Advanced Mathematics 1. The concept and representation of functions, the boundedness, monotonicity, periodicity and parity of limit and continuous examination content functions, the properties of basic elementary functions of inverse functions, piecewise functions and implicit functions, and the establishment of functional relationships of graphic elementary functions; The definitions and properties of sequence limit and function limit: the concepts of left limit, right limit and infinitesimal quantity of function and their relations; The nature of infinitesimal and four operational limits of comparison limit of infinitesimal: two important limits: monotone boundedness criterion and pinching criterion, the concept of function continuity, the type of function discontinuity, the continuity of elementary function, and the property requirements of continuous function on closed interval 1. Understanding the concept of function and mastering the expression of function 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 concepts of left limit 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. 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 you will find the limit of infinitesimal equivalence. 9. Understand the concept of function continuity (including left continuity and right continuity), and you can judge the type of function discontinuity. 10. Understand the properties of continuous functions and the continuity of elementary functions. Understand the properties of continuous functions on closed intervals (boundedness, maximum theorem, minimum theorem, intermediate value theorem) and apply these properties. Second, the relationship between the geometric meaning and physical meaning of derivative and differential of unary function and the derivability and continuity of function, the four operations of tangent and normal derivative of plane curve and differential, derivative compound function, inverse function, implicit function and differential method of function determined by parametric equation, the first-order differential mean value theorem of L'H?pital's law function, monotonicity, convexity, inflection point and asymptote of extreme function graph, maximum and minimum value of graphic depiction function, circular arc differential curvature concept, curvature circle and curvature radius test requirements/kl 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 differentiability and continuity of function. 2. Master the four algorithms of derivative and the derivation formula of basic elementary function. Understand the four algorithms of differential and the invariance of first-order differential form. Know how to distinguish functions. 3. Understand the concept of higher derivative, and know the solution of higher derivative of simple function. 4. Know how to 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 and Taylor theorem. Understand and apply Cauchy mean value theorem. 6. Master the method of using L'H?pital's law to find the limit of indefinite form. 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. Use derivative to judge the concavity and convexity of the function graph (Note: In the interval, let the function have the second derivative. When, the figure is concave; When the graph is convex, the inflection point and horizontal, vertical and oblique asymptotes of the function graph will be found to describe the function graph. 9. Understand the concepts of curvature, curvature circle and curvature radius. Curvature and radius of curvature can be calculated. Third, the content of one-variable function integration examines the concept of original function and indefinite integral, the basic properties of indefinite integral, the concept and basic properties of definite integral, the mean value theorem of definite integral, the function of the upper limit of integral and its derivative Newton-Leibniz formula, the substitution integration method of indefinite integral and definite integral, the rational number formula of partial integral and trigonometric function and the application of integral, and the abnormal (generalized) integral of simple and unreasonable function. 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. Master substitution integral method and integration by parts. 3. Know the integral of rational function, rational trigonometric function and simple unreasonable function. 4. Understand the function with upper integral limit, know its derivative and master Newton-Leibniz formula. 5. Understand the concept of generalized integral. You can calculate the improper integral. 6. Master the expression and calculation 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, the area of parallel section, the volume, work, gravity, pressure, center of mass, center of mass, etc. ) and the average value of the function. 4. In the examination of vector algebra and spatial analytic geometry, the quantitative product of the linear operation vector of the concept vector and the mixed product of the cross product vector are vertically parallel. Coordinate expression of included angle vector of two vectors and its operation unit vector direction number and direction cosine surface equation and space curve equation are quadratic equations of conceptual plane equation, straight line equation, plane-to-plane, plane-to-straight line angle and parallel and vertical conditions, distance from point to plane and point to straight line, parametric equation of graphic space curve and general equation, projection curve equation, space curve equation, examination requirements on coordinate plane 1. Understand the spatial rectangular coordinate system, understand the concept and representation of vectors. 