The third outline of postgraduate mathematics is the examination outline of postgraduate mathematics (subject code 303), including calculus, linear algebra, probability theory and mathematical statistics. It is required to understand the concept and master the representation, so that the functional relationship of the application problem will be established.
Syllabus of Mathematics Level 3 Examination and Related Requirements:
calculus
Function, limit, continuity
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 limit function and left limit and right limit.
6. Understand the nature of limit and two criteria for the existence of limit, master four algorithms of limit, and master the method of finding limit by using two important limits.
7. Understand the concepts of infinitesimal and infinitesimal, master the comparison method of infinitesimal, and find the limit with equivalent infinitesimal.
8. Understanding the concept of function continuity (including left continuity and right continuity) will distinguish the types of function discontinuity points.
9. 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 concept of derivative and the relationship between derivability and continuity, understand the geometric and economic significance of derivative (including the concepts of allowance and elasticity), and find the tangent equation and normal equation of plane curve.
2. Mastering the derivation formula of basic elementary function, four arithmetic rules of derivation and the derivation rule of compound function, we can obtain the derivation of piecewise function, inverse function and implicit function.
If you understand the concept of higher derivative, you will find the higher derivative of a simple function.
4. Understand the concept of differential, the relationship between derivative and differential, and the invariance of first-order differential form, and you will find the differential of function.
5. Understand and apply Rolle theorem, Lagrange mean value theorem, Taylor theorem, and Cauchy mean value theorem.
6. Master the method of using L'H?pital's law to find the limit of indefinite form.
7. Master the method of judging monotonicity of function, understand the concept of function extreme value, and master the solution and application of function extreme value, maximum value and minimum value.
8. We can judge the concavity and convexity of the function graph by derivative, find the inflection point and horizontal, vertical and oblique asymptotes of the function graph, and describe the function graph.
Integral calculus of unary function
Examination requirements
1. Understand the concepts of original function and indefinite integral, master the basic properties and basic integral formula of indefinite integral, and master the substitution integral method and integration by parts of indefinite integral.
2. Understand the concept and basic properties of definite integral, understand the mean value theorem of definite integral, understand the function of upper limit of integral and find its derivative, and master Newton-Leibniz formula, method of substitution and integration by parts of definite integral.
3. Will use definite integral to calculate the area of plane figure, the volume of rotating body and the average value of function, and will use definite integral to solve simple economic application problems.
4. Understand the concept of improper integral, understand the comparative discrimination method of improper integral convergence, and calculate improper integral.
Multivariate function calculus
Examination requirements
1. Understand the concept of multivariate function and the geometric meaning of bivariate function.
2. Understand the concepts of limit and continuity of binary function and the properties of binary continuous function in bounded closed region.
3. Knowing the concepts of partial derivative and total differential of multivariate function, we can find the first and second partial derivatives of multivariate composite function, total differential, existence theorem of implicit function and partial derivative of multivariate implicit function.
4. Understand the concepts of extreme value and conditional extreme value of multivariate function, master the necessary conditions of extreme value of multivariate function, understand the sufficient conditions of extreme value of binary function, find the extreme value of binary function, find the conditional extreme value by Lagrange multiplier method, find the maximum value and minimum value of simple multivariate function, and solve the simple application problem.
5. Understand the concept, basic properties and mean value theorem of double integral, and master the calculation method of double integral (rectangular coordinates. Polar coordinates), understand the simple abnormal double integral of unbounded region and calculate it.
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 convergence and divergence conditions of geometric series and P series.
3. Master the comparison discrimination method, ratio discrimination method and root value discrimination method of convergence of positive series, and use integral 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 concept of convergence radius of power series and master the solution of convergence radius, convergence interval and convergence domain of power series.
7. 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 some series.
8. We have mastered the Maclaurin expansions of the power of e to x, sin x, cos x, ln( 1+x) and (1+x) to a, and we will use them to indirectly expand some simple functions into power series.
Ordinary differential equations and difference equations
Examination requirements
1. Understand differential equations and their concepts such as order, solution, general solution, initial condition and special solution.
2. Master the differential equation with separable variables. Solutions of homogeneous differential equations and first order linear differential equations.
3. Understand the properties and structure of solutions of linear differential equations.
4. 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.
5. 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.
6. Understand the concepts of difference and difference equation and their general and special solutions.
7. Understand the solution method of the first-order linear difference equation with constant coefficients.
8. Can use differential equations to solve simple economic 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 requirements
1. Understand the concept of matrix, the definitions and properties of identity matrix, quantitative matrix, diagonal matrix and triangular matrix, and the definitions and properties of symmetric matrix, antisymmetric matrix and orthogonal matrix.
