[examination subjects]
Advanced mathematics, linear algebra, probability theory and mathematical statistics
Advanced mathematics
I. Function, Limit and Continuity
Examination content
The concept of function, the representation of boundedness, monotonicity, periodicity and parity of normal function, the properties of basic elementary functions of compound function and implicit function, the establishment of functional relationship of simple application of graphic elementary function, the definitions of sequence limit and function limit, and the four operational limits of left and right limits of property function are infinitesimal and infinitesimal. There are two important limits: monotone bounded 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 (Maximum, Minimum and Intermediate Value Theorems)
Examination requirements
1. Understand the concept of function and master the representation method of function.
2. Understand the parity, monotonicity, periodicity and boundedness of functions.
3. Understand the concept of composite function and the concepts of inverse function and implicit function.
4. Master the nature and graphics of basic elementary functions.
5. The functional relationship in simple application problems will be established.
6. Understand the concept of limit, the concept of left and right limit of function, and the relationship between the existence of limit and left and right limit.
7. Master the nature of limit and four algorithms.
8. 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.
9. To understand the concepts of infinitesimal, infinitesimal and infinitesimal order, we use equivalent infinitesimal to find the limit.
10. If you understand the concept of function continuity, you can identify the type of function discontinuity.
1 1. Understand the continuity of elementary functions and the properties of continuous functions on closed intervals (maximum, minimum and intermediate value theorems), and apply these properties.
Second, the differential calculus of unary function
Examination content
The relationship between the geometric meaning of derivative and differential concept and the derivability and continuity of physical meaning function; The four operations of derivative and differential of basic elementary functions of tangent and normal of plane curve are functions determined by the concept of differential method of inverse function, composite function and implicit function and parameter equation; Application of invariant differential of n-order derivatives of some simple functions in approximate calculation: Rolle theorem; Lagrange mean value theorem; Cauchy mean value theorem; Tyler. Theorem L'HOSPITAL rule: extreme value of function and its solution: determination of function increase and decrease and concavity of function graph; Inflexion point of function graph and its solution; The solution of the maximum and minimum value of the graphic function described by the asymptote: and the approximate solution of the intersection equation of two curvature radius curves is obtained by simply applying the concept of arc differential curvature and dichotomy and tangent method.
Examination requirements
1. Understand the concepts of 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 method of compound function, master the derivative formula of basic elementary function, understand the four algorithms of differential and the invariance of first-order differential form, and understand the application of differential in approximate calculation.
3. If you understand the concept of higher derivative, you will find the n derivative of a simple function.
4. Find the first and second derivatives of piecewise function.
5. Can find the first and second derivatives of implicit functions and functions determined by parametric equations, and can find the derivatives of inverse functions.
6. Understand and apply Rolle theorem, Lagrange mean value theorem and Taylor theorem.
7. Understand and use Cauchy mean value theorem.
8. Understand the concept of extreme value of function, master the method of judging monotonicity of function and finding extreme value of function with derivative, master the method of finding maximum and minimum value of function and its simple application.
9. We can judge the concavity and convexity of the function graph by derivative, find the inflection point of the function graph, find the horizontal, vertical and oblique asymptotes, and describe the function graph.
10. Master the method of finding the limit of indefinite form with L'H?pital's law.
1 1. Understand the concepts of curvature and radius of curvature, calculate curvature and radius of curvature, and find the intersection angle of two curves.
12. understand the dichotomy and tangent method of approximate solution of equation.
3. Integral calculus of unary function
Examination content
The concept of original function and indefinite integral The basic properties of indefinite integral The basic integral formula The mean value theorem of definite integral The concept and properties of definite integral and its derivatives are defined on definite integral variables Newton-Leibniz formula The substitution integral method of indefinite integral and definite integral and the concept of integral generalized integral of partial integral Rational function, trigonometric function and unary function and their approximate calculation methods Application of definite integral
Examination requirements
1. Understand the concept of original function, the concepts of indefinite integral and definite integral, and the mean value theorem of definite integral.
2. Master the basic formula of indefinite integral, the properties of indefinite integral and definite integral, method of substitution and integration by parts.
3. Can find the integral of rational function, rational formula of trigonometric function and simple meta-function.
4. Understand the function of variable upper bound definite integral and its derivative theorem, and master Newton-Leibniz formula.
5. Understand the concept of generalized integral and calculate generalized integral.
6. Understand the approximate calculation method of definite integral.
7. Grasp the representation and calculation of some geometric physical quantities by definite integral (the area of plane figure, the arc length of plane curve, the volume of rotating body and lateral area, and the area of parallel section are the average values of known solid volume, variable force, gravity, pressure and function, etc.). ).
