Advanced mathematics
I. Function, Limit and Continuity
Examination content
The concept and expression of function: boundedness, monotonicity, periodicity and parity of function, properties of composite function, inverse function, piecewise function and implicit function, and the establishment of functional relationship of graphic elementary function. The definitions of sequence limit and function limit, left limit and right limit of property function, the concepts of infinitesimal and infinitesimal and their relationship, four operational limits of infinitesimal comparison limit, and two important limits: monotone boundedness criterion and pinch criterion;
Concept of Function Continuity Types of Discontinuous Points of Functions Continuity of Elementary Functions Properties of Continuous Functions on Closed Interval
Examination requirements
1. Understand the concept of function and master the expression of function, and you will establish the functional relationship of application problems.
2. Understand the boundedness, monotonicity, periodicity and parity of functions.
3. Understand the concepts of compound function and piecewise function, and the concepts of inverse function and implicit function.
4. Grasp the nature and graphics of basic elementary functions and understand the concept of elementary functions.
5. Understand the concept of limit, the concept of left and right limit of function and the relationship between the existence of function limit and left and right limit.
6. Master the nature of limit and four algorithms.
7. Master two criteria for the existence of limit, and use them to find the limit, and master the method of using two important limits to find the limit.
8. Understand the concepts of infinitesimal and infinitesimal, master the comparison method of infinitesimal, and find the limit with equivalent infinitesimal.
9. Understanding the concept of function continuity (including left continuity and right continuity) will distinguish the types of function discontinuity points.
10. Understand the properties of continuous functions and the continuity of elementary functions. 1. Understand the properties of continuous functions on closed intervals (boundedness, maximum theorem, mean value theorem) and apply these properties.
Second, the differential calculus of unary function
Examination requirements
1. Understand the concepts of derivative and differential, understand the relationship between derivative and differential, understand the geometric meaning of derivative, find the tangent equation and normal equation of plane curve, understand the physical meaning of derivative, describe some physical quantities with derivative, and understand the relationship between function derivability and continuity.
2. Master the four algorithms of derivative and the derivative rule of compound function, and master the derivative formula of basic elementary function. Knowing the four algorithms of differential and the invariance of first-order differential form, we can find the differential of function.
3. If you understand the concept of higher derivative, you will find the higher derivative of simple function.
4. We can find the derivative of piecewise function, implicit function, function determined by parametric equation and inverse function.
5. Understand and apply Rolle theorem, Lagrange mean value theorem, Taylor theorem, and Cauchy mean value theorem.
6. Master the method of finding the indefinite limit by L'H?pital method.
7. Understand the concept of extreme value of function, master the method of judging monotonicity of function and finding extreme value of function with derivative, and master the method of finding maximum and minimum value of function and its application.
8. Will judge the concavity and convexity of the function graph by derivative (note: in the interval (a, b), let the function f(x) have the second derivative. When > 0, the graph of f(x) is concave; When < 0, the graph of f(x) is convex), the inflection point and horizontal, vertical and oblique asymptotes of the function graph will be found, and the function graph will be portrayed.
9. Understand the concepts of curvature, circle of curvature and radius of curvature, and calculate curvature and radius of curvature.
3. Integral calculus of unary function
Examination contents: concepts of original function and indefinite integral, basic properties of indefinite integral, concept and basic properties of definite integral formula, mean value theorem of definite integral, Newton-Leibniz formula of upper limit function of integral and its derivative, substitution integration method of indefinite integral and definite integral, rational formula of integral of partial rational function and trigonometric function, application of integral anomaly (generalized) definite integral of simple unreasonable function.
Examination requirements
1. Understand the concepts of original function and indefinite integral and definite integral.
2. Master the basic formula of indefinite integral, the properties of indefinite integral and definite integral and the mean value theorem of definite integral, and master the integration methods of method of substitution and integration by parts.
3. Know the integral of rational function, rational trigonometric function and simple unreasonable function.
4. Understand the function of the upper limit of integral, find its derivative and master Newton-Leibniz formula.
5. Understand the concept of generalized integral and calculate generalized integral.
6. Master the expression and calculation of the average value of some geometric and physical quantities (the area of plane figure, the arc length of plane curve, the volume and lateral area of rotating body, and the area of parallel section are known solid volume, work, gravity, pressure, centroid, centroid, etc.). ) and definite integral function.
Four, multivariate function calculus
Examination requirements
1. Understand the concept of multivariate function and the geometric meaning of bivariate function.
2. Understand the concepts of limit and continuity of binary function and the properties of binary continuous function in bounded closed region.
3. Knowing the concepts of partial derivative and total differential of multivariate function, we can find the first and second partial derivatives of multivariate composite function, total differential, existence theorem of implicit function and partial derivative of multivariate implicit function.
