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