4. Vector Algebra and Spatial Analytic Geometry
Examination content
Concept of vector, concept of quantitative product and cross product of linear operational vector and mixed product of operational vector, condition that two vectors are vertically parallel, coordinate representation and direction cosine surface equation and space curve equation, plane equation, straight line equation and its normal plane and plane, plane and straight line, straight line and straight line are parallel, vertical condition and distance between included angle point and plane and straight line; Equation of surface of revolution with spherical generatrix parallel to coordinate axis and cylindrical rotation axis as coordinate axis; Commonly used quadratic equation and parametric equation of its graphic space curve and projection curve equation of general equation space curve on coordinate plane.
Examination requirements
1. Understand the spatial rectangular coordinate system and the concept and representation of vectors.
2. Master the operation of vectors (linear operation, quantitative product, cross product, mixed product) and understand the conditions for two vectors to be vertical and parallel.
3. Master the coordinate expressions of unit vector, direction number, direction cosine and vector, and the method of vector operation with coordinate expressions.
4. Master the plane equation and straight line equation and their solutions, and use the relationship between plane and straight line (parallel, vertical, intersecting, etc.). ) to solve related problems.
5. Understand the concept of surface equation, understand the equation of common quadric surface and its figure, and find the revolving surface of the cylinder with the coordinate axis as the rotation axis and the generatrix parallel to the coordinate axis.
Surface equation.
6. Understand the parametric equation and general equation of space curve. People can understand the projection of the space curve on the coordinate plane and solve its equation.
Verb (abbreviation of verb) Differential calculus of multivariate functions
Examination content
The concept of multivariate function, the concept of limit and continuity of binary function, the nature of continuous function on bounded closed domain, the concept of total differential, the necessary and sufficient conditions for the existence of total differential, the application of total differential in approximate calculation, the concept of derivative of second-order partial derivative and gradient of compound function and implicit function, and the second-order Taylor formula for calculating tangent plane and normal of space curve, the necessary conditions for the extreme value of multivariate function, and the solution of sufficient and conditional extreme value of multivariate function by Lagrange multiplier method.
Examination requirements
1. Understand the concept of multivariate function.
2. Understand the concepts of limit and continuity of binary functions and the properties of continuous functions on bounded closed fields.
3. Understand the concepts of partial derivative and total differential, the necessary and sufficient conditions for the existence of total differential, and the application of total differential in approximate calculation.
4. Understand the concepts of directional derivative and gradient, and master their calculation methods.
5. Master the solution of the first and second partial derivatives of composite functions.
6. Find the partial derivative of implicit function (including implicit function determined by equation).
7. Understand the concepts of tangent plane, normal plane of curve and tangent plane and normal plane of surface, and work out their equations.
8. Understand the second-order Taylor formula of binary function.
9. 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.
Six, multivariate function integral calculus
Examination content
Concepts and properties of double integral and triple integral, calculation and application of double integral and triple integral, calculation of concepts, properties and relations of two kinds of curve integral, Green's formula, conditions that plane curve integral has nothing to do with path, concepts and properties of two kinds of surface integral and calculation of their relations, Gauss formula, Stokes formula, concepts of divergence and curl, and application of calculating curve integral and surface integral.
Examination requirements
1. Understand the concept, properties and mean value theorem of double integral.
2. Master the calculation method of double integrals (rectangular coordinates and polar coordinates), and be able to calculate triple integrals (rectangular coordinates, cylindrical coordinates and spherical coordinates).
3. Understand the concepts, properties and relationships of two kinds of curve integrals.
4. Master the calculation methods of two kinds of curve integrals.
5. Master Green's formula and use the conditions of plane curve integral and path elements to find the original function of total differential.
6. Understand the concepts, properties and relations of two kinds of surface integrals, master the calculation methods of two kinds of surface integrals, understand Gaussian formula and Stokes formula, and calculate surface integrals by using Gaussian formula.
7. The concepts of dissolution and rotation are introduced and calculated.
8. We can use multiple integral, curve integral and surface integral to find some geometric physical quantities (area, volume, surface area, arc length, mass, center of gravity, moment of inertia, gravity, work, flow, etc.). ).
Seven, infinite series
Examination content
Convergence and divergence of constant series, basic properties and necessary conditions of conceptual series and convergence and divergence of conceptual series; Comparison between positive series of P series and geometric series: ratio convergence method; Root convergence method; Leibniz theorem of staggered series; Absolute convergence and conditional convergence; Convergence domain of function series and convergence radius of sum function concept power series: convergence interval (open interval) and basic properties of convergence domain power series in its convergence interval. Necessary and sufficient conditions for the solution function of simple power series sum function to be expanded into Taylor series. Application of Maclaurin expansion power series in approximate calculation, Fourier coefficient of function and sine of Fourier series function on Dlrichlei theorem function of Fourier series.
