Multiple integrals, field theory and series: an advanced course in higher mathematics

Contents

CHAPTER 1. DOUBLE INTEGRALS

§ 1. Auxiliary Notions. Area of a Plane Figure
1. Interior and Boundary Points. Domain
2. Distance Between Two Sets
3. Area of a Plane Figure
4. Basic Properties of Area
5. The Concept of Measure

§ 2. Definition and Basic Properties of Double Integral
1. Definition of Double Integral
2. Conditions for Existence of Double Integral. Upper and Lower Darboux Sums
3. Some Important Classes. of Integrable Functions
4. Properties of Double Integrals

§ 3. Additive Set Functions. Derivative of a Set Function with Respect to the Area
1. Point Functions and Set Functions
2. Double Integral as an Additive Function of its Domain of Integration
3. Derivative of a Set Function with Respect to the Area
4. Derivative of a Double Integral with Respect to the Area of its Domain of Integration
5. Reconstruction of an Additive Set Function from its Derivative
6. Definite Integral of a Function of One Argument as a Function of Its Interval of Integration
7. Extension of Additive Set Functions

§ 4. Some Physical and Geometrical Applications of the Double integral
1. Evaluating Volumes
2. Computing Areas
3. Mass of a Plate
4. Coordinates of the Centre of Gravity of a Plate
5. Moments of Inertia of a Plate
6. Luminous Flux Incident on a Plate
7. Flux of a Fluid Through the Cross Section of a Channel

§ 5. Reducing Double Integral to a Twofold Iterated Integral
1. Heuristic Considerations
2. The Case of a Rectangular Domain of Integration
3. The Case of a Curvilinear Domain

§ 6. Change of Variables in Double Integral
1. Mapping of Plane Figures
2. Curvilinear Coordinates
3. Polar Coordinates
4. Statement of the Problem of Changing Variables in the Double Integral
5. Computing Area in Curvilinear Coordinates
6. Change of Variables in Double Integral
7. Comparison with One-Dimensional Case. Integral Over an Oriented Domain

CHAPTER 2. TRIPLE INTEGRALS AND MULTIPLE INTEGRALS
OF HIGHER ORDER

§ 1. Definition and Basic Properties of Triple Integral
1. Preliminary Observations. Volume of a Space Figure
2. Definition of Triple Integral
3. Conditions for Existence of Triple Integral. Integrability of Continuous Functions
4. Properties of Triple Integral
5. Triple Integral as an Additive Set Function

§ 2. Some Applications of Triple Integral in Physics and Geometry
1. Computing Volumes
2. Finding the Mass of a Solid from Its Density
3. Moment of Inertia
4. Determining the Coordinates of the Centre of Gravity
5. Gravitational Attraction of a Material Point by a Solid

§ 3. Evaluating Triple Integral
1. Reducing Triple Integral Over a Rectangular Parallelepiped to an Iterated Integral
2. Reducing Triple Integral Over a Curvilinear Domain to an Iterated integral

§ 4. Change of Variables in Triple Integral
1. Mapping of Space Figures
2. Curvilinear Coordinates in Space
3. Cylindrical and Spherical Coordinates
4. Element of Volume in Curvilinear Coordinates
5. Change of Variables in Triple Integral. Geometric Meaning of the Jacobian

§ 3. Multiple Integrals of Higher Order
1. General Remarks
2. Examples

CHAPTER 3. ELEMENTS OF DIFFERENTIAL GEOMETRY

§ 1. Vector Function of a Scalar Argument
1. Definition of a Vector Function. Limit. Continuity
2. Differentiation of a Vector Function
3. Hodograph. Singular Points
4. Taylor’s Formula
5. Integral of a Vector Function with Respect to Scalar Argument
6. Vector Function of Several Scalar Arguments