2. Master the operation of vectors (linear operation, quantitative product, cross product, mixed product). Understand the condition that two vectors are vertical and parallel. 3. Understand the coordinate expressions of unit vector, direction number, direction cosine and vector, and master the method of vector operation with coordinate expressions. 4. Principal plane equation and straight line equation and their solutions. 5. Know the included angles between planes, planes and straight lines, and straight lines. And will use the relationship between plane and straight line (parallel, vertical, intersecting, etc.). ) to solve related problems. 6. Will find the distance from a point to a straight line and a point to a plane. 7. Understand the concepts of surface equation and space curve equation. 8. Understand the common equations of quadric surface and its figure, and you will find the equations of simple cylinder and rotating surface. 9. Understand the parametric equation and general equation of space curve. Understand the projection of space curve on coordinate plane. And the equation of projection curve is obtained. V. Concept of Multivariate Function Geometric Meaning of Multivariate Function Limit and Continuity of Multivariate Function in Bounded Closed Region Necessary and Sufficient Conditions for the Existence of Partial Derivatives and Total Differentials of Multivariate Function; Derivation Method of Second-order Partial Derivatives Directional Derivatives and Second-order Taylor Formula of Tangents and Tangents of Gradient Space Curves; Extremes and Conditional Extremes of Plane and Surface Bivariate Functions and Their Simple Application Examination Requirements 1. Understand the concept of multivariate function and the geometric meaning of binary function. 2. Understand the concepts of limit and continuity of binary functions and the properties of continuous functions in bounded closed regions. 3. Understand the concepts of partial derivative and total differential of multivariate function, and you will seek total differential, understand the necessary and sufficient conditions for the existence of total differential, and understand the invariance of total differential form. 4. Understand the concepts of directional derivative and gradient. And master its calculation method. 5. Master the solution of the first and second partial derivatives of multivariate composite functions. 6. Understand the existence theorem of implicit function, and you will find the partial derivative of multivariate implicit function. 7. Understand the concepts of tangent plane and normal plane of space curve and tangent plane and normal plane of surface, and you will find their equations. 8. Understand the second-order Taylor formula of binary function. 9. Understand the concepts of multivariate function extremum and conditional extremum, and master the necessary conditions for the existence of multivariate function extremum. Knowing the sufficient conditions for the existence of extreme value of binary function, we can find the extreme value of binary function, and use Lagrange multiplier method to find the conditional extreme value, the maximum value and minimum value of simple multivariate function. And will solve some simple application problems. The concepts, properties, calculation and application of double integral and triple integral of intransitive verbs; The concepts, properties and calculation of two kinds of curve integrals: Green's formula of plane curve integral; Conditional binary function independent of path; Original function; The concept, properties and calculation of two kinds of surface integrals: Gaussian. The concepts of divergence and curl of Stokes formula and the application of calculating curve integral and surface integral require 1. Understand the concepts of double integral and triple integral, understand the properties of double integral and understand the mean value theorem of double integral. 2. Master the calculation method of double integral (rectangular coordinates and polar coordinates). Can calculate triple integrals (rectangular coordinates, cylindrical coordinates, 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 condition that the plane curve integral has nothing to do with the path to find the original function of the total differential of binary function. 6. Understand the concepts, properties and relationships of two kinds of surface integrals. Master the calculation methods of two kinds of surface integrals, master the method of calculating surface integrals with Gaussian formula and calculate curve integrals with Stokes formula. And can calculate. 