Examination requirements
1. Understand the concepts of convergence and sum of convergent constant series, and master the basic properties of series and the necessary conditions for convergence.
2. Master the convergence of geometric series and P series.
3. I will master the ratio method of positive series by comparison method and root value method.
4. Leibniz theorem of staggered series can be used.
5. Understand the concepts of absolute convergence and conditional convergence of infinite series, and the relationship between absolute convergence and conditional convergence.
6. Understand the convergence domain of function term series and the concept of function.
7. Master the solution of convergence radius, convergence interval and convergence domain of power series.
8. To understand some basic properties of power series in its convergence domain, we will find the sum function of some power series in the convergence domain, and then we will find the sum of some polynomial series.
9. Understand the necessary and sufficient conditions for the function to expand into Taylor series.
10. Master the maclaurin expansions of some functions and use them to indirectly expand some simple functions into power series.
1 1. Understand the simple application of power series in approximate calculation.
12. Understand the concept of Fourier series and Dirichlet's theorem of expanding functions into Fourier series. Expand the functions defined on [- 1, 1] into Fourier series, expand the functions defined on [0, 1] into sine series and cosine series, and write the expression of the sum of Fourier series.
Eight, ordinary differential equations
Examination content
The concept, general solution and initial conditions of solutions of ordinary differential equations can be separated, and the equations with special variables are homogeneous. The first-order linear equations are BER-noulli equations, the total differential equations can be solved by simple variable substitution, and some differential equations can be simplified to higher-order linear differential equations. The properties and structure theorems of the solution of the second-order homogeneous linear differential equation with constant coefficients are higher than those of the second-order homogeneous linear differential equation with constant coefficients. The first-order linear differential equations with constant coefficients in La (Eu 1er) equation contain two unknown functions. The power series solution of differential equation is a simple application problem of differential equation (or system of equations).
Examination requirements
1. Understand the concepts of differential equations and their solutions, general solutions, initial conditions and special solutions.
2. Master the solutions of equations with separable variables and first-order linear equations.
3. I can solve homogeneous equations, Bernoulli equations and fully differential equations, and I will replace some differential equations with simple variables.
4. Some equations can be solved by order reduction method (omitted)
5. Understand the properties of solutions of linear differential equations and the structure theorem of solutions.
6. 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.
7. I can use polynomials, exponential functions, sine functions, cosine functions and their sum and product to find the special and general solutions of second-order non-homogeneous linear differential equations with constant coefficients.
8. Knowing the power series solution of differential equation, we can solve Euler equation and first-order linear differential equation with constant coefficients with two unknown functions.
9. Will use differential equations (or equations) to solve some simple application problems.
linear algebra
I. Determinants
Examination content
Definition, Properties and Calculation of Determinant
Examination requirements
1. Understand the definition and properties of determinant.
2. Mastering the calculation methods of third-order and fourth-order determinants, you can calculate simple "order determinants".
Second, the matrix
Examination content
Concepts and properties of matrix identity matrix, diagonal matrix, triangular matrix, symmetric matrix and antisymmetric matrix The concept matrix of the determinant matrix of the multiplication matrix of linear operation matrix is reversible. The necessary and sufficient conditions are the elementary transformation of adjoint matrix and the rank elementary transformation of equivalent matrix of elementary matrix.
Examination requirements
1. Understand the concept of matrix.
2. Understand identity matrix, diagonal matrix, triangular matrix, symmetric matrix and antisymmetric matrix and their properties.
3. Master linear operation, multiplication, moment transposition and its operation rules, and understand the determinant of the power of square matrix and the product of square matrix.
4. Understand the concept of inverse matrix, grasp the properties of inverse matrix, the necessary and sufficient conditions of matrix reversibility, understand the concept of adjoint matrix, and use adjoint matrix to find the inverse of matrix.
5. Master the 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 matrix rank and inverse matrix by elementary transformation.
6. Understand the block matrix and its operation.
Third, the vector
Examination content
The relationship between the linear correlation of the concept vector group of vectors and the maximal linear independent group of linear independent vector group equivalent to the rank of vector group and the rank of matrix. Concept of basic transformation and coordinate transformation of N-dimensional vector space, such as vector space, subspace, basis, dimension and coordinates, linear elements of inner product of transfer matrix vector, orthogonal normalization method and its properties of standard orthogonal basis orthogonal matrix of closed vector group.
Examination requirements
1. Understand the concept of N-dimensional vector.
2. Understand the definitions of linear correlation and linear correlation of vector groups, and understand and apply the important conclusions 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 vector group and the rank of matrix.
5. Understand the concepts of N-dimensional vector space, subspace, basis, dimension and coordinate.
6. By mastering the formulas of base transformation and coordinate transformation, we can find the transfer matrix.
7. Understand the concept of inner product, and master the Schmidt method for standardization of linear independent vector groups.
