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