I.The Basic Information of the Course
Course Number:202032020001
The English Name of the Course：Mathematics method in physics
The Chinese Name of the Course：数学物理方法A
In-class Hours and Allocation:total class hours: 48, classroom teaching: 48 class hours
Credit(s):2
Semester:Fall semester
Applied Discipline(Professional Degree Category): Science and Engineering Discipline
Course Object Oriented：Academic Master,Academic Doctor
Evaluation Mode:closed-book exam
Teaching Method:Mixed teaching
Course Opening Department: Collegeof Mathematical Sciences
II. Prerequisite Course:
Advanced mathematics；Linear algebra；College physics；Complex function；Ordinary differential equation；
III. The Objectives and Requirements of the Course
Introduce the commonly used mathematical methods in the fields of physics, mechanics, engineering technology, etc., and use the mathematical tools introduced in the course of mathematical physics methods to cultivate students' ability to scientifically and accurately describe many phenomena that appear in the field of nature and science and technology, as well as calculate the corresponding results accurately. Through the study of this course, students are required to master the theory of differentiation, analysis, series, and integration, and learn how to use the theory of complex variable functions to study the nature of functions and solve more complex real integration problems. To enable students to master the methods of solving partial differential equations commonly used in solving practical problems, such as the separation variable method, the Green function method, and the application of special functions, and also improve the ability to use mathematical physics methods to solve problems in the field of science and technology.
IV. The Content of the Course
Review the basic concepts of complex numbers, analytic function content (4class hours): including complex numbers and operation rules, geometric representation of complex numbers, complex sequence and limit concepts, complex functions, regions, continuity, derivability, analytic concepts, analytic Koshi --Riemann condition, the relationship between analysis and harmony, etc., the integration of elementary functions and complex variable functions.
Multi-valued functions (4class hours): Including root functions and logarithmic functions. Inverse trigonometric functions, power functions, etc.
Taylor expansion and Luolang expansion, isolated singularities of single-valued functions (2class hours): including singularities and their classification, without singularities, extreme points, and natural singularities.
Residual number theory and its application in integration (4class hours).
Introduction of Laplace transform and Fourier transform, series solution to linear ordinary differential equation (4class hours).
Practical examples of derivation on and basic concepts of solutions to fixed problems for partial differential equations (2class hours).
The application of discrete number solution in string vibration equation, heat conduction equation and harmonic equation (2class hours).
Solution of forced vibration equation, treatment of non-homogeneous boundary conditions, Dirichlet problem in a circle (2class hours).
Traveling wave method is used to solve the problem of infinite string vibration and the initial value problem of high-dimensional wave equation (4class hours).
The eigenvalue problem of ordinary differential equations, the separation variable of Helmholtz equation in different coordinate systems (2class hours).
Representation and properties of Legendre functions and their applications in solving mathematical physics equations (4class hours).
Representation and properties of Bessel functions and their applications in solving mathematical physics equations (6class hours)
Green function and its properties, generalized Green function, basic solution, use of generalized function to solve equations, method of finding Green function on special area (6class hours)
Introduction of the basic idea of integral transformation and variational methods, integral equations, nonlinear problems, inverse problems, etc. (2 class hours)
Course for Ideological and Political Education
This course briefly introduces the famous Chinese mathematics experts and deeds in the history of the development of this course. Combined with the actual classroom teaching, the lecturer chooses to teach according to the materials. By analyzing the patriotic feelings and scientific spirit of the experts, the students will be promoted imperceptibly patriotism education and cultural confidence.
1.Multiple complex variable function, Chinese mathematician's garden
The multiple complex variable function theory discipline was founded in China. From the beginning, it aimed at international standards, which can be regarded as a banner of the Chinese mathematics community. On June 9, 2014, the 17th Academician Conference of the Chinese Academy of Sciences, the 12th Academician Conference of the Chinese Academy of Engineering, the General Secretary of the CPC Central Committee, the State President, and the Chairman of the Central Military Commission Xi Jinping delivered a speech reviewing the technology that China has achieved since the founding of New China Achievements, when listing basic scientific breakthroughs, ranked "multiple complex variable function theory" in second place, after "two bombs and one star".
Mr. Hua Luogeng is the founder of the study of multiple complex variable function in China. His achievements on multiple complex variables were mainly summarized in "Hua Luogeng's Anthology: Multiple Complex Variable Function Theory Volume 1". In 1953, Hua Luogeng pioneered the use of group representation theory to obtain the complete orthogonal system of four types of typical domains, thus obtaining the Cauchy-Szego kernel, Bergman core and Poisson core of the four types of typical domains. As W. Rudin pointed out, until Hua Luogeng's work came out, people couldn't even write the Cauchy core of the unit ball. On this basis, Hua Luogeng and Lu Qikeng established the harmonic function theory on the typical domain, and solved the Dirichlet problem of the harmonic function. In this process, Hua Luogeng discovered a group of differential operators with properties similar to harmonic operators, which are internationally known as "Fahrenheit operators". Hua Luogeng's work was summarized in a monograph "Harmonic Analysis on Typical Fields of Multiple Complex Variables", which has been translated into Russian and English, which is not only important for function theory, but also for Lie group representation theory, differential geometric homogeneous space, and multiple complex variable auto-defense function theory.
