Matrix and Tensor Calculus: With Applications to Mechanics, Elasticity and Aeronautics Book

Matrix Calculus and Its Applications
Algebraic Preliminaries
Introduction     1
Definitions and notations     1
Elementary operations on matrices     2
Algebraic Preliminaries (Continued)
Inverse of a matrix and the solution of linear equations     8
Multiplication of matrices by numbers, and matric polynomials     11
Characteristic equation of a matrix and the Cayley-Hamilton theorem     12
Differential and Integral Calculus of Matrices
Power series in matrices     15
Differentiation and integration depending on a numerical variable     16
Differential and Integral Calculus of Matrices (Continued)
Systems of linear differential equations with constant coefficients     20
Systems of linear differential equations with variable coefficients     21
Matrix Methods in Problems of Small Oscillations
Differential equations of motion     24
Illustrative example     26
Matrix Methods in Problems of Small Oscillations (Continued)
Calculation of frequencies and amplitudes     28
Matrix Methods in the Mathematical Theory of Aircraft Flutter     32
Matrix Methods in Elastic Deformation Theory     38
Tensor Calculus and Its Applications
Space Line Element in Curvilinear Coordinates
Introductory remarks     42
Notation and summationconvention     42
Euclidean metric tensor     44
Vector Fields, Tensor Fields, and Euclidean Christoffel Symbols
The strain tensor     48
Scalars, contravariant vectors, and covariant vectors     49
Tensor fields of rank two     50
Euclidean Christoffel symbols     53
Tensor Analysis
Covariant differentiation of vector fields     56
Tensor fields of rank r = p + q, contravariant of rank p and covariant of rank p     57
Properties of tensor fields     59
Laplace Equation, Wave Equation, and Poisson Equation in Curvilinear Coordinates
Some further concepts and remarks on the tensor calculus     60
Laplace's equation     62
Laplace's equation for vector fields     65
Wave equation     65
Poisson's equation     66
Some Elementary Applications of the Tensor Calculus to Hydrodynamics
Navier-Stokes differential equations for the motion of a viscous fluid     69
Multiple-point tensor fields     71
A two-point correlation tensor field in turbulence     73
Applications of the Tensor Calculus to Elasticity Theory
Finite deformation theory of elastic media     75
Strain tensors in rectangular coordinates     77
Change in volume under elastic deformation      79
Homogeneous and Isotropic Strains, Strain Invariants, and Variation of Strain Tensor
Strain invariants     82
Homogeneous and isotropic strains     83
A fundamental theorem on homogeneous strains     84
Variation of the strain tensor     86
Stress Tensor, Elastic Potential, and Stress-Strain Relations
Stress tensor     89
Elastic potential     91
Stress-strain relations for an isotropic medium     93
Tensor Calculus in Riemannian Spaces and the Fundamentals of Classical Mechanics
Multidimensional Euclidean spaces     95
Riemannian geometry     96
Curved surfaces as examples of Riemannian spaces     98
The Riemann-Christoffel curvature tensor     99
Geodesics     100
Equations of motion of a dynamical system with n degrees of freedom     101
Applications of the Tensor Calculus to Boundary-Layer Theory
Incompressible and compressible fluids     103
Boundary-layer equations for the steady motion of a homogeneous incompressible fluid     104
Notes on Part I     111
Notes on Part II     114
References for Part I     124
References for Part II     125
Index     129