Higher Mathematics for Physics and Engineering

Transcription

1 Higher Mathematics for Physics and Engineering

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3 Hiroyuki Shima Tsuneyoshi Nakayama Higher Mathematics for Physics and Engineering 123

4 Dr. Hiroyuki Shima, Assistant Professor Department of Applied Physics Hokkaido University Sapporo , Japan Dr. Tsuneyoshi Nakayama, Professor Toyota Physical and Chemical Research Institute Aichi , Japan ISBN e-isbn DOI /b Springer Heidelberg Dordrecht London New York Library of Congress Control Number: c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: estudio Calamar Steinen Printed on acid-free paper Springer is part of Springer Science+Business Media (

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7 Preface Owing to the rapid advances in the physical sciences and engineering, the demand for higher-level mathematics is increasing yearly. This book is designed for advanced undergraduates and graduate students who are interested in the mathematical aspects of their own fields of study. The reader is assumed to have a knowledge of undergraduate-level calculus and linear algebra. There are any number of books available on mathematics for physics and engineering but they all fall into one of two categories: the one emphasizes mathematical rigor and the exposition of definitions or theorems, whereas the other is concerned primarily with applying mathematics to practical problems. We believe that neither of these approaches alone is particularly helpful to physicists and engineers who want to understand the mathematical background of the subjects with which they are concerned. This book is different in that it provides a short path to higher mathematics via a combination of these approaches. A sizable portion of this book is devoted to theorems and definitions with their proofs, and we are convinced that the study of these proofs, which range from trivial to difficult, is useful for a grasp of the general idea of mathematical logic. Moreover, several problems have been included at the end of each section, and complete solutions for all of them are presented in the greatest possible detail. We firmly believe that ours is a better pedagogical approach than that found in typical textbooks, where there are many well-polished problems but no solutions. This book is essentially self-contained and assumes only standard undergraduate preparation such as elementary calculus and linear algebra. The first half of the book covers the following three topics: real analysis, functional analysis, and complex analysis, along with the preliminaries and four appendixes. Part I focuses on sequences and series of real numbers of real functions, with detailed explanations of their convergence properties. We also emphasize the concepts of Cauchy sequences and the Cauchy criterion that determine the convergence of infinite real sequences. Part II deals with the theory of the Hilbert space, which is the most important class of infinite vector spaces. The completeness property of Hilbert spaces allows one to develop

8 VIII Preface various types of complex orthonormal polynomials, as described in the middle of Part II. An introduction to the Lebesgue integration theory, a subject of ever-increasing importance in physics, is also presented. Part III describes the theory of complex-valued functions of one complex variable. All relevant elements including analytic functions, singularity, residue, continuation, and conformal mapping are described in a self-contained manner. A thorough understanding of the fundamentals treated is important in order to proceed to more advanced branches of mathematical physics. In the second half of the volume, the following three specific topics are discussed: Fourier analysis, differential equations, and tensor analysis. These three are the most important subjects in both engineering and the physical sciences, but their rigorous mathematical structures have hardly been covered in ordinary textbooks. We know that mathematical rigor is often unnecessary for practical use. However, the blind usage of mathematical methods as a tool may lead to a lack of understanding of the symbiotic relationship between mathematics and the physical sciences. We believe that readers who study the mathematical structures underlying these three subjects in detail will acquire a better understanding of the theoretical backgrounds associated with their own fields. Part IV describes the theory of Fourier series, the Fourier transform, and the Laplace transform, with a special emphasis on the proofs of their convergence properties. A more contemporary subject, the wavelet transform, is also described toward the end of Part IV. Part V deals with ordinary and partial differential equations. The existence theorem and stability theory for solutions, which serve as the underlying basis for differential equations, are described with rigorous proofs. Part VI is devoted to the calculus of tensors in terms of both Cartesian and non-cartesian coordinates, along with the essentials of differential geometry. An alternative tensor theory expressed in terms of abstract vector spaces is developed toward the end of Part VI. The authors hope and trust that this book will serve as an introductory guide for the mathematical aspects of the important topics in the physical sciences and engineering. Sapporo, November 2009 Hiroyuki Shima Tsuneyoshi Nakayama

9 Contents 1 Preliminaries Basic Notions of a Set Set and Element Number Sets Bounds Interval Neighborhood and Contact Point Closed and Open Sets Conditional Statements OrderofMagnitude Symbols O, o, and Asymptotic Behavior Values of Indeterminate Forms l Hôpital s Rule Several Examples Part I Real Analysis 2 Real Sequences and Series Sequences of Real Numbers Convergence of a Sequence Bounded Sequences Monotonic Sequences Limit Superior and Limit Inferior Cauchy Criterion for Real Sequences Cauchy Sequence Cauchy Criterion InfiniteSeriesofRealNumbers Limits of Infinite Series Cauchy Criterion for Infinite Series

