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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