I. Function, Limit and Continuity

Examination content

The concept and representation of function, boundedness, monotonicity, periodicity and parity of function, the properties of basic elementary functions of inverse function, piecewise function and implicit function, and the establishment of functional relationship of graphic elementary function.

Definitions and properties of sequence limit and function limit, left limit and right limit of function, concepts and relationships of infinitesimal and infinitesimal, properties of infinitesimal and four operational limits of infinitesimal, 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

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, 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 limit 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 function and continuity of elementary function, understand the properties of continuous function on closed interval (boundedness, maximum theorem, mean value theorem), and apply these properties.

Second, the differential calculus of unary function

Examination content

The relationship between the geometric meaning of derivative and differential concepts and the derivability and continuity of physical meaning function; Four operations of tangent, normal derivative and differential of plane curve; Derivative compound function, inverse function and implicit function of basic elementary function; And the invariant differential mean value theorem of the first-order differential form of the function determined by the parametric equation; L'H?pital (l? Distinguish the convexity, inflection point and asymptote of monotone extreme value function graph of regular function, describe the maximum and minimum values of the concept curvature circle of function graph and the curvature radius of arc differential curvature.

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 limit of infinitive with L'H?pital's law.

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.

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.

4. Vector Algebra and Spatial Analytic Geometry

Examination content

The mixed product of the number product and the cross product vector of the linear operation vector of the concept vector of the vector is the condition that the two vectors are vertically parallel; Coordinate expression of included angle vector and its operation unit vector direction number and direction cosine surface equation and space curve equation; Conceptual plane equation; Linear equation; Angle between plane and plane, plane and straight line; The distance under parallel and vertical conditions points to a plane and a straight line; The common quadric equation of spherical cylindrical surface of revolution and the parametric equation of its graphic space curve and the projection curve equation of the general equation space curve on the 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 vertically 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. Will find the included angle between plane, plane and straight line, straight line and straight line, and will use the relationship between plane and straight line (parallel, vertical, intersecting, etc.). ) to solve related problems.

6. You can find the distance from a point to a straight line and the distance from a point to a plane.

7. Understand the concepts of surface equation and space curve equation.

8. Knowing the equation of quadric surface and its graph, we can find out the equation of simple cylindrical surface and revolving surface.

9. Understand the parametric equation and general equation of space curve. Understand the projection of space curve on the coordinate plane, and find the equation of projection curve.

Verb (abbreviation of verb) Differential calculus of multivariate functions

Examination content

The concept of multivariate function, the geometric meaning of binary function, the concepts of limit and continuity of binary function, the properties of multivariate continuous function in bounded closed region, and the necessary and sufficient conditions for the existence of partial derivative and total differential of multivariate function.

Derivation method of multivariate composite function and implicit function Second-order partial derivative directional derivative and second-order Taylor formula of tangent plane and normal plane of gradient space curve Extreme value and conditional extreme value of multivariate function Maximum and minimum value of multivariate function and its simple application

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 functions and the properties of continuous functions in bounded closed regions.

3. By understanding the concepts of partial derivative and total differential of multivariate functions, we can find the total differential, understand the necessary and sufficient conditions for the existence of the total differential, and understand the invariance of the total differential form.

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 multivariate composite functions.

6. Knowing the existence theorem of implicit function, we can find the partial derivative of multivariate implicit function.

7. Understand the concepts of tangent and normal plane of space curve and tangent 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

The concepts, properties, calculation and application of double integral and triple integral; The concept, properties and calculation of two kinds of curve integrals: Green's formula; The condition that the plane curve integral is independent of the path; Original function of binary function; The concept, properties and calculation of two kinds of surface integrals: Gaussian formula; Stokes formula; The concepts of divergence and curl; And the calculation of 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 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 relations 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.

7. The concepts of dissolution and rotation are introduced and calculated.

8. Some geometric and physical quantities (area, volume, surface area, arc length, mass, centroid, centroid, moment of inertia, gravity, work and flow, etc.). ) can be obtained by using multiple integral, curve integral and surface integral.

Seven, infinite 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 conditions of geometric series and convergence and divergence of series.

3. To master the comparison method and ratio method of positive series convergence, the root value method will be used.

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 term series and the concept of function.

7. Understand the concept of convergence radius of power series and master the solution of convergence radius, convergence interval and convergence domain of power series.

8. Knowing the basic properties of power series in its convergence interval (continuity of sum function, item-by-item derivation, item-by-item integration), we will find the sum function of some power series in its convergence interval, and then find the sum of several terms of some series.

