Mathematical Preliminarymi@se.edu.cn
Mathematical Preliminary
Outline·Scalar，Vector and Matrix（标量、矢量、矩阵） Indicial Notation and Summation Convention （指标记法与求和约定)KroneckerDelta（克罗内克）Levi-Civita symbol（轮换记号）CoordinateTransformation（坐标变换）Tensor（张量）PrincipalValuesandDirections（特征值与特征向量）TensorAlgebra（张量代数）Tensor Calculus（张量微分）Integral Theorems（积分定理）·Cylindrical and Spherical Coordinates（柱与球坐标）2
Outline • Scalar, Vector and Matrix（标量、矢量、矩阵） • Indicial Notation and Summation Convention（指标记法 与求和约定） • Kronecker Delta（克罗内克δ） • Levi-Civita symbol（轮换记号） • Coordinate Transformation（坐标变换） • Tensor（张量） • Principal Values and Directions（特征值与特征向量） • Tensor Algebra（张量代数） • Tensor Calculus（张量微分） • Integral Theorems（积分定理） • Cylindrical and Spherical Coordinates（柱与球坐标） 2
Scalar: Scalar: representing a single magnitude at eachpoint in spaceExamples:. Material densityYoung's modulusPoisson's ratioShear modulusTemperature3
Scalar • Scalar: representing a single magnitude at each point in space • Examples: • Material density • Young’s modulus • Poisson’s ratio • Shear modulus • Temperature 3
Vector: Vector: representing physical quantities thathave both magnitude and directionExamples:.Force: Displacement of material pointsRotation of material points4
Vector • Vector: representing physical quantities that have both magnitude and direction • Examples: • Force • Displacement of material points • Rotation of material points 4
Matrix (Array). Matrix: a rectangular array of numbersm-by-nmatrixaijncolumnschangesmrowsai,1a1.2a1.3iuhangesa2.1a2.2a2.3a3.1a3.2a3.3Array: a data structure in which similar elementsof data are arranged in a table5
Matrix (Array) • Matrix: a rectangular array of numbers • Array: a data structure in which similar elements of data are arranged in a table 5
Indicial Notation=X3: Orthogonal unit vectors: ix j = kVector decomposition:a=OP=(a.i)i+(a·j)j+(a·k)k=e23Za,e,=ae, +a,ez +a,e, =-i=1EX Indicial notation: a shorthand scheme whereby awhole set of components is represented by asingle symbol with subscriptsxiaeiaa12a13a22a2a21a23X.X2e2a;x.e3asasa32a336
3 1 2 3 1 i i i a a a a 1 2 3 a OP a i i a j j a k k e e e e j＝e2 P y＝x2 e1＝i k＝e3 x ＝x1 O a z ＝x3 • Orthogonal unit vectors: • Vector decomposition: i j k • Indicial notation: a shorthand scheme whereby a whole set of components is represented by a single symbol with subscripts 1 1 1 11 12 13 2 2 2 21 22 23 3 3 3 3 32 33 , , , i i i ij x a a a a x x a a a a a a x a a a a e e e e Indicial Notation 6
Indicial Notation. Addition and subtraction[ai ±bi[αi ± bia13 ±b13a12±b12ai±b; =a21 ± b21a22 ± b22 a23 ±b23a2 ± b2，aij±bij=La3 ± b3 a31 ± b31a32±b32a33 ±b33? Scalar multiplication[a[2a112a122a13lai =Na2,Naij =1a22Za23a21a312a3La32Za33. Outer multiplication[abiab2 ab3α2b1 a2b2 azb3a;b; =La3biasb2a3b3.7
Indicial Notation • Addition and subtraction • Scalar multiplication • Outer multiplication 7
Indicial Notation: Commutative, associative and distributive lawsai+bi=bi+aia;(bikc) = (aibjk)claijbk = bkaijai;(bk + Ck) = aijbk +aijCkai+(bi+ci)=(ai+bi)十CiEquality of two symbols[a = b,ai2 = b2a13 = br3au =bu2α23 =b23a, = b, =αz = b2,aj =b, α21 = b21 α22 = b22a33 = b3[a31 = b31 a32 = b32[a, = b,. Avoida,=b8
Indicial Notation • Commutative, associative and distributive laws • Equality of two symbols 1 1 11 11 12 12 13 13 2 2 21 21 22 22 23 23 3 3 31 31 32 32 33 33 , i i ij ij a b a b a b a b a b a b a b a b a b a b a b a b a b a b • Avoid , i j ij kl a b a b 8
Summation Convention: Summation convention: if a subscript appears twice inthe same term, then summation over that subscript fromone to three is implied333aii =ai=a11+a22+a33ayx,x, =ZZi-1a,xxj3i=1 j=laijb;=Zaijbj = aiibi + aizb2 +ai3b3j-1αi=αj=αk In a single term, no index can appear more than twice. Dummy (repeated) indices vs. free (distinct) indices. Among terms, index property must match.9
Summation Convention • Summation convention: if a subscript appears twice in the same term, then summation over that subscript from one to three is implied ii jj kk a a a • In a single term, no index can appear more than twice. • Dummy (repeated) indices vs. free (distinct) indices • Among terms, index property must match. 3 3 1 1 ij i j ij i j i j a x x a x x 9
Contraction and Symmetry. Contraction: for example, ai, is obtained from a., bycontraction on i and j? Outer multiplication → contraction → inner producta,b, (contraction on ij) a,b,a,bu (contraction on jk) aibu Symmetric vs. antisymmetric (skewsymmetric) w.r.t. twoindices, i.e. m and naij..m..... = aij....m...aij...m.....k = -aij....m...k. Useful identity: aij =(aij_(aij -aji)10
Contraction and Symmetry • Contraction: for example, aii is obtained from aij by contraction on i and j • Outer multiplication → contraction → inner product contraction on contraction on i j i i ij kl ik kl a b ij a b a b jk a b • Symmetric vs. antisymmetric (skewsymmetric) w.r.t. two indices, i.e. m and n • Useful identity: 10