Mathematical analysis and advanced mathematics are two basic courses that are compulsory for mathematics majors, both of which involve the theory and methods of calculus, but there are obvious differences and connections between them. This paper analyzes and compares the differences between mathematical analysis and advanced mathematics from the aspects of teaching purpose, content scope, thinking methods, and application fields, aiming to help mathematics students better understand and learn these two courses and improve their mathematical literacy and ability.
Mathematical analysis and higher mathematics have different teaching purposes, reflecting their different emphases and requirements.
The purpose of the teaching of mathematical analysis is to cultivate students' mathematical analysis ideas and abilities, so that students can master the basic concepts, theorems, methods and skills of mathematical analysis, be able to use the tools and language of mathematical analysis, rigorously reason and prove mathematical problems, be able to find and solve mathematical problems, be able to read and understand mathematical literature, and be able to carry out mathematical innovation and research.
The purpose of teaching advanced mathematics is to cultivate students' mathematical application ideas and abilities, so that students can master the basic concepts, theorems, methods and skills of advanced mathematics, be able to use the tools and languages of advanced mathematics, carry out mathematical modeling and analysis of practical problems, be able to solve and apply mathematical models, be able to use computers for mathematical calculations and mathematics, and be able to comprehensively use mathematical knowledge to solve engineering and scientific problems.
The content range of mathematical analysis and higher mathematics differs, reflecting their different depth and breadth.
The content of mathematical analysis covers a wide range of real number systems, sequences and series, function limits and continuity, unary function differentiation, unary function integration, multivariate function differentiation, multivariate function integration, infinite series, Fourier series, ordinary differential equations, functional analysis, etc., some of which are not covered by advanced mathematics, such as the construction of real number systems, the properties of function spaces, the variation of functions, etc.
The content scope of advanced mathematics is narrow, mainly involving functions, limits, differentiation, integration, series, ordinary differential equations, vector analysis, multivariate function differentiation, double integration, curve integration, surface division, infinite series, etc., some of which are not focused on in mathematical analysis, such as the calculation of indefinite integrals, the application of integral transformations, and the solution of differential equations.
Mathematical analysis and the different methods of thought in higher mathematics reflect their different characteristics and styles.
The ideological method of mathematical analysis is abstract, rigorous and innovative, which emphasizes starting from the most basic axioms and definitions, through logical reasoning and proof, establishing a theoretical system of mathematical analysis, revealing the essence and laws of mathematical analysis, exploring the boundaries and expansion of mathematical analysis, creating new concepts and new theorems of mathematical analysis, and developing new fields and directions of mathematical analysis.
The thinking method of advanced mathematics is concrete, intuitive and applied, which emphasizes starting from practical problems, through mathematical modeling and analysis, using the tools and skills of mathematical analysis, solving mathematical models and applying mathematical results, solving engineering and scientific problems, demonstrating the effect and value of mathematical analysis, and promoting the application and development of mathematical analysis.
Mathematical analysis and higher mathematics have different fields of application, reflecting their different functions and roles.
The application fields of mathematical analysis are mainly mathematics itself, as well as some advanced theoretical physics, theoretical computers, theoretical biology and other fields, which provide basic mathematical theories and methods for these fields, provide support and guidance for research and discovery in these fields, and provide possibilities and impetus for innovation and breakthroughs in these fields.
The application fields of advanced mathematics are mainly various branches of engineering and science, such as machinery, electronics, communications, chemical engineering, materials, energy, environment, biology, medicine, economics, management and other fields, which provide practical mathematical models and technologies for these fields, provide basis and tools for the design and experiment of these fields, and provide ways and means for the optimization and improvement of these fields.
Conclusion. Mathematical analysis and advanced mathematics are two different but interrelated courses, both of which are compulsory courses for mathematics majors, and both are important parts of mathematics. Mathematics students should arrange their study plans reasonably according to their own interests and goals, make full use of textbooks, teachers, networks and other resources to fully master the knowledge and skills of mathematical analysis and advanced mathematics, improve their mathematical literacy and ability, and lay a solid foundation for future study and work.