2, the specific situation:
(1) Advanced Mathematics (the fraction accounts for 78% of the total score) Tongji Sixth Edition Advanced Mathematics, except for the Bernoulli equation with * in the differential equation in Chapter 7, the rest are not tested; All "approximate" questions are not tested; The fourth chapter is the use of indefinite integral, without integral table; Don't take chapter 8, spatial analytic geometry and vector algebra; Chapter 9, section 5, the case of not taking the equation test; Until the tenth chapter, the application of double integral and multiple integral will not be tested later.
(2) Linear Algebra (the fraction accounts for 22% of the total score) Tongji Fifth Edition Linear Algebra, chapter 1-5: determinant, matrix and its operation, elementary transformation of matrix and its equations, linear correlation of vector groups, similar matrix and quadratic form.
Extended data:
Advanced Mathematics in the Second Outline of Postgraduate Mathematics
I. Function, Limit and Continuity
1, exam content
Concept and representation of function, boundedness, monotonicity, periodicity and parity of function, properties and graphs of basic elementary functions of inverse function, piecewise function and implicit function; The establishment of elementary function function relationship, the definition and properties of sequence limit and function limit;
The concepts of left limit and right limit of infinitesimal and infinitesimal and their relationship, the properties of infinitesimal and the comparison of infinitesimal; Four operations of limit; There are two criteria for the existence of limit: monotone bounded criterion and pinch criterion, and two important limits: the concept of function continuity; The types of function discontinuity and the continuity of elementary function; Properties of continuous functions on closed intervals.
2. Examination requirements
(1), understanding the concept of function and mastering the representation 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 composite function and piecewise function, and understand the concepts of inverse function and implicit function.
(4) Grasp the basic elementary function and its graphic properties, and understand the concept of elementary function.
(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) Grasp the nature of limit and four algorithms.
(7) Grasp two criteria for the existence of limit, and use them to find limit, and master the method of finding limit by using two important limits.
(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.
(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 one-variable differential function
1, test 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 the first-order differential form, you will find the differential of the function.
(3) If you understand the concept of higher derivative, you will find the higher derivative of a simple function.
(4), will find the derivative of piecewise function, will find the derivative of implicit function and function determined by parameter equation and inverse function.
(5) Understand and apply Rolle's theorem, Lagrange's mean value theorem and Taylor's theorem, and understand and apply Cauchy's 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 value and minimum value of function and its application.
(8) The concavity and convexity of the function graph can be judged by the derivative (note: in the interval (a, b), let the function f(x) have the second derivative. When f'' (x) >; =0, the graph of f(x) is concave; When f'' (x) < =0, the graph of f(x) is convex), you will find the inflection point and the horizontal, vertical and oblique asymptotes of the function graph, thus depicting the function graph.
(9) Understand the concepts of curvature, curvature circle and curvature radius, and calculate curvature and curvature radius.
Third, the integral of unary function.
1, exam content
The concepts of primitive function and indefinite integral; The basic properties of indefinite integral, the concept of basic integral formula and the basic properties of definite integral; The function and derivative of the upper limit of the mean value theorem of definite integral: Newton-Leibniz formula;
Substitution integration method of indefinite integral and definite integral and application of partial integral; rational formula of rational function, trigonometric function and integral anomaly (generalized) definite integral of simple irrational function.
2. Examination requirements
(1), understand the concept of original function, and understand the concepts of indefinite integral and definite integral.
(2) Master the basic formula of indefinite integral, the properties of indefinite integral and definite integral, the mean value theorem of definite integral, and the substitution integral method and integration by parts.
(3), can find the integral of rational function, rational formula of trigonometric function and simple unreasonable function.
(4) Knowing the function of the upper limit of integral, we will find its derivative and master Newton-Leibniz formula.
(5) Knowing the concept of generalized integral, we can calculate generalized integral.
(6) Mastering to express and calculate the average values of some geometric and physical quantities (the area of a plane figure, the arc length of a plane curve, the volume of a rotating body and the lateral area, the area of a parallel section, the volume, work, gravity, pressure, centroid, centroid, etc. of a known solid. ) and definite integral function.
