Advanced Mathematics

Author: Zhang Wanguo
Publisher:
Publish Date: 2003-01-01
Features: This book is a supplementary reader for the "Advanced Mathematics" course, written for non-mathematics majors at various universities. It provides in-depth analysis and discussion of some important and challenging concepts in advanced mathematics, helping readers firmly grasp these fundamental concepts. At the same time, through a large number of example analyses and the introduction of problem-solving methods, it enhances readers' ability to analyze and solve problems, broadening their perspectives. The book contains 8 chapters, covering single-variable calculus, spatial analytical geometry, multivariable calculus, infinite series, ordinary differential equations, etc. Each chapter is followed by a large number of exercises for further practice. The answers or hints to some exercises are provided at the end of the book. This book can be used as a reference for students at various universities studying advanced mathematics, as well as a preparatory review book for those taking the "Advanced Mathematics" course for postgraduate entrance exams. It also holds certain reference value for mathematics teachers at universities.
Preface
Over 10 years ago, several teachers in our department compiled a series titled "University Mathematics Learning Guide." After its publication, the series was very well-received and quickly sold out. Subsequently, colleagues from other institutions and many young students wrote to us requesting copies, and the publisher also contacted us multiple times about a second edition. However, the authors, who have long been burdened with heavy teaching and research tasks, were unable to revise the book. In recent years, with the development of the discipline, curriculum construction has once again become a priority. New textbooks for some important foundational courses in our department have been published, while many teachers have also expressed the desire to organize and summarize their teaching experiences. With the encouragement of Fudan University Press, it has become timely and feasible to launch this brand-new series.
The development of mathematical science is currently in an extraordinary period. Advances in science and technology, increased practical applications, the influence of computers, and the progress of mathematical science itself have greatly expanded the scope and fields of mathematical science. In many contexts, mathematics has moved from the background of scientific research to the forefront of technological application, becoming the key to unlocking numerous opportunities. This has led to ever-increasing societal demands for mathematical proficiency, expecting more mathematically grounded individuals with strong innovative abilities, broad knowledge, and outstanding comprehensive qualities. Correspondingly, the goals of mathematical education have evolved beyond merely imparting professional knowledge to guiding students to master a scientific language, learn a rational mode of thinking, and receive training in mathematical qualities such as deduction, induction, analysis, and analogy. Effective mathematical training will prepare students to fully participate in future competition.
The theory of mathematics is wonderful and fascinating; its methods are exquisite and diverse. However, mastering mathematics requires arduous effort. In teaching, we often encounter students who can recite basic formulas but struggle with slightly modified calculations, or who can remember basic theorems but fail to provide well-structured reasoning. Some students still cling to the "pattern-based teaching" they received in early years, which was inappropriately exaggerated for exam purposes, and they do not understand why this teaching method has disappeared from university classrooms. These students' approaches to learning mathematics are often immature, limiting their understanding to superficial knowledge rather than achieving a deep grasp. In reality, compared to most other disciplines, mathematics offers students more opportunities for independent and critical thinking. In the learning process of any mathematics course, the primary role is not the teacher but the student. The process of learning mathematics should be one of re-creation. Students should interpret and analyze the content based on their own understanding, transform their existing knowledge with new perspectives, and thus leave their unique in their personal repository of mathematical knowledge. Only through deep reflection, by organically integrating newly acquired knowledge into their existing knowledge structure and experiencing mathematics through the creative power of their minds, can they truly master the abstract concepts and develop intuitive understanding.
We hope this series can provide students with valuable references and inspiration in their mathematical learning. Although methods of learning mathematics vary from person to person, some fundamental aspects of the course are worth common attention. First, to excel in a mathematics course, it is undeniable that one must master the basic mathematical ideas it encompasses. This means not only deeply understanding the concepts and properties of the main objects but also scientifically integrating a series of definitions and theorems, grasping the overall development, progression, and advancement of the knowledge system, and comprehensivelying the mathematical ideas and spirit that permeate the course. Second, mathematical ideas are realized through specific mathematical methods. The methods embedded in each course provide the primary tools for constructing corresponding theoretical frameworks and offer the basis for making analytical, judgmental, transformative, and problem-solving strategies. From the formation of conjectures to the expansion of analysis, and from the implementation of calculations and reasoning to the refinement and broadening, mathematical methods demonstrate their value in the process of problem-solving. Third, each mathematics course contains many special mathematical techniques. These not only showcase the flexibility of operations and arguments but are also essential elements of successful mathematical methods. A profound technique often stems from rich imagination and keen observation. The introduction and training of mathematical techniques are of great significance for stimulating, broadening, and deepening students' thinking.
In summary, mathematical ideas, methods, and techniques form an integral whole, collectively constituting the rich and vibrant essence of each mathematics course. Therefore, to excel in mathematics, one must focus on understanding ideas, mastering methods, and becoming proficient in techniques. Based on this understanding, each book in this series generally includes introductions and summaries of concepts and properties, analyses and discussions of main methods and typical examples, along with a certain number of exercises. We hope readers can follow this tripartite structure—thinking and internalizing mathematical ideas, methods, and techniques—to elevate their problem-solving abilities to a new level.
The authors of this series all have extensive teaching experience and have taken care to address the diverse needs of readers: whether used synchronously while studying a course, as a reference for comprehensive review after completing a course, or as preparation for postgraduate entrance exams, readers will gain significant benefits. We are also willing to use this series as a platform for broad teaching exchanges with colleagues from other institutions. The Department of Mathematics at Fudan University has included the compilation of this series in its plan to strengthen undergraduate teaching. Many professors from the Department and Research Institutes have offered valuable suggestions on how to compile this series, creating favorable conditions for improving its quality. Ms. Fan Renmei of Fudan University Press has devoted considerable effort to the planning and editing of this series. We extend our sincere gratitude to all the aforementioned individuals.
Given our limitations, it is inevitable that this series contains errors and shortcomings. We sincerely hope that readers will kindly point out any issues. Through the joint efforts of authors and readers, we hope that subsequent revisions will make this series increasingly mature.