10 X Contents Absolute and Conditional Convergence Rearrangements Convergence Tests for Infinite Real Series Limit Tests Ratio Tests Root Tests Alternating Series Test Real Functions FundamentalProperties Limit of a Function Continuity of a Function Derivative of a Function Smooth Functions Sequences of Real Functions Pointwise Convergence Uniform Convergence Cauchy Criterion for Series of Functions Continuity of the Limit Function Integrability of the Limit Function Differentiability of the Limit Function Series of Real Functions Series of Functions Properties of Uniformly Convergent Series of Functions Weierstrass M-test ImproperIntegrals Definitions Convergence of an Improper Integral Principal Value Integral Conditions for Convergence Part II Functional Analysis 4 Hilbert Spaces HilbertSpaces Introduction Abstract Vector Spaces Inner Product Geometry of Inner Product Spaces Orthogonality Completeness of Vector Spaces Several Examples of Hilbert Spaces HierarchicalStructureofVectorSpaces Precise Definitions of Vector Spaces

11 Contents XI Metric Space Normed Spaces Subspaces of a Normed Space Basis of a Vector Space: Revisited Orthogonal Bases in Hilbert Spaces Hilbert Spaces of l 2 and L Completeness of the l 2 Spaces Completeness of the L 2 Spaces Mean Convergence Generalized Fourier Coefficients Riesz Fisher Theorem Isomorphism between l 2 and L Orthonormal Polynomials Polynomial Approximations Weierstrass Theorem Existence of Complete Orthonormal sets of Polynomials Legendre Polynomials Fourier Series Spherical Harmonic Functions Classification of Orthonormal Functions General Rodrigues Formula Classification of the Polynomials The Recurrence Formula Coefficients of the Recurrence Formula Roots of Orthogonal Polynomials Differential Equations Satisfied by the Polynomials Generating Functions (I) Generating Functions (II) Chebyshev Polynomials Minimax Property A Concise Representation Discrete Orthogonality Relation Applications in Physics and Engineering Quantum-Mechanical State in an Harmonic Potential Electrostatic potential generated by a multipole Lebesgue Integrals Measure and Summability Riemann Integral Revisited Measure The Probability Measure Support and Area of a Step Function α-summability Properties of α-summable functions

12 XII Contents 6.2 LebesgueIntegral Lebesgue Measure Definition of the Lebesgue Integral Riemann Integrals vs. Lebesgue Integrals Properties of the Lebesgue Integrals Null-Measure Property of Countable Sets The Concept of Almost Everywhere ImportantTheoremsforLebesgueIntegrals Monotone Convergence Theorem Dominated Convergence Theorem (I) Fatou Lemma Dominated Convergence Theorem (II) Fubini Theorem The Lebesgue Spaces L p The Spaces of L p Hölder Inequality Minkowski Inequality Completeness of L p Spaces Applications in Physics and Engineering Practical Significance of Lebesgue Integrals Contraction Mapping Preliminaries for the Central Limit Theorem Central Limit Theorem Proof of the Central Limit Theorem Part III Complex Analysis 7 Complex Functions Analytic Functions Continuity and Differentiability Definition of an Analytic Function Cauchy Riemann Equations Harmonic Functions Geometric Interpretation of Analyticity Complex Integrations Integration of Complex Functions Cauchy Theorem Integrations on a Multiply Connected Region Primitive Functions Cauchy Integral Formula and Related Theorem Cauchy Integral Formula Goursat Formula Absence of Extrema in Analytic Regions Liouville Theorem

13 Contents XIII Fundamental Theorem of Algebra Morera Theorem Series Representations Circle of Convergence Singularity on the Radius of Convergence Taylor Series Apparent Paradoxes Laurent Series Regular and Principal Parts Uniqueness of Laurent Series Techniques for Laurent Expansion Applications in Physics and Engineering Fluid Dynamics Kutta Joukowski Theorem Blasius Formula Singularity and Continuation Singularity Isolated Singularities Nonisolated Singularities Weierstrass Theorem for Essential Singularities Rational Functions Multivaluedness Multivalued Functions Riemann Surfaces Branch Point and Branch Cut Analytic Continuation Continuation by Taylor Series Function Elements Uniqueness Theorem Conservation of Functional Equations Continuation Around a Branch Point Natural Boundaries Technique of Analytic Continuations The Method of Moment Contour Integrals Calculus of Residues Residue Theorem Remarks on Residues Winding Number Ratio Method Evaluating the Residues Applications to Real Integrals Classification of Evaluable Real Integrals

