Applications of Vector Analysis and Complex Variables in Engineering

Contents
Preface
	Acknowledgemenents
Chapter 1 Vectors in Three-Dimensional Space
	1.1 Unit Vectors
	1.2 Comparison with Symbolic Notation
	1.3 The Dot Product
		1.3.1 Projection of a Vector onto a Given Direction
	1.4 The Cross Product
	1.5 Free Indices and Dummy Indices
	1.6 Base Vectors
	1.7 The Kronecker Delta
	1.8 The Levi-Civita Symbol and the Kronecker Delta
Chapter 2 Vector Fields
	2.1 Field Lines: Path Lines and Stream Lines
		2.1.1 The Lagrangian Description
		2.1.2 The Eulerian Description
Chapter 3 Fundamental Equations for Fluid Mechanics
	3.1 The Euler and Bernoulli Equations
	3.2 The Mass Balance Equation
	3.3 Rotational Flow
	3.4 The Bernoulli Equation for Irrotational Flow
		3.4.1 Conservation of Zero Rotation
		3.4.2 Transient Flow of a Compressible Inviscid Fluid with Rotation
	3.5 Objects Moving through the Fluid
		3.5.1 Euler’s Equation for Moving Coordinates; Irrotational Flow
		3.5.2 The Energy Head
	3.6 Flow with Divergence and Rotation; the General Field
	3.7 Irrotational and Divergence-Free Flow
	3.8 Three-Dimensional Flow
		3.8.1 The Point Sink
		3.8.2 The Dipole
		3.8.3 Higher-Order Singularities
		3.8.4 Other Harmonic Functions
		3.8.5 An Impermeable Sphere in a Field of Uniform Flow
		3.8.6 The Line-Sink
Chapter 4 Integral Theorems
	4.1 Introduction
	4.2 The Integral Theorem of Stokes
		4.2.1 Stokes’s Integral Theorem Applied to a Vector Potential
	4.3 The Divergence Theorem
	4.4 Green’s Integral Theorem
	4.5 Boundary Integral Equations
Chapter 5 Coordinate Transformations: Definitions of Vectors and Tensors
	5.1 Coordinate Transformation
	5.2 Definition of Vectors and Tensors
		5.2.1 Tensors of the Second Rank
		5.2.2 The Gradient Tensor
		5.2.3 Principal Directions
	5.3 Non-Cartesian Transformations with Straight Axes
		5.3.1 Invariants
		5.3.2 The Gradient
	5.4 The Stress Tensor
		5.4.1 The Normal and Shear Stress Components
		5.4.2 Principal Directions and Principal Values of the Stress Tensor in Two Dimensions.
		5.4.3 Stress Deviator
		5.4.4 Invariants
		5.4.5 The Equilibrium Conditions
	5.5 The Displacement Gradient and Strain Tensors
		5.5.1 The Displacement Gradient Tensor
		5.5.2 The Strain Tensor and the Rotation Tensor
	5.6 Integrability; The Compatibility Conditions
		5.6.1 The Gradient of a Scalar
		5.6.2 The Gradient of a Vector
	5.7 The Velocity Gradient Tensor
Chapter 6 Partial Differential Equations of the First Order
	6.1 The General Differential Equation of the First Order
		6.1.1 Advective Contaminant Transport
Chapter 7 Partial Differential Equations of the Second Order
	7.1 Types of Partial Differential Equations
	7.2 Two Real Characteristics
	7.3 Two Coinciding Characteristics
	7.4 Complex Characteristic Directions
Chapter 8 The Elliptic Case: Two Complex Characteristics
	8.1 Boundary-Value Problems
	8.2 The Helmholtz Decomposition Theorem
	8.3 Vectors and Tensors in Complex Space
		8.3.1 The Bases in Complex Space
		8.3.2 Transformation of Vectors
		8.3.3 Transformation of Tensors
		8.3.4 A Combined Kronecker Delta and Alternating Tensor: The Combination Matrix
		8.3.5 The Potential
		8.3.6 The Stream Function
		8.3.7 The General Vector Field
