东南大学：《固体力学基础》课程教学课件（英文讲稿）01 Mathematical Preliminary

Kronecker Delta: Kronecker delta: a useful special symbol commonly usedin index notational schemes·Symmetric[100[ 1,if i=j(no sum)001Qiio,ifij001 .Replacement propertyOj = jid=3,d=1Qjqj = ai, Oijai = ajOijajk=aik,Ojkaik=aijOjaj=ai,Ojdj=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(+l, if ijk is an even permutation of 1,2,3-1, if ijk is an odd permutation of 1,2,3SikO, otherwise8123= 8231 = 8312 =1, 8321 = 8132= 8213=-1, 8112= 8131 = 8222=. ..=0. Cross product of two vectorsa×b=(a,e,)x(b,e,)=a,b, (e, xe,)=a,b, (kek)= Sab,ek? Scalar triple product of three vectorsa×b.c=Sa,b,ex*Cmem=Ska,b,cm(ex-em)= Ska,b,cmOm = Syka,b,C12

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) Symbo): Determinant of a matrix (easily verifiable)aa12a13det[a, ] =ai|==a21a22a23=Sjka;a2j3k=Sjkaiaj2ak3a3a32a33: By the definition of Levi-Civita and note the followingSijkGrstairajsak=Sijkaaj2ak3-Sijkaiaj3ak2+Sijkai2aj3ak1Sikai2ajiak3+Sijki3ajiak28jkai3aj2aklikrstairajsau=6detajSjkaijajmakm=Smndet.浙rst13

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? &- property8.8.S.1i2i3SsXuk =e, e, xex =(8pep)(8ea)x(er) G一SiikiikJilj3OkSk3lS8m pgr = [A][B] = [4][B] =[A][B]].182808.00S.oSSSSplrlimimpnqm2rm80SSSSSS8;2SS=p292rmDgn8sSSS.SLok1k2SoS.k3p393r3pmcmrmkmgnK88.8.S8.Sikiriip19ips8SSSSS=S一A8CikjrJp1gJp19ip1iqPASSOkSkdkrkpkqRP14

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]0720443.b. =dii122[o]1Determine the following quantities, and indicate whether theyare a scalar, vector or matrix.au,a,ay,a,ajik,a,b,a,bb,,b,b,b,b,,asymm.,aa.antiuiiSolution:at = ai +a22 +a33 = 7 (scalar)a,a =aaii+a12a12+a13a13+a21a21 +a22a22+a23a23+a3ia31 +a32a32 +a33a33=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 bb bb bb 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