8. Some geometric physical quantities (area, volume, surface area, arc length, mass, center of mass, center of mass, moment of inertia, gravity, work, flow, etc.) can be obtained by using multiple integrals, curve integrals and surface integrals. ). Infinite Series Examination Contents Convergence and Divergence of Constant Term Series, Basic Properties and Necessary Conditions of Convergence and Divergence of Conceptual Series of Geometric Series and Sum of Series, and Discrimination of Convergence and Divergence of Positive Term Series, Absolute Convergence and Conditional Convergence of Staggered Series, and Basic Properties of Concept Power Series and Its Convergence Radius and Convergence Interval (refers to the open interval) of the Convergence Interval of Sum Function of Simple Power Series; Fourier coefficient of power series expansion function of elementary function and Dirichlet theorem function of Fourier series need 1 to examine sine series and cosine series. The concepts of divergent series and convergent series, the basic properties of series and the necessary conditions for convergence. 2. Conditions of convergence and divergence of geometric series and series. 3. For comparison of convergence of positive series and ratio method, I can use root value discrimination method. 4. Master the Leibniz discriminant method of staggered series. 5. Understand the concepts of absolute convergence and conditional convergence of arbitrary series and the relationship between absolute convergence and convergence. 6. Understand the convergence domain of function series and the concept of function. 7. Understand the concept of convergence radius of power series and master the convergence radius of power series. Solution of convergence interval and convergence domain. 8. Understand the basic properties of power series in its convergence interval (continuity of sum function, item-by-item derivation, item-by-item integration), and we will find some sum functions of power series in the convergence interval. We can get the sum of some series. 9. Understand the necessary and sufficient conditions for the function to expand into Taylor series. 10. Master the Maclaurin expansions of exp(x), sinx, cosx, ln( 1+x) and (1+x)a, which will be used to indirectly expand some simple functions into power series. 1 1. Understand the concept of Fourier series and Dirichlet convergence theorem, and expand the functions defined on the ground into Fourier series, sine series and cosine series. I will write the expressions of Fourier series and functions. Eight. The basic conceptual variables of ordinary differential equations can be separated. The differential equation is homogeneous. The first order linear differential equation is Bernoulli equation. The fully differential equation of the equation can be solved by simple variable substitution. Some differential equations can be simplified to Euler equations. Properties and structure theorems of solutions of higher order linear differential equations. The second order homogeneous linear differential equation with constant coefficients is higher than the second order homogeneous linear differential equation with constant coefficients. Simple application of second-order non-homogeneous linear differential equations with constant coefficients 1. Understand the concepts of differential equation and its order, solution, general solution, initial value condition and special solution. 2. Master the solutions of differential equations with separable variables and first-order linear differential equations. 3. Understand homogeneous differential equation, Bernoulli equation and total differential equation, and replace some differential equations with simple variables. 4. Understand the properties and structure of linear differential equations. 6. Master the second-order homogeneous linearity with constant coefficients. And will solve some homogeneous linear differential equations with constant coefficients above the second order. 7. Polynomials, exponential functions, sine functions, cosine functions and their sum and product can be used to solve second-order homogeneous linear differential equations with constant coefficients. 8. Can solve Euler equation. 9. Will use differential equations to solve some simple application problems. Linear algebra I. Determinant test content The concept and basic property of determinant are rows (columns). The expansion theorem test requires 1. Understand the concept of determinant and master the properties of determinant. 2. Use the properties of determinant to calculate determinant through row (column) expansion theorem. 2. Matrix Examination Contents Matrix Concept Matrix Linear Operation Matrix Multiplication Matrix Power Matrix Product Determinant Matrix Concept and Property Matrix Reversible Necessary and Sufficient Conditions Adjoint Matrix Elementary Transformation Elementary Matrix Rank Matrix Equivalent Block Matrix and Its Operation Examination Requirements 1. Understand the concept of matrix, identity matrix, quantitative matrix, diagonal matrix, triangular matrix, symmetric matrix and antisymmetric matrix, and their properties. 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, the necessary and sufficient conditions for matrix reversibility, and understand the concept of adjoint matrix. Will use the adjoint matrix to find the inverse matrix. 4. Understand the concept of matrix elementary transformation, the properties of elementary matrix and the concept of matrix equivalence, and the concept of matrix rank. Master the method of finding the rank and inverse matrix of matrix by elementary transformation. 5. Understand the block matrix and its operation. 