		8.3.8 Helmholtz’s Decomposition Theorem
		8.3.9 Areal Integration
		8.3.10 Integral Theorems
	8.4 Irrotational and Divergence-Free Vector Fields
		8.4.1 Summary
	8.5 Functions of a Single Complex Variable
	8.6 Basic Complex Functions
		8.6.1 The Taylor Series
		8.6.2 The Asymptotic Expansion
		8.6.3 The Laurent Series
	8.7 Cauchy Integrals
Chapter 9 Applications of Complex Variables
	9.1 Flow of an in-viscid fluid
		9.1.1 Uniform Flow
		9.1.2 The Point Sink
		9.1.3 The Vortex
		9.1.4 The Dipole
	9.2 Superposition of Elementary Solutions
		9.2.1 Flow Around an Impermeable Cylinder
		9.2.2 A Moving Cylinder
		9.2.3 A Rotating Stationary Cylinder in a Moving Fluid
		9.2.4 Flow with Many Cylindrical Impermeable Objects
		9.2.5 Two-Dimensional Horizontal Flow toward a Sink
		9.2.6 Rotational Flow with a Forced Vortex
	9.3 Conformal Mapping
		9.3.1 Transforming an Ellipse into a Circle
		9.3.2 An Elliptical Impermeable Object in Uniform Flow
	9.4 Groundwater flow
		9.4.1 Horizontal Confined Flow
	9.5 Linear Elasticity
		9.5.1 Basic Equations
		9.5.2 General Solution
		9.5.3 Stresses and Strains
		9.5.4 Tractions and Displacements Acting on Arbitrary Planes
		9.5.5 Stresses Acting On a Half Plane
		9.5.6 A Half-Space Loaded Along a Section
		9.5.7 Flamant’s Problem
		9.5.8 An Analytic Element for Gravity
		9.5.9 A Pressurized Crack
		9.5.10 An Analytic Element for Gravity in a Half-Space
Chapter 10 The Parabolic Case: Two Coinciding Characteristics
	10.1 The Diffusion Equation
		10.1.1 Characteristics
	10.2 Conduction of Heat in Solids
	10.3 Transient Groundwater Flow
	10.4 Consolidation of Soils
	10.5 The Laplace Transform
		10.5.1 The Unit Step Function
		10.5.2 The Dirac Delta Function
	10.6 Applications of the Laplace transform
		10.6.1 One-Dimensional Consolidation
		10.6.2 Transient Groundwater Flow
	10.7 Separation of Variables
		10.7.1 Response to a Sinusoidal Tidal Fluctuation
		10.7.2 Initial Condition
Chapter 11 The Hyperbolic Case: Two Real Characteristics
	11.1 Longitudinal Vibration in a Bar
	11.2 Transverse Vibration in a String
	11.3 The Differential Equation Along the Characteristics
	11.4 Initial Value ProblemWith Discontinuous Shape
	11.5 Initial Value ProblemWith Smooth Shape
		11.5.1 Solution Using Characteristics
		11.5.2 Solution Using the Laplace Transform
Chapter 12 Hyperbolic Quasi Linear Partial Differential Equations
	12.1 Quasi Linear Partial Differential Equations
	12.2 Granular Soils at Impending Failure
		12.2.1 Conditions for Limit Equilibrium
	12.3 Differential equations for impending failure of granular media
		12.3.1 Examination of the Type of the Differential Equations
		12.3.2 The Characteristics
		12.3.3 Characteristic Directions
		12.3.4 Equations Along the Characteristics
		12.3.5 Application: Prandtl’s Wedge
		12.3.6 Logarithmic Spirals
		12.3.7 Width of the Failure Zones Next to the Strip Load
Chapter 13 The Navier-Stokes Equations
	13.1 Energy Transfer in a Fluid
		13.1.1 Turbulence
	13.2 The Bernouilli Equation with Energy Losses; A Simplified Form of the Energy Equation
Appendices
	Appendix A Numerical Integration of the Cauchy Integral
	Appendix B List of Problems with Page Numbers
Bibliography
Index