# Mathematics

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Continuum Physics: Volume 1 — Mathematics is a collection of papers that discusses certain selected mathematical methods used in the study of continuum physics. Papers in this collection deal with developments in mathematics in continuum physics and its applications such as, group theory functional analysis, theory of invariants, and stochastic processes. Part I explains tensor analysis, including the geometry of subspaces and the geometry of Finsler. Part II discusses group theory, which also covers lattices, morphisms, and crystallographic groups. Part III reviews the theory of invariants that includes isotrophy, transverse isotrophy, and nonpolynomial invariants. Part IV explains functional analysis that also includes set theory, vector spaces, topological spaces, and topological vector spaces. Part V deals with analytic function theory and covers topics, such as Cauchy's theorem, the residue theorem, and the Plemelj formulas. Part VI reviews the elements of stochastic processes and cites some examples where stochastic theory is applied. This book can be valuable for researchers and scientists involved in nuclear physicists, students, and academicians in the field of advanced physics.

## Table of Contents

List of Contributors

Preface

Part I. Tensor Analysis

Introduction

1. Tensor Algebra

1.1. Scope of the Section

1.2. Curvilinear Coordinates

1.3. Affine Geometry

1.4. Vector Space

1.5. Geometric Objects

1.6. Scalars, Vectors, and Tensors

1.7. Base Vectors and Reciprocal Bases

1.8. Tensor Algebra

1.9. Multivectors

1.10. Second-Order Tensors

1.11. Normal Form of Symmetric Tensors

1.12. Normal Form of a Bivector

2. Tensor Analysis

2.1. Scope of the Section

2.2. Metric Tensor

2.3. Anholonomic Components of Tensors

2.4. Physical Components of Tensors

2.5. Covariant Differentiation

2.6. Invariant Differential Operators

2.7. Intrinsic Differentiation

2.8. The Lie Derivative

2.9. The Riemann-Christoffel Tensor

3. Geometry of Subspaces

3.1. Scope of the Section

3.2. Curvilinear Coordinates for a Surface in ε3

3.3. Subspace Xm of Xn

3.4. Sections and Reductions of Tensors

3.5. Total Covariant Differentiation

3.6. Curves in Space

3.7. Hypersurface Xn-1 in Rn

3.8. The Volume of Xm in Xn

3.9. Stokes' Theorem

4. Nonriemannian Geometry

4.1. Scope of the Section

4.2. Affine Connection

4.3. Geodesies

4.4. Curvature

4.5. Some Identities Involving the Curvature Tensor

4.6. Covariant Differentiation and Curvature Tensor in Anholonomic Coordinates

5. Geometry of Finsler

5.1. Scope of the Section

5.2. Finsler Spaces

5.3. Metric Tensor Derivable from a Function

5.4. Covariant Differentiation

5.5. Torsion and Curvature Tensor

References

Part II. Group Theory

Introduction

1. Groups and Semigroups

2. Lattices and Morphisms

3. Lie Groups

4. Linear Algebras, Frobenius and Lie

5. Crystallographic Groups

References

Part III. Theory of Invariants

1. Introduction

1.1. Invariants of Vectors and Tensors

1.2. Reducible and Irreducible Invariants; Integrity Bases

1.3. Results from Classical Theory

1.4. The Orthogonal Groups and Certain Subgroups

2. Isotropy

2.1. Integrity Bases for Vectors

2.2. Isotropic Tensors

2.3. Invariants of Vectors and Second-Order Tensors; General Forms

2.4. Results concerning Traces of Matrix Polynomials

2.5. Invariants of Symmetric Second-Order Tensors

2.6. Invariants of Vector and Symmetric Second-Order Tensors; Proper Orthogonal Group

2.7. Full Orthogonal Group; Invariants of Vectors and Second-Order Tensors

3. Transverse Isotropy

3.1. Invariants of Vectors and Tensors; General Forms

3.2. Relations for Matrix Polynomials in 2X2 Matrices

3.3. Invariants of Symmetric Second-Order Tensors

3.4. Invariants of Symmetric Second-Order Tensors and Vectors

3.5. Syzygies for the Invariants

4. The Crystal Classes

4.1. Theorems concerning Integrity Bases

4.2. Invariants of a Symmetric Tensor

4.3. Syzygies between the Invariants of a Symmetric Tensor

4.4. Invariants of a Symmetric Tensor and a Vector

4.5. Invariants of an Arbitrary Number of Vectors

5. Tensor Polynomial Functions of Vectors and Tensors

5.1. General Statements

5.2. Examples of Tensor and Vector Polynomial Functions; Proper Orthogonal Transformation Group

5.3. Examples of Tensor and Vector Polynomial Functions; Full Orthogonal Transformation Group

6. Invariant Functionals; Vector and Tensor Functionals

6.1. General Statements

6.2. Differential Approximations to Functionals

6.3. Integral Approximations to Functionals

7. Minimality of the Integrity Bases

7.1. The Number of Linearly Independent Invariants

7.2. Method of Demonstrating Minimality of an Integrity Basis

7.3. Minimality of Integrity Bases. The Crystal Classes

7.4. Minimality of Integrity Bases. Full and Proper Orthogonal Groups

8. Nonpolynomial Invariants

8.1. Nonpolynomial Invariants

References

Part IV. Functional Analysis

Introduction

1. Set Theory

1.1. Notation

1.2. The Algebra of Sets

1.3. Mappings

1.4. Countable Sets

1.5. Classes of Subsets

1.6. Set Functions

2. Vector Spaces

2.1. Definition of a Vector Space

2.2. Linear Mappings

2.3. The Algebraic Dual of a Vector Space

