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Tsinghua Edition Mathematical Physics Method
This book is revised on the basis of Methods of Mathematical Physics (for graduate students) published by Beijing University of Posts and Telecommunications Press. This revision not only adjusts the contents of some chapters to make them more suitable for teaching, but also mainly increases the application of computer software Maple in solving definite solutions and visualizing some results with Maple. After the book 1 was published in June 2003, it was printed for the second time in September 2005 thanks to the care and love of teachers, friends and readers. This revision mainly adds the applied mathematical software Maple to help solve the definite solution problem of mathematical physics, and visualizes some results with Maple. Because Mathematical Physics Method is one of the basic courses for many science and engineering students, it has many applications in the follow-up courses and scientific research after completing their studies. Students are required to clearly understand the concepts, master the problem-solving methods and understand the physical meaning of the results. However, the course itself is difficult because of its many contents and complex topics, which is recognized as a difficult course, mainly reflected in the derivation of many formulas and the calculation of complex integrals or series in problem-solving exercises. With the popularization of computers, powerful mathematical software (such as Maple, etc. ) provides a powerful tool for solving complex mathematical problems. The purpose is: (1) complex mathematical operations, such as solving ordinary differential equations, calculating integrals, solving complex algebraic equations, etc. All of them are completed with the help of computers, so that readers can pay more attention to the establishment of models (mathematical and physical equations), the formation of physical ideas and the application of mathematical methods to the theoretical system of physical processes; (2) With the help of the powerful visual function of the computer, it is undoubtedly beneficial for readers to understand and master some abstract but very useful knowledge by turning it into vivid and "vivid" physical images. Maple, a mathematical software, has powerful symbolic operation function. Its greatest advantage is that it can directly perform symbolic operation without programming, so readers do not need to learn programming knowledge, let alone learn based on programming, which will bring great convenience to readers.

In addition to the above contents, this revision has also made the following adjustments to the original contents: Chapter 1 "Preliminary field theory" has been changed to "Preliminary vector analysis and field theory", the content of vector analysis has been added, and the gradient and tensor of vector field and their calculation and dyadic analysis have been deleted; The fifth chapter "special function" is divided into two chapters: "special function (1)- legendre polynomials" and "special function (2)- Bessel function". In the chapter of "Variational Method", the derivation of complex functional Euler equation is added, because the problem of solving complex variational problems is often encountered in mathematical and physical problems; In the chapter "General properties and solutions of integral equations", the corresponding solutions are explained according to the types of integral kernels, which makes the content clearer and more systematic. The text of the book has been rewritten or modified, and several typographical errors in the 1 version have been corrected. The content marked with * in the book can be used as elective content, and readers can choose according to their needs.

The author is very grateful to Tsinghua University Publishing House for its strong support and help to the second edition of this book, and especially thanks to two editors, Liu Ying and Wang Haiyan, whose rigorous and hardworking professionalism is admirable. 1 chapter vector analysis and preliminary field theory.

Vector function of 1. 1 and its derivatives and integrals

1. 1. 1 vector function

Limit and continuity of vector function 1. 1.2

Derivative and Integral of Vector Function 1. 1.3

Expressions of 1.2 gradient, divergence and curvature in orthogonal curvilinear coordinate system

"Three Degrees" and Hamiltonian in 1.2. 1 Cartesian Coordinate System

1.2.2 "three degrees" in orthogonal curvilinear coordinate system

1.2.3 "three degrees" operation formula

Laplace operator, Green's first and second formulas in 1.3 orthogonal curvilinear coordinate system.

1.4 operator equation

Chapter II Problems of Definite Solutions in Mathematical Physics

2. 1 Establishment of basic equation

2. 1. 1 Small transverse vibration of uniform string

2. Small transverse vibration of1.2 uniform membrane

2. 1.3 transmission line equation

2. 1.4 electromagnetic field equation

2. 1.5 heat conduction equation

2.2 definite solution conditions

2.2. 1 initial conditions

boundary conditions

2.3 formulation of definite solution problem

2.4 Classification and simplification of second-order linear partial differential equations

2.4. 1 Classification and simplification of two independent variable equations

2.4.2 Further simplification of partial differential equations with constant coefficients

2.4.3 superposition principle of linear partial differential equations

Chapter III Separation of Variables Method

3. 1( 1+ 1) Vezi sub-equation.

3. 1. 1 Free vibration of bounded strings

3. 1.2 Heat conduction on a finite rod

The definite solution of 3.22-dimensional Laplace equation

3.3 High-dimensional Fourier series and its application in high-dimensional definite solution problems

3.4 Solutions of nonhomogeneous equations

3.4. 1 Intrinsic Function Method

impulse method

Special solution

3.5 Treatment of Non-homogeneous Boundary Conditions

The fourth chapter is the series solution of the eigenvalue problem of second order ordinary differential equations.