14 XIV Contents Type 1: Integrals of f(cos θ, sin θ) Type 2: Integrals of Rational Function Type 3: Integrals of f(x)e ix Type 4: Integrals of f(x)/x α Type 5: Integrals of f(x) log x More Applications of Residue Calculus Integrals on Rectangular Contours Fresnel Integrals Summation of Series Langevin and Riemann zeta Functions ArgumentPrinciple The Principle Variation of the Argument Extentson of the Argument Principle RouchéTheorem Dispersion Relations Principal Value Integrals Several Remarks Dispersion relations Kramers Kronig Relations Subtracted Dispersion Relation Derivation of Dispersion Relations Conformal Mapping Fundamentals Conformal Property of Analytic Functions Scale Factor Mapping of a Differential Area Mapping of a Tangent Line The Point at Infinity Singular Point at Infinity Elementary Transformations Linear Transformations Bilinear Transformations Miscellaneous Transformations Mapping of Finite-Radius Circle Invariance of the Cross ratio Applications to Boundary-Value Problems Schwarz Christoffel Transformation Derivation of the Schwartz Christoffel Transformation The Method of Inversion Applications in Physics and Engineering Electric Potential Field in a Complicated Geometry Joukowsky Airfoil

15 Contents XV Part IV Fourier Analysis 11 Fourier Series Basic Properties Definition Dirichlet Theorem Fourier Series of Periodic Functions Half-range Fourier Series Fourier Series of Nonperiodic Functions The Rate of Convergence Fourier Series in Higher Dimensions Mean Convergence of Fourier Series Mean Convergence Property Dirichlet and Fejér Integrals Proof of the Mean Convergence of Fourier Series Parseval Identity Riemann Lebesgue Theorem Uniform Convergence of Fourier series Criterion for Uniform and Pointwise Convergence Fejér theorem Proof of Uniform Convergence Pointwise Convergence at Discontinuous Points Gibbs Phenomenon Overshoot at a Discontinuous Point Applications in Physics and Engineering Temperature Variation of the Ground String Vibration Under Impact Fourier Transformation Fourier Transform Derivation of Fourier Transform Fourier Integral Theorem Proof of the Fourier Integral Theorem Inverse Relations of the Half-width Parseval Identity for Fourier Transforms Fourier Transforms in Higher Dimensions Convolution and Correlations Convolution Theorem Cross-Correlation Functions Autocorrelation Functions Discrete Fourier Transform Definitions Inverse Transform Nyquest Frequency and Aliasing

16 XVI Contents Sampling Theorem Fast Fourier Transform Matrix Representation of FFT Algorithm Decomposition Method for FFT Applications in Physics and Engineering Fraunhofer Diffraction I Fraunhofer Diffraction II Amplitude Modulation Technique Laplace Transformation Basic Operations Definitions Several Remarks Significance of Analytic Continuation Convergence of Laplace Integrals Abscissa of Absolute Convergence Laplace Transforms of Elementary Functions Properties of Laplace Transforms First Shifting Theorem Second Shifting Theorem Laplace Transform of Periodic Functions Laplace Transform of Derivatives and Integrals Laplace Transforms Leading to Multivalued Functions Convergence Theorems for Laplace Integrals Functions of Exponential Order Convergence for Exponential-Order Cases Uniform Convergence for Exponential-Order Cases Convergence for General Cases Uniform Convergence for General Cases Distinction Between Exponential-Order Cases and General Cases Analytic Property of Laplace Transforms Inverse Laplace Transform The Two-Sided Laplace Transform Inverse of the Two-Sided Laplace Transform Inverse of the One-Sided Laplace Transform Useful Formula for Inverse Laplace Transformation Evaluating Inverse Transformations Inverse Transform of Multivalued Functions Applications in Physics and Engineering Electric Circuits I Electric Circuits II

17 Contents XVII 14 Wavelet Transformation Continuous Wavelet Analyses Definition of Wavelet The Wavelet Transform Correlation Between Wavelet and Signal Actual Application of the Wavelet Transform Inverse Wavelet Transform Noise Reduction Technique Discrete Wavelet Analysis Discrete Wavelet Transforms Complete Orthonormal Wavelets Multiresolution Analysis Orthogonal Decomposition Constructing an Orthonormal Basis Two-Scale Relations Constructing the Mother Wavelet Multiresolution Representation Fast Wavelet Transformation Generalized Two-Scale Relations Decomposition Algorithm Reconstruction Algorithm Part V Differential Equations 15 Ordinary Differential Equations Concepts of Solutions Definition of Ordinary Differential Equations Explicit Solution Implicit Solution General and Particular Solutions Singular Solution Integral Curve and Direction Field Existence Theorem for the First-Order ODE Picard Method Properties of Successive Approximations Existence Theorem and Lipschitz Condition Uniqueness Theorem Remarks on the Two Theorems Sturm Liouville Problems Sturm Liouville Equation Conversion into a Sturm Liouville Equation Self-adjoint Operators Required Boundary Condition Reality of Eigenvalues