§ 2. Space Curves
1. Vector Equation of a Curve
2. Moving Trihedron
3. Frenet-Serret Formulas
4. Evaluating Curvature and Torsion
5. Coordinate System Connected with Moving Trihedron
6. The Shape of a Curve in the Vicinity of Its Point
7. Curvature of a Plane Curve
8. Intrinsic Equations of a Curve
9. Some Applications to Mechanics

§ 3. Parametric Equations of a Surface
1. The Concept of a Surface
2. Parametrization of a Surface
3. Parametric Equations of a Surface
4. Curves on a Surface
5. Tangent Plane
6. Normal to a Surface
7. Coordinate Systems in Tangent Planes

§ 4. Determining Lengths, Angles and Areas on a Curvilinear Surface.
1. First Fundamental Quadratic Form of a Surface
2. Affine Coordinate System in the Plane
3. Are Length of a Curve on a Surface. First Fundamental
4. Quadratic Form
5. Angle Between Two Curves
6. Definition of Area of a Surface. The Schwarz Example
7. Computing Area of a Smooth Surface


§ 5. Curvature of Curves on a Surface. Second Fundamental Quadratic Form of a Surface
1. Normal Sections of a Surface and Their Curvature
2. Second Fundamental Quadratic Form of a Surface
3. Dupin Indicatrix
4. Principal Directions and Principal Curvatures of a Surface. Equation of Euler
5. Determining Principal Curvatures
6. Total Curvature and Mean Curvature
7. Classification of Points on a Surface
8. The First and the Second Fundamental Quadratic Forms as Invariants of a Surface

§ 6. Intrinsic Properties of a Surface
1. Applicable Surfaces. Necessary and Sufficient Condition for Applicability
2. Intrinsic Properties of a Surface
3. Surfaces of Constant Curvature

CHAPTER 4. LINE INTEGRALS

§ 1. Line Integrals of the First Type
1. Definition of Line Integral of the First Type
2. Properties of Line Integrals
3. Some Applications of Line Integrals of the First Type
4. Line Integrals of the First Type in Space

§ 2. Line Integrals of the Second Type
1. Statement of the Problem. Work of a Field of Force
2. Definition of Line Integral of the Second Type
3. Connection Between Line Integrals of the First and the Second Type
4. Evaluating Line Integral of the Second Type
5. Dependence of Line Integral of the Second Type on the Orientation of the Path of Integration
6. Line Integrals Along Self-Intersecting and Closed Paths
7. Line Integral of the Second Type Over a Space Curves

§ 3. Green's Formula
1. Derivation of Green’s Formula
2. Application of Green’s Formula to Compute Areas

§ 4. Conditions for a Line Integral of the Second Type Being Path Independent. Integrating Total Differentials
1. Statement of the Problem
2. The Case of a Simply Connected Domain
3. Reconstructing a Function from its Total Differential
4. Line Integrals in a Multiply Connected Domain


CHAPTER 5. SURFACE INTEGRALS

§ 7. Surface Integral of the First Type
1. Definition of Surface Integral of a Scalar Function
2. Reducing Surface Integral to Double Integral
3. Some Applications of Surface Integrals to Mechanics .
4. Surface Integral of a Vector Function. General Concept of Surface Integral of the First Type

§ 2. Surface Integral of the Second Type
1. One-Sided and Two-Sided Surfaces
2. Definition of Surface Integral of the Second type
3. Reducing Surface Integral of the Second Type to Double Integral




§ 3. Ostrogradsky Theorem
1. Derivation of Ostrogradsky Theorem
2. Application of Ostrogradsky Theorem to Evaluating Surface Integrals.
3. Expressing Volume of a Space Figure in the Form of a Surface Integral

§ 4. Stokes’ Theorem
1. Derivation of Stokes’ Formula
2. Application of Stokes’ Theorem to Investigating Line integrals in Space

CHAPTER 6. FIELD THEORY

§ 1. Scalar Field
1. Definition and Examples of Scalar Field
2. Level Surfaces and Level Lines
3. Various Types of Symmetry of Field
4. Directional Derivative
5. Gradient of Scalar Field

