Since childhood, mathematics has been growing with us, from 1+ 1 to four operations, to solving equations, to function geometry, to limit and infinity. Our cognition has progressed step by step from simple arithmetic, and has basically reached a more abstract functional world under the urging of the educational system of the motherland. But what is mathematics? Maybe, we don't have a very macro concept yet.

Mathematics, in the eyes of many people, still stays at the level of calculation, but it is not. At present, there are at most three majors in domestic mathematics: computational mathematics, mathematics and applied mathematics, and statistics. Since up itself is a mathematical correspondence, strictly speaking, only numbers should be the most suitable for the whole mathematical theoretical system, so this paper only involves mathematical correspondence and talks about basic mathematics in detail.

Basic mathematics is divided into several parts, and the classical classification is: geometry, algebra, analysis and differential equations.

First, geometry (study shape)

Gas refining period: elementary geometry (properties of two-dimensional and three-dimensional classical graphics)

Basic period: analytic geometry (considering the coordinate representation of points, lines and surfaces in three dimensions and extending it to the nature of N-dimensional graphic coordinates)

Differential geometry (using parametric variables to represent two-dimensional and three-dimensional smooth graphics, exploring the curvatures of several strange shapes, and extrapolating straight lines on the plane to geodesics on the surface to represent the shortest distance)

Yuan Ying Period: Riemannian Geometry (When a surface is used as a canvas and geodesic is used as the shortest path, how should we consider the differential properties of the graphics on it? Manifold: when it moves, how will it consider the changing relationship before and after moving? )

Deification period: geometric research

Second, algebra (multi-dimensional study of everything, representing everything)

Gas refining period: understand polynomials and the significance of setting unknowns.

Basic period (1): linear algebra (understanding determinant, matrix operation and matrix transformation, eigenvalue)

This topic can be said to be an abstract starting point, but unlike primary schools, 1+ 1=2 can be compared to 1 apple+1 apple = two apples first; Starting from 1+ 1=2. In fact, in order to study more practical problems, we should first learn the mathematical tool of matrix, which is similar to understanding 1+ 1=2 first, and then considering the objective facts corresponding to this 1, 2.

But because the facts behind the matrix are very complicated, in order to save time, we should learn its operation and characteristics first. When learning this stage, we can first consider the N-dimensional vector and its changes in the N-dimensional space as abstract considerations.

Basic period (2): advanced algebra (consistent with linear algebra, with more proof components)

The stage of knot Dan: abstract algebra (group, ring, field)

Among them, groups are produced through the study of symmetry. For example, a regular triangle has three axisymmetries and three rotational changes (120 degrees, 240 degrees, 360 degrees). These transformations cover all symmetries (self-sealing) and are reversible. 360-degree rotation is an identity transformation, so the set of these transformations constitutes a group.

Rings are born for modules, the set of matrices is rings, and rings are also modules.

Fields are numeric fields, such as real number fields and complex number fields.

Yuan's infancy: algebraic representation theory

Sometimes an algebra (a set of elements and operations) is difficult to represent, so we use linear space to represent it. Because many times the operation itself is important, homomorphism (preserving operation) is used to reasonably link the two.

Deification period: algebraic research

Third, analysis (learning continuous function and continuous mapping)

Gas refining period: function, mapping, set (function is from counting to counting; The mapping from set to set is a sign of change)

Basic stage: calculus (gradient differential, area integral and volume integral)

Nodding period: learn real variable function ten times (give the general set distance and map it to real number field with continuous function; Consider defining calculus in this sense)

Yuan's infancy: complex variable function (understand complex domain continuous function, complex number to complex number; Similar to the mapping from two-dimensional plane to two-dimensional plane, but there are some interesting conclusions due to the definition of complex numbers)

Deification period: chilling of functional analysis (consider the mapping from arbitrary set to real number field or complex number field, because mapping itself is the representation of transformation, so it is also called "operator" and "functional")

Next field: analysis and research

Fourth, differential equations.

Gas refining period: solving equations

Basic stage: ordinary differential equation

Node critical period: partial differential equation