作者：Vladimir I. Arnold[著
页数：157页
出版社：世界图书出版公司
出版日期：2024
ISBN：9787519296681
电子书格式：pdf/epub/txt
内容简介
本书是苏联/俄罗斯数学家阿诺德为本科生写的讲义，内容简明扼要，读者只需掌握线性代数、基础分析和常微分方程知识。主要包括以下内容：单一阶方程的一般理论；波传播理论中的Huygens原理；弦振动；傅里叶方法；振荡理论和振动原理；调和函数特性；拉普拉斯基本解及位势；双层位势；球函数、麦克斯韦定理和可去奇点定理；用拉普拉斯方程解边界值问题；线性方程和线性系统理论。
作者简介
弗拉基米尔·阿诺德（Vladimir Igorevich Arnold，1937~2010），20世纪最伟大的数学家之一，动力系统和古典力学等方面的大师。俄罗斯科学院院士，1982年获首届Crafoord奖，2001年获Wolf奖，2008年获Shaw奖。
目录
Preface to the Second Russian Edition
1.The General Theory for One First-Order Equation
Literature
2.The General Theory for One First-Order Equation (Continued)
Literature
3.Huygens’ Principle in the Theory of Wave Propagation
4.1.The Vibrating String (d’Alembert’s Method)
4.1.The General Solution
4.2.Boundary-Value Problems and the Cauchy Problem
4.3.The Cauchy Problem for an Infinite String. d’Alembert’s Formula
4.4.The Semi-Infinite String
4.5.The Finite String Resonance
4.6.The Fourier Method
5.The Fourier Method (for the Vibrating String)
5.1.Solution of the Problem in the Space of Trigonometric Polynomials
5.2.A Digression
5.3.Formulas for Solving the Problem of Section 5.1
5.4.The General Case
5.5.Fourier Series
5.6.Convergence of Fourier Series
5.7.Gibbs’ Phenomenon
6.The Theory of Oscillations.The Variational Principle.adro 41
Literature
7.The Theory of Oscillations.The Variational Principle (Continued)
8.Properties of Harmonic Functions
8.1.Consequences of the Mean-Value Theorem
8.2.The Mean-Value Theorem in the Multidimensional Case
9.The Fundamental Solution for the Laplacian.Potentials
9.1.Examples and Properties
9.2.A Digression.The Principle of Superposition
9.3.Appendix.An Estimate of the Single-Layer Potential
10.The Double-Layer Potential
10.1.Properties of the Double-Layer Potential
11.Spherical Functions.Maxwell’s Theorem.The Removable Singularities Theorem
12.Boundary-Value Problems for Laplace’s Equation.Theory of Linear Equations and Systems
12.1.Four Boundary-Value Problems for Laplace’s Equation
12.2.Existence and Uniqueness of Solutions
12.3.Linear Partial Differential Equations and Their Symbols
A.The Topological Content of Maxwell’s Theorem on the Multifeld Representation of Spherical Functions
A.1.The Basic Spaces and Groups
A.2.Some Theorems of Real Algebraic Geometry
A.3.From Algebraic Geometry to Spherical Functions
A.4.Explicit Formulasa
A.5.Maxwell’s Theorem and CP2/conj≈S4
A.6.The History of Maxwell’s Theorem Literature
B.Problems
B.1.Material from the Seminars
B.2.Written Examination Problems
1.The General Theory for One First-Order Equation
Literature
2.The General Theory for One First-Order Equation (Continued)
Literature
3.Huygens’ Principle in the Theory of Wave Propagation
4.1.The Vibrating String (d’Alembert’s Method)
4.1.The General Solution
4.2.Boundary-Value Problems and the Cauchy Problem
4.3.The Cauchy Problem for an Infinite String. d’Alembert’s Formula
4.4.The Semi-Infinite String
4.5.The Finite String Resonance
4.6.The Fourier Method
5.The Fourier Method (for the Vibrating String)
5.1.Solution of the Problem in the Space of Trigonometric Polynomials
5.2.A Digression
5.3.Formulas for Solving the Problem of Section 5.1
5.4.The General Case
5.5.Fourier Series
5.6.Convergence of Fourier Series
5.7.Gibbs’ Phenomenon
6.The Theory of Oscillations.The Variational Principle.adro 41
Literature
7.The Theory of Oscillations.The Variational Principle (Continued)
8.Properties of Harmonic Functions
8.1.Consequences of the Mean-Value Theorem
8.2.The Mean-Value Theorem in the Multidimensional Case
9.The Fundamental Solution for the Laplacian.Potentials
9.1.Examples and Properties
9.2.A Digression.The Principle of Superposition
9.3.Appendix.An Estimate of the Single-Layer Potential
10.The Double-Layer Potential
10.1.Properties of the Double-Layer Potential
11.Spherical Functions.Maxwell’s Theorem.The Removable Singularities Theorem
12.Boundary-Value Problems for Laplace’s Equation.Theory of Linear Equations and Systems
12.1.Four Boundary-Value Problems for Laplace’s Equation
12.2.Existence and Uniqueness of Solutions
12.3.Linear Partial Differential Equations and Their Symbols
A.The Topological Content of Maxwell’s Theorem on the Multifeld Representation of Spherical Functions
A.1.The Basic Spaces and Groups
A.2.Some Theorems of Real Algebraic Geometry
A.3.From Algebraic Geometry to Spherical Functions
A.4.Explicit Formulasa
A.5.Maxwell’s Theorem and CP2/conj≈S4
A.6.The History of Maxwell’s Theorem Literature
B.Problems
B.1.Material from the Seminars
B.2.Written Examination Problems
