1 2 3 1 3 2 2 3 1 2 1 3 3 1 2 3 2 1 6det , det ijk rst ir js kt ijk i j k ijk i j k ijk i j k ijk i j k ijk i j k ijk i j k ijk rst ir js kt ij ijk il jm kn lmn ij a a a a a a a a a a a a a a a a a a a a a a a a a a a a a                              Levi-Civita (Permutation) Symbol • By the definition of Levi-Civita and note the following

Levi-Civita (Permutation) Symbol ε- propertyo68138Suk =e,-e, ×ex =(Spe,).(Sjge.)×(13SkSk3ux pgr = [4][B] = [4][B]| = [A][B]S8.[o08S...88...SSq15plDngn8.8SSilS i3SS8..S3-=P2r2am92Pm8S8LokiS0k3SSSk2SSp393r3kmkmpmqmrm8.88.8.d8.idipiq11ips808=8.8.-8.8SSLUGik8SjkpakTjpjqppq!1010mpkr0kpSO kaOkkkq14

14 • ε-δ property Levi-Civita (Permutation) Symbol                  1 2 3 1 2 3 1 2 3 1 2 3 1 1 1 1 2 3 2 2 2 1 2 3 3 3 3 i i i ijk i j k ip p jq q kr r ijk j j j k k k T T ijk pqr i i i p q r im pm im qm im rm j j j p q r jm pm jm qm jm rm k k k p q r km A B A B A B                                                                            e e e e e e pm km qm km rm ip iq ir ip iq i ijk pqr jp jq jr ijk pq jp jq j ip jq iq jp kp kq kr k k k k k p q kk                                     

Sample Problem: The matrix ai, and vector b, are specified by[2]0[124043b, =aij22110Determine the following quantities, and indicate whether theyare a scalar, vector or matrix.ai,a,aj ,a,ajk,a,b,,a,b,b,,bb,b,b,,asymm.,aanti. Solution:ai =ai +a22 +a33 =7 (scalar)a,ay=a1ia1+a12a12+a1313+a21a21+a2222+a2323+a3ia31+a32a32+a33a3=1+4+0+0+16+9+4+1+4=39(scalar)15

1 2 0 2 0 4 3 , 4 2 1 2 0 ij i a b               symm. anti. , , , , , , , , ii ij ij ij jk ij j ij i j i i i j a a a a a a b a b b b b b b a a • The matrix aij and vector bi are specified by • Determine the following quantities, and indicate whether they are a scalar, vector or matrix. Sample Problem • Solution: 11 22 33 11 11 12 12 13 13 21 21 22 22 23 23 31 31 32 32 33 33 7 (scalar) 1 4 0 0 16 9 4 1 4 39 (scalar) ii ij ij a a a a a a a a a a a a a a a a a a a a a a a a                        15