It was his student Mr. Lu Qikeng who succeeded Mr. Hua Luogeng. He proved that the complete boundary analysis of constant curvature is equivalent to the unit hypersphere. This result is called "Lu Qikeng's theorem" and is generally recognized by the international mathematics community. On this basis, Lu Qikeng proposed to be known as the "Lu Qikeng Conjecture" internationally, which is the first conjecture named after Chinese mathematicians in the international mathematics community since the founding of the People's Republic of China. In addition, "Lu Qikeng Invariant" and "Lu Qikeng Constant" are named after Lu Qikeng.
2.A pioneer in the study of partial differential equations in China
Wu Xinmou is a mathematician. He is one of the main founders of partial differential equation research in China, and established the differential equation laboratory of the Institute of Mathematics, Chinese Academy of Sciences. He compiled the first textbook of partial differential equation theory in my country, and also hosted a nationwide large-scale workshop, making a major contribution to building a research team in this field in my country.
Professor Wu Xinmou studied abroad in France in his early years and was engaged in the research of fluid mechanics, focusing on the stability study of viscous fluid motion. The results he obtained are of great significance for overcoming the difficulties faced by the small motion method commonly used in classical fluid motion stability theory. Since the 1950s, Professor Wu Xinmou has focused on the research of mixed equations and created a "zero integral" method. This paper was used by the Courant Institute of the United States as a designated reading document for graduate students.
Professor Wu Xinmou returned to China with his family in 1951. He devoted himself enthusiastically to the cause of mathematics in the people's Republic of China. Under almost blank conditions, a seminar on PDE was set up, which is the first large-scale seminar in China with the theme of modern PDE theory. About 100 teachers were sent to the seminar by colleges and universities across the country, including Gu Chaohao, Qi Minyou, Xiao Shutie, Wu Zhuoqun, Dong Guangchang, etc. The handout prepared by Wu Xinmou and used in the workshop was officially published soon. It is the first specialized textbook of partial differential equation theory after the establishment of the people's Republic of China, and also a distinctive research reference book. Since then, many mathematics departments of colleges and universities have offered the course of partial differential equation theory, and the backbone team of partial differential equation work nationwide has begun to form.
Through the analysis of the fruitful achievements of this course, many Chinese contemporary mathematicians have made contributions to enable students to understand the history and current situation of the development of this course, learn the hard work spirit of the older generation of mathematicians, increase cultural self-confidence and pride, and provide spiritual impetus for the efforts to study scientific and cultural knowledge.
3. A mathematician with a physical disability, played the strongest sound on the broken piano.
Academician of the Chinese Academy of Sciences Lu Qikeng has made great contributions to the development of multiple complex variable functions and mathematical physics in China. His life is quite legendary. He was disabled by poliomyelitis and legs when he was young, and his family was poor.
Under the guidance of Hua Luogeng, Lu Qikeng trained the first team in the field of multiple complex variable function research in China by organizing a seminar on multiple complex function theory in the Institute of Mathematics and a special class on multiple complex variable functions in the Department of Mathematics of Peking University. Their research achievements have also reached international leadership, especially Lu Qikeng. Lu Qikeng and Zhong Tongde cooperated to publish the "Promotion of Privalov's Theorem" and independently completed "Schwarz lemma and analytic invariants" ("Schwarz lemma and analytic invariants") and other papers, which attracted the attention of the international mathematical community, and some of the results were written in the monograph of the Soviet mathematician. The results he achieved on the proof of Schwartz's lemma are now called "Lu Qikeng's lemma". On the occasion of the tenth anniversary of National Day in 1959, Lu Qikeng summarized the study of multiple complex changes in China over the past decade as a gift, which attracted the attention of the American Mathematical Society. It was Hua Luogeng and Lu Qikeng's extensive pioneering work that enabled the multiple complex variable function subject to be initiated in New China.
In the 1960s, Lu Qikeng made breakthrough progress in the field of multiple complex variable function theory. His paper "K? Hler Manifolds about Constant Curvature" proves that the complete boundary resolution of constant curvature is equivalent to the unit hypersphere. This result is called "Lu Qikeng's Theorem" and is widely accepted by the international mathematics community. It is recognized that it has been leading Western counterparts for nearly two decades and is still widely cited today. On this basis, Lu Qikeng put forward a conjecture, that is, whether the kernel function of the bounded domain has a zero point as a function of two points. It is called the "Lu Qikeng conjecture" in the world, and the domain with no zero point of the nuclear function is called "Lu Qikeng domain". "Lu Qikeng's conjecture" is the first conjecture named by Chinese mathematicians in the international mathematics community after the founding of the People's Republic of China. In addition, "Lu Qikeng's invariants" and "Lu Qikeng's constant" are named after Lu Qikeng. A scientist's name appears in successive mathematical concepts, enough to illustrate his contribution in mathematics.
V. Reference Books, Reference Literatures, and Reference Materials
1.郭敦仁.《数学物理方法》.人民教育出版社,1993。
2．管平,计国君,黄骏.数学物理方法.高等教育出版社，2001。
3．胡嗣柱,倪光炯.数学物理方法.高等教育出版社,2002。
4．陆全康,赵慧芬.数学物理方法.高等教育出版社，2007。
5．刘连寿,王正清.数学物理方法.高等教育出版社，1990。
6．吴崇试.数学物理方法(第2版).北京大学出版社，2018。
7．姚端正.数学物理方法(第3版).科学出版社，2013。
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