4. Understand the concepts of extreme value and conditional extreme value of multivariate function, master the necessary conditions of extreme value of multivariate function, understand the sufficient conditions of extreme value of binary function, find the extreme value of binary function, find the conditional extreme value by Lagrange multiplier method, find the maximum value and minimum value of simple multivariate function, and solve some simple application problems.
5. Understand the concept and basic properties of double integral, and master the calculation methods of double integral (rectangular coordinates and polar coordinates).
Verb (abbreviation of verb) ordinary differential equation
Examination content
The basic concepts of ordinary differential equations are separated from variable differential equations, homogeneous differential equations, properties and structural theorems of solutions of first-order linear differential equations, which can be reduced to higher order. Some simple applications of second-order homogeneous linear differential equations with constant coefficients are higher than second-order homogeneous linear differential equations with constant coefficients.
Examination requirements
1. Understand differential equations and their concepts such as order, solution, general solution, initial condition and special solution.
2. Mastering the solutions of differential equations with separable variables and first-order linear differential equations can solve homogeneous differential equations.
3. The following differential equation will be solved by order reduction method.
4. Understand the properties of the solution of the second-order linear differential equation and the structure theorem of the solution.
5. Master the solution of second-order homogeneous linear differential equations with constant coefficients, and be able to solve some homogeneous linear differential equations with constant coefficients higher than the second order.
6. Polynomials, exponential functions, sine functions, cosine functions and their sum and product can be used to solve second-order non-homogeneous linear differential equations with constant coefficients.
7. Can use differential equations to solve some simple application problems.
linear algebra
I. Determinants
Examination content
The concept and basic properties of determinant The expansion theorem of determinant by row (column)
Examination requirements
1. Understand the concept of determinant and master its properties.
2. The properties of determinant and determinant expansion theorem will be applied to calculate determinant.
Second, the matrix
Examination content
Concept of matrix, linear operation of matrix, multiplication of matrix, concept and properties of transposed inverse matrix of determinant matrix, necessary and sufficient conditions for matrix reversibility, equivalent block matrix of elementary transformation of matrix and rank matrix of elementary matrix and its operation.
Examination requirements
1. Understand the concepts and properties of matrix, identity matrix, quantization matrix, diagonal matrix, triangular matrix, symmetric matrix, antisymmetric matrix and orthogonal matrix.
2. Master the linear operation, multiplication, transposition and its operation rules of matrix, and understand the determinant properties of square matrix power and square matrix product.
3. Understand the concept of inverse matrix, master the properties of inverse matrix and the necessary and sufficient conditions for matrix reversibility. Understand the concept of adjoint matrix and use adjoint matrix to find the inverse matrix.
4. Understand the concept of elementary transformation of matrix, understand the properties of elementary matrix and the concept of matrix equivalence, understand the concept of matrix rank, and master the method of finding the rank and inverse matrix of matrix by elementary transformation.
Third, the vector
Examination content
The linear combination of concept vectors of vectors and the linear representation of linear correlation of vector groups and the maximum linear independence of linear independent vector groups are equivalent to the orthogonal normalization method of inner product linear independent vector groups between the rank of rank vector groups and the rank of matrix.
Examination requirements
1. Understand the concepts of n-dimensional vectors, linear combinations of vectors and linear representations.
2. Understand the concepts of linear correlation and linear independence of vector groups, and master the related properties and discrimination methods of linear correlation and linear independence of vector groups.
3. Understand the concepts of maximal linearly independent group and rank of vector group, and find the maximal linearly independent group and rank of vector group.
4. Understand the concept of vector group equivalence and the relationship between the rank of matrix and the rank of its row (column) vector group.
5. Understand the concept of inner product and master the Schmidt method of orthogonal normalization of linear independent vector groups.
Fourth, linear equations.
Examination content
Cramer's Law of Linear Equations Necessary and Sufficient Conditions for Homogeneous Linear Equations to Have Non-zero Solutions Necessary and Sufficient Conditions for Non-homogeneous Linear Equations to Have Solutions Properties and Structures of Solutions of Linear Equations Basic System of Solutions and General Solutions of Non-homogeneous Linear Equations
Examination requirements
1. Cramer's law can be used.
2. Understand the necessary and sufficient conditions for homogeneous linear equations to have nonzero solutions and nonhomogeneous linear equations to have solutions.
3. Understand the concepts of basic solution system and general solution of homogeneous linear equations, and master the solution of basic solution system and general solution of homogeneous linear equations.
4. Understand the structure of solutions of nonhomogeneous linear equations and the concept of general solutions.
5. We can use elementary line transformation to solve linear equations.
Eigenvalues and eigenvectors of verb (abbreviation of verb) matrix
Examination content
The concepts of eigenvalues and eigenvectors of matrices, the concepts of property similarity matrices and the necessary and sufficient conditions for similar diagonalization of property matrices, and the eigenvalues and eigenvectors of similar diagonal matrices and their real symmetric matrices.
