Vector Analysis and Cartesian Tensors

✍ Scribed by Bourne, Donald Edward

• Publisher: Chapman and Hall/CRC
• Year: 2018
• Tongue: English
• Leaves: 315
• Edition: 3rd
• Category: Library

✦ Table of Contents

Content: Cover
Title Page
Copyright Page
Table of Contents
Preface
Preface to second edition
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
2: Scalar and vector algebra
2.1 Scalars
2.2 Vectors: basic notions
2.3 Multiplication of a vector by a scalar 2.4 Addition and subtraction of vectors2.5 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
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 kinematics
4: Scalar and vector fields 4.15 Method of Steepest Descent5: 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
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
7: Applications in potential theory
7.1 Connectivity
7.2 The scalar potential 7.3 The vector potential7.4 Poisson's equation
7.5 Poisson's equation in vector form
7.6 Helmholtz's theorem
7.7 Solid angles
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
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