Matrix and Tensor Calculus: With Applications to Mechanics, Elasticity and Aeronautics
This volume offers a working knowledge of the fundamentals of matrix and tensor calculus that can be applied to a variety of fields, particularly scientific aeronautical engineering. Mathematicians, physicists, and meteorologists as well as engineers will benefit from its skillful combination of mathematical statements and immediate practical applications. 1947 edition.
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This volume offers a working knowledge of the fundamentals of matrix and tensor calculus. Relevant to several fields, particularly aeronautical engineering, the text skillfully combines mathematical statements with practical applications. 1947 edition.
Matrix Calculus and Its ApplicationsAlgebraic PreliminariesIntroduction 1Definitions and notations 1Elementary operations on matrices 2Algebraic Preliminaries (Continued)Inverse of a matrix and the solution of linear equations 8Multiplication of matrices by numbers, and matric polynomials 11Characteristic equation of a matrix and the Cayley-Hamilton theorem 12Differential and Integral Calculus of MatricesPower series in matrices 15Differentiation and integration depending on a numerical variable 16Differential and Integral Calculus of Matrices (Continued)Systems of linear differential equations with constant coefficients 20Systems of linear differential equations with variable coefficients 21Matrix Methods in Problems of Small OscillationsDifferential equations of motion 24Illustrative example 26Matrix Methods in Problems of Small Oscillations (Continued)Calculation of frequencies and amplitudes 28Matrix Methods in the Mathematical Theory of Aircraft Flutter 32Matrix Methods in Elastic Deformation Theory 38Tensor Calculus and Its ApplicationsSpace Line Element in Curvilinear CoordinatesIntroductory remarks 42Notation and summationconvention 42Euclidean metric tensor 44Vector Fields, Tensor Fields, and Euclidean Christoffel SymbolsThe strain tensor 48Scalars, contravariant vectors, and covariant vectors 49Tensor fields of rank two 50Euclidean Christoffel symbols 53Tensor AnalysisCovariant differentiation of vector fields 56Tensor fields of rank r = p + q, contravariant of rank p and covariant of rank p 57Properties of tensor fields 59Laplace Equation, Wave Equation, and Poisson Equation in Curvilinear CoordinatesSome further concepts and remarks on the tensor calculus 60Laplace's equation 62Laplace's equation for vector fields 65Wave equation 65Poisson's equation 66Some Elementary Applications of the Tensor Calculus to HydrodynamicsNavier-Stokes differential equations for the motion of a viscous fluid 69Multiple-point tensor fields 71A two-point correlation tensor field in turbulence 73Applications of the Tensor Calculus to Elasticity TheoryFinite deformation theory of elastic media 75Strain tensors in rectangular coordinates 77Change in volume under elastic deformation 79Homogeneous and Isotropic Strains, Strain Invariants, and Variation of Strain TensorStrain invariants 82Homogeneous and isotropic strains 83A fundamental theorem on homogeneous strains 84Variation of the strain tensor 86Stress Tensor, Elastic Potential, and Stress-Strain RelationsStress tensor 89Elastic potential 91Stress-strain relations for an isotropic medium 93Tensor Calculus in Riemannian Spaces and the Fundamentals of Classical MechanicsMultidimensional Euclidean spaces 95Riemannian geometry 96Curved surfaces as examples of Riemannian spaces 98The Riemann-Christoffel curvature tensor 99Geodesics 100Equations of motion of a dynamical system with n degrees of freedom 101Applications of the Tensor Calculus to Boundary-Layer TheoryIncompressible and compressible fluids 103Boundary-layer equations for the steady motion of a homogeneous incompressible fluid 104Notes on Part I 111Notes on Part II 114References for Part I 124References for Part II 125Index 129