2.4. Convex Sets in a Vector Space

2.5. Maximal Subspaces and Hyperplanes

2.6. Linear Transformations from Rn into Rm in Terms of Coordinates

3. Topological Spaces

3.1. Open Sets

3.2. Closed Sets

3.3. Metric Spaces

3.4. Continuous Mappings

3.5. Hausdorff Spaces

3.6. Compact Sets

3.7. Complete Metric Spaces

3.8. Homeomorphisms

4. Topological Vector Spaces

4.1. Definition of a Topological Vector Space

4.2. Normed Vector Spaces

4.3. Vector Subspaces

4.4. Banach Spaces

4.5. Finite-Dimensional Normed Spaces

4.6. Hilbert Spaces

4.7. Locally Convex Spaces

5. Spectral Theory of Linear Operators

5.1. The Spectrum of an Operator

5.2. Normal Operators in a Hilbert Space

5.3. Self-Adjoint Operators in a Hilbert Space

5.4. Compact Symmetric Operators in a Hilbert Space

6. Differential Calculus

6.1. The Gateaux Derivative

6.2. The Frechet Derivative

6.3. The Chain Rule

6.4. Newton's Method

6.5. Higher Derivatives

6.6. Differentiable Manifolds

7. Distributions

7.1. The Space of Test Functions

7.2. Distributions

7.3. Examples of Distributions

7.4. Differentiation of Distributions

7.5. Multiplication of Distributions

7.6. Distributions with Compact Support

7.7. Tensor Product of Distributions

7.8. Convolution of Distributions

7.9. Fourier Transforms

7.10. Sobolev Spaces

Appendix: The Lebesgue Integral

A.l. Lebesgue Measure

A.2. The Lebesgue Integral

A.3. Properties of the Lebesgue Integral

A.4. The Lp-Spaces

References

Part V. Analytic Function Theory

1. Cauchy Integrals

1.1. Introduction

1.2. Cauchy's Theorem and Integral Formula

1.3. Singularities: The Residue Theorem

1.4. The Point at Infinity

1.5. Cauchy Principal Values

1.6. The Riemann-Stieltjes Integral: The Delta Function

1.7. The Plemelj Formulas

1.8. Contour Passing through the Point at Infinity

1.9. Inversion Formulas: The Poincare-Bertrand Formula

1.10. Singularities of the Contour C

2. The Fundamental Problems of Potential Theory

2.1. The Dirichlet and Neumann Problems

2.2. Solutions of the Fundamental Problems for a Circle

2.3. The Reflection Principle

2.4. Solutions of the Fundamental Problems for an Infinite Strip

2.5. Simply Mixed Boundary Conditions for a Strip

2.6. Periodic Boundary Conditions

2.7. Periodic Boundary Conditions for the Infinite Strip

2.8. The Solution of the Fundamental Problems for the Rectangle

2.9. Uniqueness of Solutions to the Dirichlet Problem

2.10. Reduction of the Dirichlet Problem to an Integral Equation

2.11. Green's Function

3. Conformal Mapping

3.1. General Principles

3.2. The Schwarz-Christoffel Mapping Theorem

3.3. An Integral Equation for Mapping on the Upper Half Plane

3.4. Generalizations of the S-C Mapping Theorem

3.5. The Mapping of Nearly Circular Domains

3.6. The Mapping of Doubly Connected Regions

3.7. Green's Function and Conformal Mapping

3.8. The Kernel Function

3.9. Mapping and Fluid Dynamics

3.10. Practical Methods of Constructing Conformal Maps

4. The Hilbert Problem and Applications

4.1. The Hilbert Problem

4.2. The Riemann-Hilbert Problem for Single-Connected Domains

4.3. An Alternative Treatment of the Riemann-Hilbert Problem

4.4. The Riemann-Hilbert Problem with Discontinuous Coefficients

4.5. Inversion Formulas for Arcs

4.6. Singular Integral Equations

4.7. The Poincare Problem

4.8. The Wiener-Hopf Technique

References

Part VI. Elements of Stochastic Processes

1. Introduction

1.1. Some Applications of the Theory of Stochastic Processes

1.2. Historical Development of the Mathematical Theory of Stochastic Processes

1.3. Scope of This Work

1.4. Physical Description of Random Processes

1.5. Mathematical Description of Random Processes

1.6. Random Functions of Several Independent Variables

1.7. Random Tensors

1.8. Classification of Random Functions

2. Calculus and Second-Order Properties of Random Functions

2.1. Stochastic Limits and Modes of Convergence

2.2. Regular Random Functions

2.3. Continuity of Random Functions

2.4. Differentiability of Random Functions

2.5. Integration of Random Functions

2.6. Theorems on Stochastic Differentiation and Integration

2.7. Autocovariance and Autocorrelation Functions

2.8. Spectral Density Function

2.9. Orthogonal Expansions of Random Function

2.10. Fourier Analysis of Stationary Processes

2.11. The Gaussian Random Process, Definitions

2.12. Theorems and Results

3. Differential Equations for Distribution Functions of Stochastic Processes and Some Special Properties of Stochastic Processes

3.1. Differential Equation for the Characteristic and Distribution Functions of Physical Processes (Sufficiently Smooth Processes)

3.2. Markoff Processes

3.3. Expected Number of Zeros per Unit of Time

3.4. Expected Number of Maxima per Unit of Time

3.5. Other Properties of Interest

4. Stochastic Boundary and Initial Value Problems

4.1. Formulation of Boundary and Initial Value Problems under Stochastic Conditions

4.2. Stochastic Ordinary Differential Equations

4.3. Stochastic Stability Theory

References

Author Index

Subject Index

## Product details

- No. of pages: 694
- Language: English
- Copyright: © Academic Press 1971
- Published: January 28, 1971
- Imprint: Academic Press
- eBook ISBN: 9781483277165