§ 2. Vector Field
1. Definition and Examples of Vector Field
2. Vector Lines and Vector Surfaces
3. Types of Symmetry of Vector Field
4. Field of Gradients. Potential Field

§ 3. Flux of Vector Field. Divergence
1. Flux of Vector Field Across a Surface
2. Divergence
3. Physical Meaning of Divergence for Various Types of Field. Examples
4. Solenoidal Field
5. Equation of Continuity
6. Plane Fluid Flow. Ostrogradsky Theorem for Plane Field

§ 4. Circulation. Rotation
1. Circulation of Vector Field
2. Rotation of Vector Field. Stokes’ Formula in Vector Notation
3. Symbolic Formula for Rotation
4. Physical Meaning of Rotation
5. More on Potential and Solenoidal Fields

§ 5. Hamiltonian Operator
1. Symbolic Vector
2. Operations with Vectors

§ 6. Repeated Operations Involving ¥. Laplacian Operator
1. Repeated Differentiation
2. Heat Conductivity Equation
3. Stationary Distribution of Temperature. Harmonic Functions

§ 7. Expressing Field Operations in Curvilinear Orthogonal Coordinates
1. Statement of the Problem
2. Curvilinear Orthogonal Coordinates in Space
3. Cylindrical and Spherical Coordinates
4. Gradient
5. Divergence
6. Rotation
7. Laplace’s Operator
8. Basic Field Operations in Cylindrical and Spherical Coordinates

§ 8. Variable Fields in Continuous Media
1. Partial and Total Time Derivatives
2. Eulerian Equations of Motion of Ideal Liquid
3. Derivative with Respect to Time of an Integral Over a Fluid Volume
4. Application to Deriving Equation of Continuity

CHAPTER 7. TENSORS

§ 1. Orthogonal Affine Tensor
1. Transformation of Orthonormal Bases
2. Definition of Orthogonal Affine Tensor

§ 2. Connection Between Tensors of Second Rank and Linear Operators
1. Linear Operator as a Tensor of Second Rank
2. Tensor of Second Rank as a Linear Operator

§ 3. Connection Between Tensors and Invariant Multilinear Forms
1. Tensors of Rank One and Invariant Linear Forms
2. Tensors of Rank Two and Invariant Bilinear Forms
3. Tensors of Arbitrary Rank p and Invariant Multilinear Forms

§ 4. Strain Tensor

§ 5. Stress Tensor
1. Definition of Stress Tensor
2. Stress Tensor as a Linear Operator

§ 6. Algebraic Operations on Tensors
1. Addition, Subtraction and Multiplication of Tensors
2. Multiplying Tensor by Vector
3. Contraction
4. interchanging Indices
5. Resolution of Tensor of Second Rank into ‘Symmetric and Antisymmetric Parts

§ 7. Tensor of Relative Displacements

§ 8. Tensor Field
1. Tensor Field. Divergence of Tensor
2. Ostrogradsky Theorem for Tensor Field
3. Equations of Motion of a Continuous Medium

§ 9. Principal Axes of Symmetric Tensor of Second Rank

§ 10. General Tensors
1. Reciprocal Bases
2. Covariant and Contravariant Components of Vectors
3. Summation Convention
4. Transformation of Base Vectors
5. Transformation of Covariant and Contravariant Components of Vector
6. General Definition of Tensor
7. Operations on Tensors
8. Some Further Generalizations

Appendix to Chapter 7. On Multiplication of Matrices

CHAPTER 8 FUNCTIONAL SEQUENCES AND SERIES

§ 1. Uniform Convergence. Tests for Uniform Convergence
1. Convergence and Uniform Convergence
2. Tests for Uniform Convergence