3. Vector tests the linear combination of concept vectors of content vectors and the relationship between the rank of linear correlation vector group and the rank of matrix of equivalent vector group of linear representation vector group and maximum linear independent vector group, the base transformation and coordinate transformation of vector space of related concept dimension, the orthogonal normalization method of orthogonal base orthogonal matrix of inner product linear independent vector group and its property test requirements are 65 438+0. Understand the concepts of dimension vector, linear combination of vectors and linear representation. 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 maximum linear independence and rank of vector groups, and you will find the maximum linear independence and rank of vector groups. 4. Understand the concept of vector group equivalence. Understand the relationship between the rank of a matrix and the rank of its row (column) vector group. 5. Understand the concepts of dimension vector space, subspace, basis, dimension and coordinates. 6. Understand the formulas of base transformation and coordinate transformation, and find the transformation matrix. 7. Understand the concept of inner product and master the Schmidt method of orthogonal normalization of linear independent vector groups. 8. Understand the concepts and properties of normalized orthogonal basis and orthogonal matrix. 4. Examination content of linear equations: Clem 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 Structure of Solutions; Basic Solution System of Homogeneous Linear Equations and General Solution in General Solution Space; Examination requirements L. Cramer's Law will be used. 2. Understand the necessary and sufficient conditions for homogeneous linear equations to have nonzero solutions and for solving nonhomogeneous linear equations. 3. Understand the concepts of a necessary and sufficient condition basic solution system, general solution and solution space of homogeneous linear equations, and master the solution methods 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. Master the method of solving linear equations with elementary line transformation. 5. Eigenvalues and eigenvectors of matrices: the concepts of eigenvalues and eigenvectors of matrices, the transformation of similar properties, the concept of similar matrices and the necessary and sufficient conditions for similar diagonalization of property matrices, and the eigenvalues of real symmetric matrices of similar diagonal matrices. Test requirements for eigenvectors and their similar diagonal matrices: 1. Understand the concepts and properties of eigenvalues and eigenvectors of matrices, and you will find the eigenvalues and eigenvectors of matrices. 2. Understand the concept and properties of similarity matrix and the necessary and sufficient conditions for matrix similarity diagonalization. Master the method of transforming a matrix into a similar diagonal matrix. 3. Master the properties of eigenvalues and eigenvectors of real symmetric matrices. 6. The quadratic form of the second examination content and its matrix represent the rank inertia theorem of the contract transformation and the quadratic form of the contract matrix. Using orthogonal transformation and matching method, the standard form and standard form of quadratic form are transformed into standard quadratic form and its matrix. The requirement of positive test is 1. Principal quadratic form and its matrix representation. Understand the concepts of quadratic rank, contract transformation and contract matrix, canonical form, canonical form and inertia theorem of quadratic form. 2. Master the method of transforming quadratic form into standard form by orthogonal transformation, and will transform quadratic form into standard form by matching method. 3. Understand the concepts of positive definite quadratic form and positive definite matrix. And master its identification method. Probability theory and mathematical statistics 1. The relationship between random events and events in sample space and the basic properties of the concept of complete operational probability. The probability basic formula of classical probability of event group probability is independent and repetitive. The test requirement is 1. Understand the concept of sample space (basic event space) and the concept of random events. Master the relationship and operation of events. 2. Understand the concepts of probability and conditional probability, master the basic properties of probability, calculate classical probability and geometric probability, and master the addition formula, subtraction formula, multiplication formula, total probability formula and Bayesian formula of probability. 3. Understand the concept of event independence and master the probability calculation with event independence; Understand the concept of independent repeated test and master the calculation method of related event probability. Second, the concepts and properties of random variables and random variable distribution functions: random variables, probability distribution of discrete random variables, probability density of continuous random variables, distribution of common random variables and distribution requirements of random variable functions 1. Understand the concept of random variables and the concept and properties of distribution function. The probability of an event related to a random variable can be calculated. 2. Understand the concept and probability distribution of discrete random variables, and master 0- 1 distribution, binomial distribution, geometric distribution, hypergeometric distribution and Poisson distribution and their applications. 