18 XVIII Contents 16 System of Ordinary Differential Equations Systems of ODEs Systems of the First-Order ODEs Column-Vector Notation Reducing the Order of ODEs Lipschitz Condition in Vector Spaces Linear System of ODEs Basic Terminology Vector Space of Solutions Fundamental Systems of Solutions Wronskian for a System of ODEs Liouville Formula for a Wronskian Wronskian for an nth-order Linear ODE Particular Solution of an Inhomogeneous System Autonomous Systems of ODEs Autonomous System Trajectory Critical Point Stability of a Critical Point Linear Autonomous System Classification of Critical Points Improper Node Saddle Point Proper Node Spiral Point Center Limit Cycle Applications in Physics and Engineering Van der Pol Generator Partial Differential Equations Basic Properties Definitions Subsidiary Conditions Linear and Homogeneous PDEs Characteristic Equation Second-Order PDEs Classification of Second-Order PDEs The Laplacian Operator Maximum and Minimum Theorem Uniqueness Theorem Symmetric Properties of the Laplacian The Diffusion Operator The Diffusion Equations in Bounded Domains Maximum and Minimum Theorem Uniqueness Theorem

19 Contents XIX 17.4 The Wave Operator The Cauchy Problem Homogeneous Wave Equations Inhomogeneous Wave Equations Wave Equations in Finite Domains Applications in Physics and Engineering Wave Equations for Vibrating Strings Diffusion Equations for Heat Conduction Part VI Tensor Analyses 18 Cartesian Tensors Rotation of Coordinate Axes Tensors and Coordinate Transformations Summation Convention Cartesian Coordinate System Rotation of Coordinate Axes Orthogonal Relations Matrix Representations Determinant of a Matrix Cartesian Tensors Cartesian Vectors A Vector and a Geometric Arrow Cartesian Tensors Scalars Pseudotensors Improper Rotations Pseudovectors Pseudotensors Levi Civita Symbols Tensor Algebra Addition and Subtraction Contraction Outer and Inner Products Symmetric and Antisymmetric Tensors Equivalence of an Antisymmetric Second-Order Tensor to a Pseudovector Quotient Theorem Quotient Theorem for Two-Subscripted Quantities Applications in Physics and Engineering Inertia Tensor Tensors in Electromagnetism in Solids Electromagnetic Field Tensor Elastic Tensor

20 XX Contents 19 Non-Cartesian Tensors Curvilinear Coordinate Systems Local Basis Vectors Reciprocity Relations Transformation Law of Covariant Basis Vectors Transformation Law of Contravariant Basis Vectors Components of a Vector Components of a Tensor Mixed Components of a Tensor Kronecker Delta Metric Tensor Definition Geometric Role of Metric Tensors Riemann Space and Metric Tensor Elements of Arc, Area, and Volume Scale Factors Representation of Basis Vectors in Derivatives Index Lowering and Raising Christoffel Symbols Derivatives of Basis Vectors Nontensor Character Properties of Christoffel Symbols Alternative Expression Covariant Derivatives Covariant Derivatives of Vectors Remarks on Covariant Derivatives Covariant Derivatives of Tensors Vector Operators in Tensor Form Applications in Physics and Engineering General Relativity Theory Riemann Tensor Energy Momentum Tensor Einstein Field Equation Tensor as Mapping Vector as a Linear Function Overview Vector Spaces Revisited Vector Spaces of Linear Functions Dual Spaces Equivalence Between Vectors and Linear Functions Tensor as Multilinear Function Direct Product of Vector Spaces Multilinear Functions Tensor Product

21 Contents XXI General Definition of Tensors Components of Tensors Basis of a Tensor Space Transformation Laws of Tensors Natural Isomorphism Inner Product in Tensor Language Index Lowering and Raising in Tensor Language Part VII Appendixes A Proof of the Bolzano Weierstrass Theorem A.1 Limit Points A.2 CantorTheorem A.3 Bolzano WeierstrassTheorem B Dirac δ Function B.1 BasicProperties B.2 Representation as a Limit of Function B.3 Remarks on Representation C Proof of Weierstrass Approximation Theorem D Tabulated List of Orthonormal Polynomial Functions Index...677