§ 2. Properties of Uniformly Convergent Functional Sequences and Series
1. Continuity and Uniform Convergence
2. Passage to Limit Under the Sign of integration and Term-wise Integration of a Series
3. Passage to Limit Under the Sign of Differentiation and Term-wise Differentiation of a Series
4. Term-by-Term Passage to Limit in Functional Sequences and Series

§ 3. Power Series
1. Interval of Convergence of Power Series. Radius of Convergence
2. On Uniform Convergence of a Power Series and Continuity of Its Sum
3. Differentiation and Integration of Power Series
4. Arithmetical Operations on Power Series

§ 4. Expanding Functions in Power Series
1. Key Theorems on Expanding Functions in Power Series. Expanding Elementary Functions
2. Some Applications of Power Series

§ 5. Power Series in Complex Argument

§ 6. Convergence in the Mean
1. Mean Square Deviation and Convergence in the Mean
2. Cauchy-Bunyakovsky Inequality
3. Integration of Sequences and Series Convergent in the Mean
4. Connection Between Convergence in the Mean and Term-by-Term Differentiation of Sequences and Series
5. Connection Between Convergence in the Mean and Other Types of Convergence

Appendix I to Chapter 8. Criterion for Compactness of a Family of Functions
Appendix 2 to Chapter 8. Weak Convergence and Delta Function

CHAPTER 9. IMPROPER INTEGRALS

§ 1. Integrals with Infinite Limits of Integration
1. Definitions. Examples
2. Reducing Improper Integral of the Form ∫ f(z)dr to Numerical Sequence and Numerical Series
3. Cauchy Criterion for Improper Integrals
4. Absolute Convergence. Tests for Absolute Convergence
5. Conditional Convergence
6. Extending Methods of Evaluating Integrals to the Case of Improper Integrals

§ 2. Integrals of Unbounded Functions with Finite and Infinite Limits of Integration


§ 3. Cauchy's Principal Value of a Divergent Improper Integral

§ 4. Improper Multiple Integrals
1. Integral of an Unbounded Function Over a Finite Domain
2. Integrals of Non-negative Functions
3. Absolute Convergence
4. Tests for Absolute Convergence
5. Equivalence of Convergence and Absolute Convergence in the Case of Improper Multiple Integral
6. Improper Integrals with Infinite Domain of integration
7. Methods of Computing Improper Multiple Integrals

CHAPTER 10. INTEGRALS DEPENDENT ON PARAMETER

§ 1. Proper and Simplest improper Integrals Dependent on Parameter
1. Proper Integrals Dependent on Parameter
2. Simplest Improper Integrals Dependent on Parameter

§ 2. Improper Integrals Dependent on Parameter
1. Uniform Convergence
2. Reducing Improper Integral Dependent on Parameter to a Functional Sequence
3. Properties of Uniformly Convergent Improper Integrals Dependent on Parameter
4. Tests for Uniform Convergence of Improper Integrals Dependent on Parameter
5. Examples of Evaluating Improper Integrals Dependent on Parameter by Means of Differentiation and Integration with Respect to a Parameter

§ 3. Euler’s integrals
1. Properties of Gamma Function
2. Properties of Beta Function

§ 4. Multiple Integrals Dependent on Parameter

CHAPTER 11. FOURIER SERIES AND FOURIER INTEGRAL

§ 1. Properties of Periodic Functions, Statement of the Key Problem
1. Periods of a Periodic Function
2. Periodic Extension of a Non-periodic Function
3. Integral of a Periodic Function
4. Arithmetical Operations on Periodic Functions
5. Superposition of Harmonics with Multiple Frequencies
6. Statement of the Key Problem
7. Orthogonality of Trigonometric System. Fourier Coefficients and Fourier Series
8. Expanding Even and Odd Functions in Fourier Series
9. Expanding Functions in Fourier Series on the Interval $[-pi,\pi]$