3. Understand the conclusion and application conditions of Poisson theorem, and use Poisson distribution to approximately represent binomial distribution. 4. Understand the concept of continuous random variables and their probability density. Master uniform distribution, normal distribution, exponential distribution and their applications, in which the probability density of exponential distribution with parameters is 5. The distribution of random variable function will be found. 3. Multidimensional Random Variables and Their Distribution Examination Contents Probability distribution, edge distribution and conditional distribution of two-dimensional discrete random variables Probability density, marginal probability density and conditional density of two-dimensional continuous random variables are commonly used. The distribution of two or more simple functions of random variables requires 1. Understand the concept of multidimensional random variables, understand the concept and properties of multidimensional random variable distribution, and understand the probability distribution, edge distribution and conditional distribution of two-dimensional discrete random variables. Understand the probability density, edge density and conditional density of two-dimensional continuous random variables, and you will find the probability of related events of two-dimensional random variables. 2. Understand the concepts of independence and irrelevance of random variables, and master the conditions of independence of random variables. 3. Grasp the two-dimensional uniform distribution, understand the probability density of the two-dimensional normal distribution, and understand the probability meaning of the parameters. 4. Find the distribution of simple functions of two random variables. I will find the distribution of simple functions of multiple independent random variables. Fourth, the digital characteristics of random variables test content The mathematical expectation (mean), variance, standard deviation and its properties of random variables The mathematical expectation moment, covariance, correlation coefficient and its properties test requirements of random variable functions are 1. Understand the concepts of numerical characteristics of random variables (mathematical expectation, variance, standard deviation, moment, covariance, correlation coefficient). Be able to use the basic properties of digital characteristics and master the digital characteristics of common distribution. 2. Know the mathematical expectation of random variable function. 5. Examination content of the law of large numbers and central limit theorem Chebyshev inequality Chebyshev law of large numbers Bernoulli law of large numbers de Morville-Laplace. The test of Levi-Lindbergh theorem needs 65,438+0. Understanding Chebyshev inequality. 2. Understand Chebyshev's law of large numbers, Bernoulli's law of large numbers and Sinchin's law of large numbers (the law of large numbers of independent and identically distributed random variable sequences). 3. Understand De Morville-Laplace theorem (binomial distribution takes normal distribution as the limit distribution). And Levi-Lindbergh theorem (central limit theorem of independent and identically distributed random variable sequence). Basic concepts of mathematical statistics of intransitive verbs examination content General individual simple random sample statistics Sample mean sample variance and sample moment distribution quantile normal general common sampling distribution examination requirements 1. Understand the concepts of population, simple random sample, statistics, sample mean, sample variance and sample moment. Sample variance is defined as 2. Understand the concepts and properties of distribution, distribution, distribution, understand the concept of upper quantile and look up the table to calculate. 3. Understand the common sampling distribution of normal population. 7. Parameter estimation test content point estimation concept estimator and estimation moment estimation method maximum likelihood estimation method estimator selection standard interval estimation concept interval estimation of mean and variance of a single normal population Interval estimation test requirements for mean difference and variance ratio of two normal populations 1. Understand the concepts of point estimation, estimator and estimated value of parameters. 2. Master moment estimation method (first-order moment, second-order moment) and maximum likelihood estimation method. 3. Understand the concepts of unbiased estimator, validity (minimum variance) and consistency (consistency), and verify unbiased estimator. 4. By understanding the concept of interval estimation, we can find the confidence interval of the mean and variance of a single normal population. Will find the confidence interval of the mean difference and variance ratio of two normal populations. Eight. The significance test of the content of the hypothesis test Two types of errors in the hypothesis test Hypothetical test Requirements for the mean and variance of a single and two normal populations 1. Understand the basic idea of significance test, master the basic steps of hypothesis test, and understand two kinds of errors that may occur in hypothesis test. 2. Master the hypothesis test of the mean and variance of single and two normal populations.

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