9. Understand the necessary and sufficient conditions for the function to expand into Taylor series.

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, quantitative matrix, diagonal matrix, triangular matrix, symmetric matrix and antisymmetric 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, grasp the properties of inverse matrix and the necessary and sufficient conditions of matrix reversibility, understand the concept of adjoint matrix, and use adjoint matrix to find 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 matrix rank and inverse matrix by elementary transformation.

5. Understand the block matrix and its operation.

Third, the vector

Examination content

The linear combination of concept vectors of vectors and the linear representation of vector groups are linearly related to the largest linear independent group of linear independent vector groups. The relationship between the rank of equivalent vector group and the rank of matrix. Base transformation and coordinate transformation in vector space and their related concepts. Orthogonal normalization of inner product linear independent vector group defines orthogonal matrix of orthogonal basis and its properties.

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 concepts of N-dimensional vector space, subspace, basis, dimension and coordinate.

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 standard orthogonal bases and orthogonal matrices.

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 Homogeneous Linear Equations and General Solutions of Non-homogeneous Linear Equations in General Solution Space

Examination requirements

The length can be used by Kramer'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 concepts of basic solution system, general solution and solution space 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. Master the method of solving linear equations with elementary line transformation.

Eigenvalues and eigenvectors of verb (abbreviation of verb) matrix

Examination content

Concepts of eigenvalues and eigenvectors of matrices, property similarity transformation, concepts of similar matrices and necessary and sufficient conditions for similar diagonalization of property matrices, 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 similar matrix and the necessary and sufficient conditions for matrix similarity diagonalization, and master the method of transforming matrix into similar diagonal matrix.

3. Master 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. Master quadratic form and its matrix representation, understand the concepts of quadratic form rank, contract transformation and contract matrix, and understand the concepts of 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 can transform quadratic form into standard form by matching method.

3. Understand the concepts of positive definite quadratic form and positive definite matrix, and master their discrimination methods.

Probability and mathematical statistics

I. Random events and probabilities

Examination content

The relationship between random events and events in sample space and the basic properties of complete operation concept probability Basic formula of classical probability of event group probability Geometric probability Conditional independent repetition test of probability events.

Examination requirements

1. Understand the concept of sample space (basic event space), understand the concept of random events, and 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.

Fourth, the numerical characteristics of random variables

Examination content

Mathematical expectation (mean), variance, standard deviation and their properties of random variables Mathematical expectation moment, covariance, correlation coefficient and their properties of random variable functions

Examination requirements

1. Understand the concept of numerical characteristics of random variables (mathematical expectation, variance, standard deviation, moment, covariance, correlation coefficient), and use the basic properties of numerical characteristics to master the numerical characteristics of common distributions.

2. Know the mathematical expectation of random variable function.

Law of Large Numbers and Central Limit Theorem

Examination content

Chebyshev Inequality Chebyshev's Law of Large Numbers Bernoulli's Law of Large Numbers Democracy-Laplace Theorem Levi-Lindbergh Theorem

Examination requirements

1. 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 moivre-Laplace Theorem (binomial distribution takes normal distribution as the limit distribution) and Levi-Lindbergh Theorem (central limit theorem of independent identically distributed random variable sequence).

Seven. parameter estimation

Examination content

Concept estimation of point estimation and estimated value Method of moment estimation Maximum likelihood estimation Method of estimation criterion Interval estimation Concept Interval estimation of mean and variance of a single normal population Interval estimation of mean difference and variance ratio of two normal populations.

Examination requirements

1. Understand the concepts of point estimation, estimator and parameter estimation.

2. Master moment estimation methods (first-order moment, second-order moment) and maximum likelihood estimation methods.

3. Understand the concepts of unbiased estimator, validity (minimum variance) and consistency (consistency), and verify unbiased estimator.

4. In order to understand the concept of interval estimation, we will find the confidence interval of the mean and variance of a single normal population, and the confidence interval of the mean difference and variance ratio of two normal populations.

Eight, hypothesis testing

Examination content

Two types of false hypothesis testing in significance testing Hypothesis testing of mean and variance of single and two normal populations.

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. Master the hypothesis test of the mean and variance of single and two normal populations.

The above is the original text of the 2020 Postgraduate Mathematics Examination Outline, including line generation, advanced mathematics and probability statistics. Pay attention to the outline information and prepare for the exam more efficiently. Everyone must make rational use of the examination syllabus and master more information related to the examination. Welcome to continue to pay attention.