4. 1 the relationship between coefficients and solutions of second-order ordinary differential equations

4.2 Series Solution of Second Order Ordinary Differential Equation

4.2. 1 series solution in the neighborhood of invariant points

4.2.2 Series Solutions in the Neighborhood of Regular Singularity

4.3 Series Solution of Legendre Equation

4.4 Series Solution of Bessel Equation

4.5Sturm? Joseph Liouville eigenvalue problem

The fifth chapter special function (1) legendre polynomials

5. 1 separation of variables in orthogonal curvilinear coordinate system

5. 1. 1 Laplace equation

5. 1.2 Helmholtz equation

5.2 legendre polynomials and its nature

5.2. Derivation of1legendre polynomials

The nature of legendre polynomials

5.3 Application of legendre polynomials

5.4 General spherical function

5.4. 1 correlation Legendre function

Spherical function

Chapter VI Special Functions (2) Bessel Functions

6. Properties and applications of1Bessel function

6. 1. 1 column function

6. Properties of1.2 Bessel Function

6. 1.3 Modified Bessel Function

6. The application of1.4 Bessel function

6.2 Spherical Bessel Function

6.3 Cylindrical Wave and Spherical Wave

6.3. 1 cylindrical wave

spherical wave

6.4 Equation that can be transformed into Bessel equation

6.5 Introduction of other special function equations

hermite polynomial

6. 5. 2 Lagrange polynomials

Chapter VII Traveling Wave Method and Integral Transformation Method

7. 1 D'Alembert formula of one-dimensional wave equation

7.2 Poisson formula of three-dimensional wave equation

7.3 Fourier integral transformation method to solve the problem

7.3. 1 Preliminary knowledge-Fourier transform and its properties

Fourier transform method

7.4 Spatial transformation method for solving definite solution problems

7. 4. 1 laplace transform and its characteristics

7. 4. 2 Platform conversion method

Chapter VIII Green's Function Method

8. 1 Introduction

8.2 Boundary Value Problems of Poisson Equation

8. 2. 1 green formula

8.2.2 Integral form of solution-Green's function method

8. 2. 3 Green's Green function is symmetric about the source point and the field point.

8.3 General solution of Green's function

8.3. Green's function of1unbounded region

8.3.2 Solving Green's Function of Boundary Value Problem by Eigenfunction Expansion Method

8.4 Find Dirichlet in some special areas by electric image method? Green's function

8.4. Dirichlet of1Poisson equation? Green's function and its physical significance

8.4.2 Solving Green's Function by Electric Image Method

*8.5 The definite solution of Green's function with time

Chapter 9 Variational Method

9. 1 functional and extreme value of functional

9. 1. 1 functionality

9. Extreme value of1.2 functional and change of functional

9. 1.3 Necessary Condition of Functional Extremum —— Euler Equation

Euler equation of 9. 1.4 complex functional

9. Conditional Extreme Value Problem of1.5 Functional

9. 1.6 A Direct Method for Finding Functional Extremum —— Ritz Method

9.2 Solving Mathematical and Physical Equations by Variational Method

9.2. 1 Relationship between eigenvalue problem and variational problem

9.2.2 Find the eigenvalue by finding the extreme value of the functional.

9.2.3 Relationship between Boundary Value Problems and Variational Problems

*9.3 Variational principle and approximate calculation related to waveguide

9.3.1* * Variational principle of vibration frequency

9.3.2 Variational principle of waveguide propagation constant γ

9.3.3 Approximate Calculation of Cut-off Frequency of Cylindrical Waveguide with Arbitrary Section

10 general properties and solutions of integral equations

The concept and classification of 10. 1 integral equation

Iterative solution of 10.2 integral equation

Iterative solution of Volterra equation of the second kind 10.2. 1

10.2.2 Iterative Solutions of Volterra Equation of the First Kind

10.2.3 Iterative solutions of Fredholm equation of the second kind.

10.2.4 overlapping kernel and pre-decomposition kernel

The solution of 10.3 degenerate kernel equation

The Solution of Abel Equation for 10.4 Weak Singular Kernel

Fredholm equation of 10.5 symmetric kernel

10.6 relationship between differential equation and integral equation

10.6. 1 the relationship between second-order linear ordinary differential equation and Volterra equation

10.6.2 relationship between eigenvalue problem of differential equation and symmetric kernel integral equation

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