§ 2. Fundamental Theorem on Convergence of Fourier Series
1. Class of Piecewise Smooth Functions
2. Formulation of Fundamental Theorem on Convergence of Fourier Series
3. Key Lemma
4. Proof of Convergence Theorem
5. Examples
6. Fourier Sine and Cosine Series ‘for ‘Functions Defined on Interval [0,l]


§ 3. Fourier Series with Respect to General Orthogonal Systems. Bessel's Inequality
1. Orthogonal Systems of Functions
2. Fourier Coefficients and Fourier Series of a Function with Respect to an Orthogonal System
3. Least Square Deviation. Bessel’s Inequality

§ 4. Speed of Convergence of Fourier Series. Acceleration of Convergence of Fourier Series
1. Conditions for Uniform Convergence of Fourier Series
2. Connection Between the Degree of Smoothness of a Function and the Speed of Convergence of Its Fourier Series
3. Acceleration of Convergence of Fourier Series

§ 5. Uniform Approximation of Continuous Function by Trigonometric and Algebraic Polynomials. Weierstrass’ Approximation Theorems

§ 6. Complete and Closed Orthogonal Systems
1. Complete Orthogonal System
2. Parseval Relation as a Necessary and Sufficient Condition for an Orthogonal System Being Complete Properties of Complete Systems
3. Completeness of Trigonometric System
4. Completeness of Same Other Classical Orthogonal Systems
5. Fourier Series in Orthogonal Systems of Complex Functions
6. Fourier Series for Functions of Several Independent Variables

§ 9. Fourier Integral
1. Formal Derivation of Fourier Integral Formula
2. Proof of Fourier Integral Theorem
3. Fourier Integral as an Expansion into a Sum of Harmonics
4. Fourier Integral in Complex Form
5. Fourier Transformation
6. Fourier Integral for Functions of Several Independent Variables

Appendix 1 to Chapter 11. On Legendre’s Polynomials
Appendix 2 to Chapter 11. Orthogonality with Weight Function and Orthogonalization Process
Appendix 3 to Chapter 11. Functional Space and Geometric Analogy
Appendix 4 to Chapter 11. Some Applications of Fourier Transforms
Appendix 5 to Chapter 11. Expanding Delta Function in Fourier Series and Further Integral
Appendix 6 to Chapter 11. Uniform Approximation of Functions with Polynomials
Appendix 7 to Chapter 11. On Stable Summation of Fourier Series with Perturbed Coefficients

SUPPLEMENT 1. ASYMPTOTIC EXPANSIONS
§ 1. Examples of Asymptotic Expansions
1. Asymptotic Expansions in the Neighbourhood of the Origin
2. Asymptotic Expansions in the Neighbourhood of the Point at Infinity

§ 2. General Definitions and Theorems
1. Order of Smallness. Asymptotic Equivalence
2. Asymptotic Expansions of Functions

§ 3. Laplace Method for Deriving Asymptotic Expansions of Some Integrals

SUPPLEMENT 2. ON UNIVERSAL DIGITAL COMPUTERS
§1. Computers
1. Introduction
2. Basic Types of Computer 603
3. Principal Components of a Computer and Their Functions Git4
4. Number Systems Used in Computers 606
5. Representing Numbers Within a Computer GON

§ 2. Basic Operations Executed by a Computer. Instructions GUS
1. Types of Operation
2. Arithmetical Operations
3. Additional Computational Operations a GIO
4. Logical Operations
5. Input and Output Operations
6. Transfer of Control
7. Realization of Operations Within a. Computer

§ 3. Elements of Programming
1. General Notions
2. Formula Programming
3. Cyclic Processes
4. Flow-Chart. Subroutines
5. Instruction Codes. Operations on Instructions
6. Automatic Programming

§ 4. Organization of Computer Work
1. Conditions for Effective Use of a Computer
2. Basic Stages of Solving a Problem on a Computer
3. Checking Computer Operation. Error Detection

Bibliography
Name Index
Subject Index