Mathematical Preliminaries
Mathematical Preliminaries
OutlineScalar, Vector and MatrixIndicial Notation and Summation ConventionKronecker DeltaLevi-Civita symbol Coordinate TransformationTensorPrincipal Values and Directions Tensor Algebra Tensor CalculusIntegral Theorems: Cylindrical and Spherical Coordinates2
Outline • Scalar, Vector and Matrix • Indicial Notation and Summation Convention • Kronecker Delta • Levi-Civita symbol • Coordinate Transformation • Tensor • Principal Values and Directions • Tensor Algebra • Tensor Calculus • Integral Theorems • Cylindrical and Spherical Coordinates 2
Scalar: Scalar: representing a single magnitude at eachpointin spaceMaterial densityMassEnergyDistance VolumeAreaTemperature3
Scalar • Scalar: representing a single magnitude at each point in space • Material density • Mass • Energy • Distance • Volume • Area • Temperature 3
Vector: Vector: representing physical quantities thathave both magnitude and directionElectric field.Force: Displacement of material pointsVelocity: Rotation of material points: Force couple (Moment)4
Vector • Vector: representing physical quantities that have both magnitude and direction • Electric field • Force • Displacement of material points • Velocity • Rotation of material points • Force couple (Moment) 4
Matrix (Array): Matrix: a rectangular array of numbersm-by-nmatrixaijncolumnschangesmrowsa1.1a1,2a1.3iuhangena2.1a2.2a2.3a3.1a3,2a3.3 Array: 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二j=e23=ae +aze2 +a,e -Za,e,i-1: Indicial notation: a shorthand scheme whereby awhole set of components is represented by asingle symbol with subscriptsx,e[a,[aia13a12X2aza21a22a23x, =e=e2,aj=a.e3asasa32X3a336
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
Lndicial Notation. Addition and subtraction[ai±bi]anl±bula12±b12a13±b13ai±bi=a2±b2，aij±bij=a21 ±b21a22 ±b22a23 ±b23La3±b3]a31 ±b31a32 ±b3233 ±b33. Scalar multiplication[2a][2a112a122a13lai =2a2，2aij=2a212a222a232a32a312a322a33Outer multiplicationaib3[aibi aib2a2b1a2b2 a2b3a;b; =La3b a3b2 a3b37
Indicial Notation • Addition and subtraction • Scalar multiplication • Outer multiplication 7
Lndicial Notation: Commutative, associative and distributive lawsai+bi=bi+aia;(bjkci) = (a;bjk)ciaijbk = bkaijai;(bk+Ck)=aijbk+aijCkai十（bi十ci)=（ai十b)+Ci: Equality of two symbolsa, = b,au = ba12 = b2ai3 = br3α23 = b23α22 = b2a, = b, =αz =b2 ,aj, =bi山α21 =b21La31 = bs1ag2 = b32a33 = b33-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 8
Summation Convention: Summation convention: if a subscript appears twice inthe same term, then summation over that subscript fromone to three is implied3V33aii =ai=a11+a22+a33ayxx, =ZZa,xx,i=13i=1 j=1aibj =Zaijbj=aibi+ai2b2+ai3b3j=1a=aij=aki. 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 • 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 ai, bycontraction on i and j? Outer multiplication → contraction → inner product(contraction on i) a,b,aa..b(contraction on jk) ax buijSymmetric vs. antisymmetric (skewsymmetric) w.r.t. twoindices, i.e. m and naij..m..... = aij....m...aij...m...n...k=-ajj...n...m...k. Useful identity: aij 十=(aj一aji)210
Contraction and Symmetry • Contraction: for example, aii is obtained from aij by contraction on i and j • Outer multiplication → contraction → inner product • Symmetric vs. antisymmetric (skewsymmetric) w.r.t. two indices, i.e. m and n • Useful identity: 10