8. Understand the concepts of standard orthogonal basis and orthogonal matrix, and their properties.
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 Basic Solution System of Homogeneous Linear Equations and General Solution of Non-homogeneous Linear Equations in General Solution Space Method of Solving Linear Equations by Elementary Transformation.
Examination requirements
1. Understand Cramer's Law.
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 basic solution system, general solution and solution space of homogeneous linear equations.
4. Understand the structure of solutions of nonhomogeneous linear equations and the concept of general solutions.
5. Master the method of finding the general solution of linear equations by row elementary transformation.
Eigenvalues and eigenvectors of verb (abbreviation of verb) matrix
Examination content
Concepts, properties and solutions of eigenvalues and eigenvectors of matrices Similar transformation, concepts of similar matrices and necessary and sufficient conditions for similar diagonalization of property matrices Similar diagonal matrices of 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 similarity matrix and the necessary and sufficient conditions for matrix similarity diagonalization.
3. Master the method of transforming real symmetric matrix into diagonal matrix by similarity transformation.
Sixth, quadratic form
Examination content
Quadratic form and its matrix represent the rank inertia theorem of quadratic form. Quadratic form is transformed into standard quadratic form by orthogonal transformation and matching method, and the positive definiteness of the corresponding matrix and its discrimination method
Examination requirements
Master quadratic form and its matrix representation, understand the concept of quadratic form rank, and understand the numerical characteristics of habitual quartic form and random variables.
Probability and Mathematical Statistics
Fourth, the numerical characteristics of random variables
Examination content
The concepts and properties of mathematical expectation (mean) and variance, and the mathematical expectation moment, covariance and correlation coefficient of mathematical expectation and variance of binomial distribution, Poisson distribution, normal distribution, uniform distribution and exponential distribution are calculated.
Examination requirements
1. Understand the concepts of mathematical expectation and variance, and master their properties and calculations.
2. Grasp the mathematical expectation and variance of binomial distribution, Poisson distribution and normal distribution, and understand the mathematical expectation and variance of uniform distribution and exponential distribution.
3. The mathematical expectation of random variable function can be calculated.
4. Understand the concepts and properties of moment, covariance and correlation coefficient, and calculate them.
Law of Large Numbers and Central Limit Theorem
Examination content
Chebyshev inequality Chebyshev theorem and Bernoulli theorem Lindberg-DevO theorem (central limit theorem of independent and identical distribution) and Demovre-LAPLACE theorem (binomial distribution takes normal distribution as the limit distribution).
Examination requirements
1. Understanding Chebyshev Inequality.
2. Understand Chebyshev Theorem and Bernoulli Theorem.
3. Understand Lindbergh-Levi theorem (central limit theorem of independent and identical distribution) and Okmov-Laplace theorem (binomial distribution takes normal distribution as the limit distribution).
Basic concepts of mathematical statistics of intransitive verbs
Examination content
The concepts of population, individual, simple random sample and statistics, the definitions of sample mean and sample variance distribution, and the distribution of some commonly used natural population statistics.
Examination requirements
1. Understand the concepts of population, individual, simple random sample and statistics, and master the calculation of sample mean, sample individual and sample flow.
2. The definitions and properties of advanced/distribution, distribution and underdistribution, and understand the concept of quantile and the calculation of yield table.
3. Understand the distribution of some commonly used statistics of normal population.
Seven. parameter estimation
Examination content
Concept moment estimation method of point estimation Maximum likelihood estimation method Select standard interval estimation concept Confidence interval of mean and variance of single normal population Confidence interval of mean difference and variance ratio of two normal populations.
Examination requirements
1. Understand the concept of point estimation.
2. Master moment estimation method (first order, second order) and maximum likelihood estimation method.
3. Understand the evaluators' selection criteria (fairness, effectiveness and consistency).
4. Understand the concept of interval estimation.
5. The confidence interval of the mean and variance of a single normal population will be found.
6. Will find the confidence interval of the mean difference and variance ratio of two normal populations.
Eight, hypothesis testing
Examination content
The basic idea and steps of significance test and two possible errors: the hypothesis of mean and variance of single and two normal populations; Test method of population distribution hypothesis
Examination requirements
1. Understand the basic idea of significance test, master the basic steps of hypothesis test, and understand two possible errors in hypothesis test.
2. Understand the hypothesis test of the mean and variance of a single and two normal populations.
3. Understand the test method of general distribution hypothesis.
[Test Paper Structure]
(1) content ratio
About 60% of advanced mathematics
Linear algebra accounts for about 20%
Probability theory and mathematical statistics are about 20%
(B) the proportion of questions
Fill in the blanks and multiple-choice questions about 30%
Answer questions (including proof questions) about 70%
I have never heard of such a thing as English.