High number, linear generation and probability are the three major difficulties in postgraduate mathematics. Only by mastering the discipline law and proposition law of mathematics can we better plan and arrange intensive reading, and we need to analyze the breakthrough of mathematics.
Laws of three disciplines
First, high numbers.
(1) Learn more
Higher mathematics can be divided into univariate function calculus and multivariate function calculus in a big way.
One-dimensional calculus includes limit, derivative, indefinite integral and definite integral; Multivariate function calculus includes multivariate function differential calculus (mainly binary function) and multivariate function integral calculus. There are also differential equations and series, which can be regarded as the application of calculus.
In addition, there are vector algebra and spatial analytic geometry. Among them, the number one is the triple integral, curve integral and surface integral in vector algebra, spatial analytic geometry and multivariate function integral, and the other is the part of the number one, which has some subtle differences.
Generally speaking, the review time of advanced mathematics is the most, and its success or failure is directly related to the success or failure of postgraduate entrance examination.
(2) A clear sense of modularity
Answer the advanced math questions one by one, and answer them in the first class. For example, the sum expansion of power series, remember several common Taylor series formulas, and transform known functions (or series) into common formulas through basic deformation or derivative quadrature, and this kind of problems will be basically solved. Line generation is not like this, you can learn basic questions, and you will have no bottom when you get deep into the exam.
Second, probability.
The knowledge structure of probability is an inverted tree structure. In the first chapter, random events and probability are the basis, and random variables are introduced on this basis, and distribution is the description of random variables. The second and third chapters introduce random variables and their distribution. Distribution describes all the information of random variables, while digital features only describe part of the information (for example, the mathematical expectation of discrete random variables can be understood as the average value of random variables in the sense of probability).
Then discuss the theoretical basis of total probability-the law of large numbers and the central limit theorem. So much for probability theory. Mathematical statistics is regarded as the application of probability theory.
Third, line generation
The knowledge structure generated by line is a network structure: there are many connections between knowledge points, which are interlaced into a network. Take the invertibility of matrix A as an example, please consider what the equivalent conditions are. From the point of view of vector group, the column vector group (or row vector group) of matrix A is linearly independent; From the perspective of determinant, the determinant of matrix A is not zero; From the point of view of linear equations, Ax=0 has zero solution (or Ax=b has unique solution); From the perspective of quadratic form, a is transposed by positive definite, and from the perspective of rank, the rank of matrix is the order of matrix; From the point of view of eigenvalue, the eigenvalue of matrix does not contain zero. It is not difficult to find that the basic concept of matrix reversibility can string together the generation of the whole line.
Review methods and difficulties of three major subjects
● Advanced Mathematics
(1) review points: the solution of limit; Application of variable limit integral: derivative application; Calculation of multiple integrals.
(2) review method:
Higher mathematics should strengthen the training of solving comprehensive test questions and application problems, and strive to make a breakthrough in solving problems. Pay attention to the investigation of comprehensive questions. Generally speaking, the content of comprehensive questions can be different chapters of the same subject or different subjects. In recent years, the common comprehensive questions in test papers are: the comprehensive questions of series and integral; The synthesis of calculus and differential equations; Comprehensive problem of seeking limit; Spatial analytic geometry and differential synthesis of multivariate functions; Synthesis of linear algebra and spatial analytic geometry; And the application of calculus and differential equations in geometry, physics and economy. When solving comprehensive problems, it is a key step to find the breakthrough point quickly, so you need to be familiar with standardized problem-solving ideas.
(3) Summary of key problems of high number
● Linear algebra
(1) Review points: determinant and matrix formula; Solutions of linear equations; Similar diagonalization problem.
(2) review method:
There are many concepts in linear algebra. Important are: algebraic cofactor, adjoint matrix, inverse matrix, elementary transformation and elementary matrix, orthogonal transformation and orthogonal matrix, rank (matrix, vector group, quadratic form), equivalence (matrix, vector group), linear combination and linear representation, linear correlation and linearity independence, maximal linearity independence, basic solution system and general solution, solution structure and solution space, etc.
There are many arithmetic rules in linear algebra, which should be sorted out and not confused. It is important to calculate determinant (number type and letter type), find the inverse matrix, find the rank of matrix, find the power of square matrix, find the rank of vector group unrelated to maximum linearity, judge or find the parameters related to linearity, find the basic solution system, find the general solution of non-homogeneous linear equations, find the eigenvalue and eigenvector (definition method, characteristic polynomial basic solution system method).
Linear algebra is criss-crossed, interlocking and interpenetrating in content, so the method of solving problems is flexible and changeable. When reviewing, you should always ask yourself if you have done it right. One more question, okay? Only by constantly summing up and trying to figure out the internal relations among them, so that the knowledge learned can be integrated, the interface and breakthrough point can be more familiar, and the thinking will naturally be broadened.
For example, if A is an m×n matrix, B is an n×s matrix, and AB = 0, then we can know that the column vectors of B are all solutions of the homogeneous equation AX = 0. According to the basic system theory and the relationship between matrix rank and vector group rank, R (b) ≤ n-R (a) means R (a)+R (b) ≤
It is precisely because the knowledge points of linear algebra are inextricably linked that algebra problems are more comprehensive and flexible, so we should pay attention to series connection, cohesion and transformation when reviewing.
(3) Summary of key problems in linear algebra.
● Probability theory and mathematical statistics
(1) review points: common distribution; Digital characteristics; Point estimation problem;
(2) review method:
In recent years, the order of the key contents of mathematics examination for science and engineering is: ① two-dimensional random variables and their probability distribution; ② Numerical characteristics of random variables; ③ Random events and probability; ④ Mathematical statistics. The order of the key contents of the third mathematics examination in recent four years is: ① the numerical characteristics of random variables; ② Two-dimensional random variables and their probability distribution; ③ Random events and probability; ④ Mathematical statistics. In recent years, the order of the key contents of economic management mathematics examination is: ① the numerical characteristics of random variables; ② Two-dimensional random variables and their probability distribution; ③ Random events and probability; ④ Law of large numbers and central limit theorem.
Different from "calculus" and "linear algebra", in probability theory and mathematical statistics, a deep understanding of basic concepts accounts for a considerable proportion, and there are not many methods to solve problems and few skills involved (even no skills). It is necessary to combine the characteristics of probability theory and mathematical statistics to carry out targeted review.
The main goal of the strengthening stage is to be familiar with the postgraduate entrance examination questions, strengthen the connection between knowledge points, distinguish the important and difficult points, shorten the review period as much as possible, master the overall knowledge system, and master theorem formulas and problem-solving skills skillfully.
(3) Summary of key issues of probability theory and mathematical statistics (1) and (2)
The outline of 202 1 postgraduate entrance examination has come out. Students can review according to the new examination syllabus and review the changed parts in a targeted manner.