First, mathematical analysis.
Also called advanced calculus, the oldest and most basic branch of analysis. Generally speaking, it refers to a relatively complete mathematical subject with the general theory of calculus and infinite series as the main content, including their theoretical basis (basic theory of real number, function and limit). It is also a basic course for college mathematics majors.
The branch of analysis in mathematics is a branch of mathematics that specializes in studying real numbers and complex numbers and their functions. Its development began with calculus and extended to the continuity, differentiability and integrability of functions. These characteristics help us to study the material world and discover the laws of nature.
Second, advanced algebra
Elementary algebra begins with the simplest one-dimensional linear equation. On the one hand, elementary algebra further discusses binary and ternary linear equations, on the other hand, it studies equations that are larger than quadratic and can be reduced to quadratic. Along these two directions, algebra discusses the linear equations with any number of unknowns, also known as linear equations, and also studies the univariate equations with higher degrees.
This stage is called advanced algebra. Advanced algebra is a general term for the development of algebra to an advanced stage, including many branches. Higher algebra offered by universities now generally includes two parts: linear algebra and polynomial algebra.
Third, topology.
Topology is the study of some properties that geometric figures or spaces can remain unchanged after continuous shape changes. It only considers the positional relationship between objects, without considering their shapes and sizes. In topology, important topological properties include connectivity and compactness.
Some contents about topology appeared as early as the eighteenth century. Some isolated problems were discovered at that time. Later it played an important role in the formation of topology. For example, the problem of the Seven Bridges in Konigsberg, the polyhedral euler theorem and the four-color problem are all important problems in the history of topology development.
Fourth, probability theory and mathematical statistics
The main contents include: the basic concept of probability theory, random variables and their probability distribution, numerical characteristics, law of large numbers and central limit theorem, statistics and their probability distribution, parameter estimation and hypothesis testing, regression analysis, variance analysis, Markov chain and so on.
Probability theory and mathematical statistics are a unique and active branch of mathematics. On the one hand, it has a unique research topic, its own unique concepts and methods, rich content and profound results; On the other hand, it is closely related to other disciplines and is an important part of modern mathematics.
Verb (abbreviation of verb) theory of real variable function
The theory of real variable function is a branch of mathematics formed at the end of 19 and the beginning of the 20th century. Originated from classical analysis, the main research object is the function of independent variables (including multivariate) taking real values. The problems studied include the basic theories of function continuity, differentiability, integrability and convergence, which are the deepening and development of calculus.
Because it studies not only the functions in calculus, but also more general functions, and draws deeper and more general conclusions than the corresponding theories in calculus, the theory of real variable functions is the basis of all branches of modern analytical mathematics.
Baidu Encyclopedia-Mathematical Analysis
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