APPLICATIONS OF TENSOR ANALYSIS

Transcription

1 APPLICATIONS OF TENSOR ANALYSIS (formerly titled: Applications of the Absolute Differential Calculus) by A J McCONNELL Dover Publications, Inc, Neiv York

2 CONTENTS PART I ALGEBRAIC PRELIMINARIES/ CHAPTER I NOTATION AND DEFINITIONS 1 The indioial notation * 1 2 The summation convention Addition, multiplication, and contraction of systems 5 4 Symmetric and skew-symmetric systems 6 5 The skew-symmetric three-systems and the Kroneoker deltas - 7 DETERMINANTS 6 The determinant formed by a double system oj The cofactors of the elements in a determinant Linear equations Corresponding formula for the system a mn Positive definite quadratic forma The determinantal equation 16 CHAPTER H TENSOR ANALYSIS 1 Linear transformations IQ 2 Invariants, contravariant and covariant vectors 20 3 Tensors of any order ' Addition, multiplication and contraction of tensors The quotient law of tensors - -» Relative or weighted tensors General functional transformations Tensors with respect to the general functional transformation r - 32 PART II ALGEBRAIC GEOMETRY CHAPTER m RECTILINEAR COORDINATES 1 Coordinates and tensors Contravariant veotors and displacements The unit points and the geometrical interpretation of rectilinear coordinates vii

3 viii CONTENTS 7-4 The distance between two points and the fundamental double' sor The e-systems 5 The angle between two directions; orthogonality - - "-; 6 Associated tensors - - i, 7 Scalar and vector products of vectors "* 8 Areas and volumes, f CHAPTER IV * THE PLANE \ 1/ The equation of a plane " j 2 The perpendicular distance from a point to a plane \i 3 The intersection of two planes '- \ 4 The intersection of three planes 5 Plane coordinates - - ^ 6 Systems of planes ', : 7 The equation of a point - *'' CHAPTER V THE STRAIGHT LINE 1 The point equations of the straight line * ~- 2 The relations of two straight lines "- 3 The six coordinates of ft straight line, ' 4 The plane equations of a straight line - - T j CHAPTER VI THE QUADRIC CONE AND THE CONIC! 1 The equation of a quadrio cone 'I 2 The equation of a conic - -» v - 3 The tangent plane'to a cone \ 4 Poles and polar planes with respect to a cono 6 The canonical equation of a cone The principal axes of a cone 7 The classification of cones ' CHAPTER VII SYSTEMS OF CONES AND CONICS? of 1 The equation of a system of cones with a common vertex - 2 The common polar directions of a family of cones - 3 The canonical forms of the equation 4 The theory of elementary divisors a family of cones - 5 Analytical discrimination of the cases - - -

4 CONTENTS a CHAPTER VHI CENTRAL QUADRICS 1 The point equation of a central quadric The tangential equation of a central quadric - ' Canonical form of the equation of a quadric Principal axes Classification of the central quadrics Confocal quadrics '110 CHAPTER IX THE GENERAL QUADRIC 1 The general equation of a quadric, The centre , The reduction of the equation of a quadrio ' CHAPTER X AFFINE TRANSFORMATIONS 1 Affine transformations The quadric of a transformation Pure strain Rigid body displacements Infinitesimal deformations 126 PART III DIFFERENTIAL GEOMETRY CHAPTER XI CURVILINEAR COORDINATES 1 General coordinate systems Tensor-fields - - ^ The line-element and the metric tensor The e-systems The angle between two directions 136 CHAPTER XH COVARIANT DIFFERENTIATION 1 A parallel field of vectors The Christoffel symbols The intrinsic and covariant derivation of vectors The intrinsic and covariant derivatives of tensors Conservation of the rules of the ordinary differential calculus Ricci's lemma The divergence and curl of a vector The Laplacian The Riemann-Christoffel tensor The Lame relations - 152

5 * CONTENTS - CHAPTER XIII CURVES IN SPACE 1 The tangent vector to a curve Normal vectors The principal normal and binormal The Frenet fsrmulae - - " Parallel vectors along* a curve The straight line "CHAPTER XIV INTRINSIC GEOMETRY OF A SURFACE 1 Curvilinear coordinates on a surface The conventions regarding Greek indices Surface tensors The element of length" and the metrio tensor Directions on a surface Angle between two directions The equations of a geodesic The transformation of the Christoffel symbols Geodesic coordinates /-"' Parallelism with respect to a surface Intrinsic and covariant differentiation of surface tensors The Riemann-Christoffel tensor The Gaussian ourvature of a surface _ The geodesic curvature of a curve on a surface Beltrami's differential parameters ~ Green's theorem on a surface ' [' CHAPTER XV THE FUNDAMENTAL FORMULAE OF A SURFACE 1 Notation - - _ - - : - ; The tangent vectors^ to a surface The first groundform of a surface The normal vector to the; surface ~- - ' jgg 5 The tensor derivation'of-tensors Gauss's formulae 'The second groundform of a surface Weingarten's formulae The'third groundform of a surface The equations of Gauss and Codazzi - - ' 203 ; CHAPTER XVI CURVES ON A SURFACE 1 The equations of a curve on a surface Meusnier's theorem " " ", ' " " The principal ourvatures Gauss s theorem The lines of curvature, ll 5 The asymptotic lines Enneper's formula - _ ' * 6 The geodesic torsion of a'curve on a surface 2 U

6 CONTENTS xi PART IV APPLIED MATHEMATICS ' CHAPTER XVH DYNAMICS OF A PARTICLE 1 The equations of motion W o r k and energy Lagrange's equations of motion Particle on a curve 223,4 Particle on a surface The principle of least action Trajectories as geodesies 228 CHAPTER DYNAMICS OF RIGID BODIES SECTION A RECTILINEAR COORDINATES 1 Moments of Inertia The equations of motion Moving axes Euler's equations SECTION B THE GEOMETRY OF DYNAMICS 4 Generalised coordinates of a dynamical system The equations of motion in generalised coordinates The manifold of configurations The kinematics! line-element The dynamical trajectories of the manifold of configurations The principle of stationary action The action line-element CHAPTER XTX ELECTRICITY AND MAGNETISM 1 Green's theorem Stokes's theorem The electrostatic field Dielectrics The magnetostatic field The electromagnetic equations CHAPTER XX MECHANICS OF CONTINUOUS MEDIA 1 Infinitesimal strain Analysis of stress Equations of motion for a perfect fluid The equations of elasticity The motion of a viscous fluid - 280

7 xii CONTENTS CHAPTER XXI THE SPECIAL THEORY OF RELATIVITY 1 The four-dimensional manifold Generalised coordinates in space-time r, The principle of special relativity The interval and the fundamental quadratic form - " Local coordinate systems and their transformations Relativistic dynamics of a particle Dynamics of a continuous medium The electromagnetic equations APPENDIX ORTHOGONAL OTEVffilNEAR COORDINATES IN MATHEMATICAL PHYSICS 1 The classical notation The physical components of vectors and tensors Dynamics Electricity Elasticity Hydrodynamics 309 BIBLIOGRAPHY INDEX - 31fi