30 1 Mathematics I Examination subjects: advanced mathematics, linear algebra, probability theory and mathematical statistics.
30 1 Mathematical Bibliography:
Advanced Mathematics Textbook: Advanced Mathematics-Tongji Edition, published by Higher Education Press;
Textbook of Line Generation: Linear Algebra-Tongji Edition, Higher Education Press;
Probability textbook: Probability Theory and Mathematical Statistics-Zhejiang University Press, Higher Education Press;
Advanced mathematics:
Function, limit, continuity
Examination requirements:
1. Understand the concept of function
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 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 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.
Differential calculus of univariate function
Examination requirements:
1. Understand the concepts of derivative and differential, the relationship between derivative and differential, and the relationship between differentiability and continuity of functions.
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.
8. The derivative will be used 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, 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.
Integral calculus of unary 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.
Vector Algebra and Spatial Analytic Geometry
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.
Multivariate differential 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 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 multivariate function extremum and conditional extremum, and solve some simple application problems.
Multivariate function integral calculus
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.
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.
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. We will find the sum function of some power series in the convergence interval, and then we will find the sum of some series with several terms.
9. Understand the necessary and sufficient conditions for the function to expand into Taylor series.
10. If you master maclaurin expansion, you will use it to indirectly expand some simple functions into power series.
1 1. Knowing the concept of Fourier series and Dirichlet's convergence theorem, we expand the functions defined on the ground into Fourier series, and expand the functions defined on the ground into sine series and cosine series, and write the expressions of Fourier series and functions.
ordinary differential equation
Examination requirements:
1. Understand differential equations and their concepts such as order, solution, general solution, initial condition and special solution.
2. Master the solutions of differential equations with separable variables and first-order linear differential equations.
3. Homogeneous differential equations, Bernoulli equations and total differential equations can be solved, and some differential equations can be replaced by simple variables.
4. The following differential equations will be solved by order reduction method:.
5. Understand the properties and structure of solutions of linear differential equations.
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. 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.
8. Euler equation can be solved.
9. Can use differential equations to solve some simple application problems.
linear algebra
decisive factor
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.
matrix
Examination contents: the concept of matrix, linear operation of matrix, the concept and properties of transposed inverse matrix of determinant matrix of multiplication matrix, necessary and sufficient conditions for matrix reversibility, elementary transformation of matrix, equivalent block matrix of 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. Understand the concept of inverse matrix, master 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 inverse matrix.
3. 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.
4. Understand the block matrix and its operation.
vectors
Examination content
The linear combination of the concept vectors of vectors is linearly related to the linear representation vector group and the equivalent vector group of the largest linearly independent vector group. What is the relationship between the rank of the rank vector group and the rank of the matrix? Base transformation and coordinate transformation of vector space and their related concepts; Orthogonal normalization method of inner product linear independent vector group of transfer matrix vector: orthogonal basis specification? Orthogonal matrix and its properties
Examination requirements:
1. 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 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 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 standard orthogonal bases and orthogonal matrices.
linear system of equations
Examination contents: Cramer's rule for linear equations, necessary and sufficient conditions for homogeneous linear equations to have non-zero solutions, necessary and sufficient conditions for nonhomogeneous linear equations to have solutions, and general solutions for spatial nonhomogeneous linear equations.
Examination requirements:
The length can be determined by 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 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 matrices
Examination contents: the concepts of eigenvalues and eigenvectors of matrices, the similarity transformation of properties, and the concepts and properties of similar 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.
square
Test content: quadratic form and its matrix representation? Contract transformation and rank of quadratic form of contract matrix? Inertia theorem? By using orthogonal transformation matching method and the positive definiteness of its matrix, the canonical form and canonical form of quadratic form are transformed into canonical quadratic form.