2. Master the linear operation, multiplication, transposition and its operation rules of matrix, and understand the determinant properties of square matrix power and square matrix product.
3. Understand the concept of inverse matrix, 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 concepts of elementary transformation of matrix and elementary matrix and matrix equivalence, understand the concept of matrix rank, and master the method of finding the inverse matrix and rank of matrix by elementary transformation.
5. Understand the concept of block matrix and master the algorithm of block matrix.
vectors
Examination requirements
1. Understand the concept of vectors and master the addition and multiplication of vectors.
2. Understand the concepts of linear combination and linear representation of vectors, 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 concept of maximal linearly independent group of vector group, and find the maximal linearly independent group and rank of vector group.
4. Understand the concept of vector group equivalence and the relationship between the rank of matrix and the rank of its row (column) vector group.
5. Understand the concept of inner product. Master the Schmidt method of orthogonal normalization of linear independent vector groups.
linear system of equations
Examination requirements
1. will use Cramer's rule to solve linear equations.
2. Master the judgment method of non-homogeneous linear equations with and without solutions.
3. Understand the concept of basic solution system of homogeneous linear equations, and master the solution and general solution of basic solution system 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 requirements
1. Understand the concepts of matrix eigenvalues and eigenvectors, master the properties of matrix eigenvalues, and master the methods of finding matrix eigenvalues and eigenvectors.
2. Understand the concept of matrix similarity, master the properties of similar matrix, understand the necessary and sufficient conditions for matrix similarity to diagonal, and master the method of transforming matrix into similar diagonal matrix.
3. Master the properties of eigenvalues and eigenvectors of real symmetric matrices.
square
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 and canonical form of quadratic form and inertia theorem.
2. Master the method of transforming quadratic form into standard form by orthogonal transformation and transforming quadratic form into standard form by matching method.
3. Understand the positive definite quadratic form. The concept of positive definite matrix, and master its discrimination method,
probability statistics
Random events and probability
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.
Random variables and their distribution
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 concept and probability distribution of discrete random variables, and master 0- 1 distribution, binomial distribution, geometric distribution, hypergeometric distribution, Poisson distribution and their applications.
3. Grasp 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, and master uniform distribution, normal distribution, exponential distribution and their applications.
5. Find the distribution of random variable function.
Multidimensional random variables and their distribution
Examination requirements
1. Understand the concept and basic properties of the distribution function of multidimensional random variables.
2. Understand the probability distribution of two-dimensional discrete random variables and the probability density of two-dimensional continuous random variables, and master the edge distribution and conditional distribution of two-dimensional random variables.
3. Understand the concepts of independence and irrelevance of random variables, master the conditions of mutual independence of random variables, and understand the relationship between irrelevance and independence of random variables.
4. Master two-dimensional uniform distribution and two-dimensional normal distribution? Understand the probability meaning of parameters.
5. The distribution of function will be found according to the joint distribution of two random variables, and the distribution of function will be found according to the joint distribution of several independent random variables.
Numerical characteristics of random variables
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.
3. Understand Chebyshev inequality.
Law of Large Numbers and Central Limit Theorem
Examination requirements
1. Understand Chebyshev's law of large numbers, Bernoulli's law of large numbers and Sinchin's law of large numbers (law of large numbers for independent and identically distributed random variable sequences).
2. Understand de moivre-Laplacian central limit theorem (binomial distribution takes normal distribution as the limit distribution) and Levi-Lindbergh central limit theorem (central limit theorem of independent identically distributed random variable sequence), and use relevant theorems to approximately calculate the probability of random events.
Basic concepts of mathematical statistics
Examination requirements
1. Understand the concepts of population, simple random sample, statistics, sample mean, sample variance and sample moment.
2. Understand variables, variables and typical patterns of variables; Understand the upper quantile of standard normal distribution, t distribution, f distribution and distribution, and look up the corresponding numerical table.
3. Grasp the sampling distribution of sample mean, sample variance and sample moment of normal population.
4. Understand the concept and properties of empirical distribution function.
parameter estimation
Examination content: the concept estimator of point estimation and the moment estimation method of maximum likelihood estimation of estimated value.
Examination requirements
1. Understand the concepts of point estimation, estimator and parameter estimation.
2. Master moment estimation method (first-order moment, second-order moment) and maximum likelihood estimation method.