Examination requirements
1. Understand the concepts and properties of eigenvalues and eigenvectors of a matrix, and you will find the eigenvalues and eigenvectors of the matrix.
2. Understand the concept and properties of matrix similarity and the necessary and sufficient conditions for matrix similarity diagonalization, and transform the matrix into a similar diagonal matrix.
3. Understand the properties of eigenvalues and eigenvectors of real symmetric matrices.
Sixth, quadratic form
Examination content
Quadratic form and its matrix represent contract transformation and rank inertia theorem of quadratic form of contract matrix. The canonical form and canonical form of quadratic form are transformed into canonical quadratic form and the positive definiteness of its matrix by orthogonal transformation and matching method.
Examination requirements
1. Understand the concept of quadratic form, express quadratic form in matrix form, and understand the concepts of contract transformation and contract matrix.
2. Understand the concept of rank of quadratic form, the concepts of standard form and standard form of quadratic form, and inertia theorem, and transform quadratic form into standard form by orthogonal transformation and collocation method.
3. Understand the concepts of positive definite quadratic form and positive definite matrix, and master their discrimination methods.
Let me tell you about the learning method of mathematics.
For graduate mathematics:
This stage is the stage of laying a solid foundation. Let's focus on the math textbook. High numbers are large blocks, and probability and line generation are relatively simple. Tilt the time to a higher number, and there will be more scores. Read more textbooks, just like learning for the first time, and have a good understanding of the knowledge points and theorems of textbooks. It is best to cooperate with last year's mathematics postgraduate entrance examination outline and focus on it. It is not recommended to do all the exercises after class. Just choose some representative exercises, think about other ideas and look at the answer book. It's a waste of time to do everything, as long as you master the methods and problem-solving skills. After that, review the whole book, 660 or something, and then the real question. Plan your time and itinerary well. Learn slowly, but don't worry. I'll post a bibliography summary for you, which I borrowed from others. I hope it helps you. I have considered my math exam, so I hope to adopt it.
1, the encyclopedia of mathematics review in Li Zhengyuan, Li Yongle * * * *, the same practical guide to mathematics review in Wendeng Chen * * *, but Wendeng pays attention to skills, the essence of which is calculus, and Yongle pays attention to foundation. Moreover, judging from the exams in the past three years, the whole book is more suitable for postgraduate entrance examination, and some contents of Wendeng are beyond the outline. If you have bought Wendeng's review guide, it is highly recommended to buy Yongle's lecture on linear algebra, because Yongle's line generation is very simple and can make up for Wendeng's lack of line generation. Comrades who want to get high marks can choose both (personally, they want both books);
2. The math foundation doesn't have to pass 660 questions, but it is already very good as a basic exercise in the early stage.
3, the real questions over the years. It is best to have two versions, one is Yongle's "Analysis of Examination Questions over the Years" * * *, which has the advantage of being classified according to chapters, with annotations behind the topics, so you can self-test the previous examination papers over the years; Another article, "Research on Mathematics Postgraduate Entrance Examination Questions over the Years" by Wu Zhongxiang of Xi Jiaotong University, has the advantages of classification by chapter, analysis of test sites and classified statistics. There are synchronization exercises at the end of each chapter. If you can't buy these two books, the real questions in any other version are the same. There is also a recommendation, you can pay a set of dedicated postgraduate questions * * * *, which is very cost-effective. As long as you buy two more books for 2 yuan, you won't lose money, because if you do the real questions several times, your score will grow a few points. Just explain it in detail.
4. "The final sprint of mathematics exceeds 135" * * * * *; Or Wendeng's "Problems and Exercises" * * as the final sprint to check and fill the gaps.
5. Li Yongle's "400 classic mathematical simulation problems" should be done at least three times * * * * *. Don't buy more other simulation questions. Although it is called sea tactics, it is too wasteful, and it will affect your mood if you don't do it. Bourne's simulation questions * * *, as well as Kaoru's simulation questions * * *, can be downloaded to the topic of Polytechnic University, and the best * * * * is close to the real question.
6. In addition, the better guidance books are "Methods and Skills of Solving Multiple-choice Questions in Postgraduate Mathematics" and "Lecture Notes on Probability Theory and Mathematical Statistics (Improved Articles)". You can download the online courseware of New Oriental if you can. This courseware is enough. It is best to go to Yongle's lecture on linear algebra in 2005, which is very classic, and Fei Yunjie's lecture on probability in 2006 is also very classic. Other Tian Genbao's line generation and probability courseware are not needed and are not recommended; Wendeng's sprint lecture is unnecessary, let alone the remedial class. In principle, if you can watch it by yourself, you don't need courseware, because attending classes is a waste of time. If the foundation is really bad, go to class.
Remember, a good book can help you reach the finish line faster. But there are not many books, so we must do them several times and sum up the methods. Courseware is a waste of time. Don't use courseware if you can understand it. Hope to adopt