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 statistics
Random events and probability
Test content: random events and sample space? What is the relationship between the event and the complete operation event group? The concept of probability, the basic properties of probability? Classical probability? Geometric probability? Basic formula of conditional probability independence of events? Independent repeat test
Examination requirements:
1. Understand the concept of sample space (basic event space)
2. Master the addition formula, subtraction formula, multiplication formula, total probability formula and Bayesian formula of probability.
3. Understand the concept of event independence
Random variables and their distribution
Test content: a random variable? The concept and properties of random variable distribution function: the probability distribution of discrete random variables; Probability density of continuous random variables; Distribution of common random variables; Distribution of functions of random variables.
Examination requirements:
1. Knowing the concept of random variables and the concept and properties of distribution functions will calculate the probability of events related to random variables.
2. Understand the conclusion and application conditions of Poisson theorem, and use Poisson distribution to approximately represent binomial distribution.
3. Understand the concept of continuous random variables and their probability density, and master the uniform distribution? Normal distribution? Exponential distribution and its application, in which the probability density of exponential distribution with parameters is
4. Find the distribution of random variable function.
Multidimensional random variables and their distribution
Test contents: multidimensional random variables and their distribution probability distribution, edge distribution and conditional distribution probability density of 2D discrete random variables, marginal probability density and conditional density of 2D continuous random variables, and distribution of two or more simple functions of commonly used 2D random variables.
Examination requirements:
1. Understand the concept of multidimensional random variables, the concept and properties of multidimensional random variable distribution, the probability distribution, edge distribution and conditional distribution of two-dimensional discrete random variables, and the probability density, edge density and conditional density of two-dimensional continuous random variables, so as to 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 mutual 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. Will find the distribution of simple functions of two random variables, and will find the distribution of simple functions of multiple independent random variables.
Numerical characteristics of random variables
Test 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 function.
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 law of large numbers Bernoulli law of large numbers De Morville-Laplace theorem Levi-Lindbergh theorem.
Examination requirements:
1. Understanding Chebyshev Inequality.
2. 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).
Basic concepts of mathematical statistics
Test content: sample variance and sample moment distribution of population sample mean Quantile normal population Simple random sample statistics of commonly used sampling distribution.
Examination requirements:
1. Understand the concepts of population, simple random sample, statistics, sample mean, sample variance and sample moment, where sample variance is defined as:
2. Understand the concept and nature of distribution, distribution and distribution, understand the concept of upper quantile and look it up.
3. Understand the common sampling distribution of normal population.
parameter estimation
Examination contents: concept of point estimation, moment estimation method of estimator and estimated value, maximum likelihood estimation method, selection criteria of estimator, concept of interval estimation, 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.
Hypothesis test
Test content: significance test, hypothesis test, two kinds of errors, hypothesis test 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.
Extended data:
First, you must use math I as your enrollment major.
1. Mechanics, mechanical engineering, optical engineering, instrument science and technology, metallurgical engineering, power engineering and engineering thermophysics, electrical engineering, electronic science and technology, information and communication engineering, control science and engineering, network engineering, electronic information engineering, computer science and technology, civil engineering, surveying and mapping science and technology, transportation engineering, ship and ocean engineering, aerospace science and technology.
2. A first-class discipline in management science and engineering, with an engineering degree.
Second, you must use the enrollment major of Mathematics II.
Textile science and engineering, light industry technology and engineering, agricultural engineering, forestry engineering, food science and engineering are all two disciplines majors.
Three, you must choose a math or math two enrollment major (decided by the admissions unit).
Among the first-class engineering disciplines, such as materials science and engineering, chemical engineering and technology, geological resources and geological engineering, mining engineering, oil and gas engineering, environmental science and engineering, there are two disciplines. Whoever has higher requirements for mathematics should choose mathematics as the major and mathematics as the minor.
Fourth, we must use the enrollment major of Mathematics III.
1. The first-level discipline of economics.
2. Management of business administration, agriculture and forestry economic management level discipline.
3. First-class discipline of management science and engineering, with a degree in management.
Reference link: Baidu Encyclopedia: Postgraduate Mathematics