Vector Analysis and Cartesian Tensors

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Front Cover; Vector Analysis and Cartesian Tensors; Copyright Page; Table of Contents; Dedication; Preface; Chapter 1. Rectangular Cartesian Coordinates and Rotation of Axes; 1.1 Rectangular cartesian coordinates; 1.2 Direction cosines and direction ratios; 1.3 Angles between lines through the origin; 1.4 The orthogonal projection of one line on another; 1.5 Rotation of axes; 1.6 The summation convention and its use; 1.7 Invariance with respect to a rotation of the axes; 1.8 Matrix notation; Chapter 2. Scalar and Vector Algebra; 2.1 Scalars; 2.2 Vectors: basic notions. 2.3 Multiplication of a vector by a scalar2.4 Addition and subtraction of vectors; 2.5 The unit vectors i, j, k; 2.6 Scalar products; 2.7 Vector products; 2.8 The triple scalar product; 2.9 The triple vector product; 2.10 Products of four vectors; 2.11 Bound vectors; Chapter 3. Vector Functions of a Real Variable. Differential Geometry of Curves; 3.1 Vector functions and their geometrical representation; 3.2 Differentiation of vectors; 3.3 Differentiation rules; 3.4 The tangent to a curve. Smooth, piecewise smooth, and simple curves; 3.5 Arc length; 3.6 Curvature and torsion. 3.7 Applications in kinematicsChapter 4. Scalar and Vector Fields; 4.1 Regions; 4.2 Functions of several variables; 4.3 Definitions of scalar and vector fields; 4.4 Gradient of a scalar field; 4.5 Properties of gradient; 4.6 The divergence and curl of a vector field; 4.7 The del-operator; 4.8 Scalar invariant operators; 4.9 Useful identities; 4.10 Cylindrical and spherical polar coordinates; 4.11 General orthogonal curvilinear coordinates; 4.12 Vector components in orthogonal curvilinear coordinates; 4.13 Expressions for grad ], div F, curl F, and 2 in orthogonal curvilinear coordinates. 4.14 Vector analysis in n-dimensional spaceChapter 5. Line, Surface, and Volume Integrals; 5.1 Line integral of a scalar field; 5.2 Line integrals of a vector field; 5.3 Repeated integrals; 5.4 Double and triple integrals; 5.5 Surfaces; 5.6 Surface integrals; 5.7 Volume integrals; Chapter 6. Integral Theorems; 6.1 Introduction; 6.2 The Divergence Theorem (Gauss's theorem); 6.3 Green's theorems; 6.4 Stokes's theorem; 6.5 Limit definitions of div F and curl F; 6.6 Geometrical and physical significance of divergence and curl; Chapter 7. Applications in Potential Theory; 7.1 Connectivity. 7.2 The scalar potential7.3 The vector potential; 7.4 Poisson's equation; 7.5 Poisson's equation in vector form; 7.6 Helmholtz's theorem; 7.7 Solid angles; Chapter 8. Cartesian Tensors; 8.1 Introduction; 8.2 Cartesian tensors: basic algebra; 8.3 Isotropic tensors; 8.4 Tensor fields; 8.5 The divergence theorem in tensor field theory; Chapter 9. Representation Theorems for Isotropic Tensor Functions; 9.1 Introduction; 9.2 Diagonalization of second order symmetrical tensors; 9.3 Invariants of second order symmetrical tensors; 9.4 Representation of isotropic vector functions