The "usefulness" of the main subjects of higher mathematics. Not to mention middle school mathematics, which is arithmetic in the eyes of mathematicians. Some branches developed purely for application, such as probability statistics, discrete mathematics, operational research, cybernetics, etc., are not mentioned, and the basic aspects are emphasized.
Mathematical analysis: mainly including calculus and series theory. Calculus is the foundation of advanced mathematics, and it has a wide range of applications. Basically, all fields involving functions need knowledge of calculus. Among them, Fourier series and Fourier transform are mainly used in the field of signal analysis, including filtering, data compression and power system monitoring. And the manufacture of electronic products is also inseparable from it.
Real variable function (real analysis): one of the enhanced versions of mathematical analysis. Mainly used in fields such as economics that focus on data analysis.
Complex function (complex analysis): the second enhanced edition of mathematical analysis. A widely used subject, widely used in aviation mechanics, fluid mechanics, solid mechanics, information engineering, electrical engineering and other fields, so engineering students should take this course.
Higher algebra mainly includes linear algebra and polynomial theory. Linear algebra can be said to be a branch of mathematics that is widely used at present. Data structure, program algorithm, mechanical design, electronic circuit, electronic signal, automatic control, economic analysis, management science, medicine, accounting, etc. Everyone needs the knowledge of linear algebra, which is a compulsory course for students majoring in economics, engineering and computer.
Advanced geometry: including spatial analytic geometry, projective geometry, spherical geometry, etc. , mainly used in architectural design and engineering drawing.
Analytics, advanced algebra and advanced geometry are the three pillars of modern mathematics.
Differential equation: including ordinary differential equation and partial differential equation, one of the important tools. Fluid mechanics, superconducting technology, quantum mechanics, stability analysis in mathematical finance, material science, pattern recognition, signal (image) processing, industrial control, power transmission and distribution, remote sensing measurement and control, infectious disease analysis, weather forecast and other fields are all needed.
Functional analysis: mainly studies functions in infinite dimensional space. Because it is abstract, it is not widely used directly in technology, and is generally used in continuum mechanics, quantum physics, computational mathematics, infinite dimensional commodity space, cybernetics, optimization theory and other theories.
Modern algebra (abstract algebra): It mainly studies various axiomatic abstract algebraic systems. There is no application in technology, but it is used more in physics, especially group theory.
Topology: the study of the invariance of sets under continuous transformation. It is widely used in natural science, such as the classification of liquid crystal structural defects in physics, molecular topological configuration in chemistry, DNA ring and topoisomerase in biology. In addition, the game theory in economics also has a very important application.
Functional analysis, modern algebra and topology are three popular branches of modern mathematics.
Non-Euclidean geometry: mainly used in physics, the most famous is the theory of relativity.