Four, multivariate function calculus
1, test requirements
(1), understand the concept of multivariate function, and understand the geometric meaning of binary 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 multivariate function extremum and conditional extremum, master the necessary conditions of multivariate function extremum, understand the sufficient conditions of bivariate function extremum, find bivariate function extremum, use Lagrange multiplier method to find conditional extremum, find the maximum and minimum of simple multivariate function, and solve some simple application problems.
(5) Understand the concept and basic properties of double integral, and master the calculation methods of double integral (rectangular coordinates and polar coordinates).
Verb (abbreviation of verb) ordinary differential equation
1, exam content
Basic concepts of ordinary differential equations; The properties of the solution of the first-order linear differential equation and the higher-order differential equation with separable variables, the homogeneous differential equation and the structural theorem of the solution; Second-order homogeneous linear differential equation with constant coefficients; Some homogeneous linear differential equations with constant coefficients above second order: simple non-homogeneous linear differential equations with constant coefficients of second order; Simple application of differential equation.
2. Examination requirements
(1), understand differential equations and their concepts such as order, solution, general solution, initial conditions and special solution.
(2) The homogeneous differential equation can be solved by mastering the solutions of differential equations with separable variables and first-order linear differential equations.
(3), will use the reduced order method to solve the differential equation.
(4) Understand the properties of the solution of the second-order linear differential equation and the structure theorem of the solution.
(5) Mastering the solution of second-order homogeneous linear differential equations with constant coefficients can solve some homogeneous linear differential equations with constant coefficients higher than the second order.
(6), can use polynomial, exponential function, sine function, cosine function and their sum and product to solve second-order non-homogeneous linear differential equations with constant coefficients.
(7), can use differential equations to solve some simple application problems.
The second outline of postgraduate mathematics linear algebra
I. Determinants
1, exam content
The concept and basic properties of determinant The expansion theorem of determinant by row (column)
2. Examination requirements
(1), understand the concept of determinant and master the properties of determinant.
(2), will apply the properties of determinant and determinant expansion theorem to calculate determinant.
Second, the matrix
1, exam content
The concept of matrix; Linear operation of matrix; Matrix multiplication; Power of square matrix; Determinant of square matrix product; Transposition of matrix; The concept and properties of inverse matrix; Necessary and sufficient conditions for matrix reversibility: elementary transformation of adjoint matrix: elementary matrix; Rank of matrix; Equivalence of matrices; Block matrix and its operation.
2. Examination requirements
(1), understand the concept of matrix, identity matrix, quantitative matrix, diagonal matrix, triangular matrix, symmetric matrix, antisymmetric matrix, orthogonal matrix and their properties.
(2) Master the linear operation, multiplication, transposition and its operation rules of matrices, 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 and the necessary and sufficient conditions for matrix reversibility. Understand the concept of adjoint matrix and use adjoint matrix to find the 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
1, exam content
The concept of vector; Linear combination and linearity of vectors; It means that the linear correlation of vector groups has nothing to do with linearity; Maximal linear independent group equivalent vector group of vector group; Rank of vector group; The relationship between the rank of vector group and the rank of matrix; Inner product linearity of vector; Orthogonal Normalization Method for Irrelevant Vector Groups
2. Examination requirements
(1). Solve the concepts of N-dimensional 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) Knowing the concepts of the rank of maximal linearly independent group and vector group, we can find the rank of maximal linearly independent group and 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 and master the Schmidt method of orthogonal normalization of linear independent vector groups.
Fourth, linear equations.
1, exam content:
Cramer's law for linear equations: Necessary and sufficient conditions for homogeneous linear equations to have nonzero solutions: necessary and sufficient conditions for nonhomogeneous linear equations to have solutions: properties and structure of solutions of linear equations; Basic solution system and general solution of homogeneous linear equations; General solution of nonhomogeneous linear equations.
2. Examination requirements
(1), you can use 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 and general solution 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) Linear equations can be solved by elementary line transformation.
Eigenvalues and eigenvectors of verb (abbreviation of verb) matrix
1, exam content
The concepts of eigenvalues and eigenvectors of matrices; The concept and properties of attribute similarity matrix: the necessary and sufficient conditions of matrix similarity diagonalization and the eigenvalue of real symmetric matrix of similar diagonal matrix; Eigenvectors and their similar diagonal matrices.
2. Examination requirements
(1), understand the concepts and properties of eigenvalues and eigenvectors of matrices, and find the eigenvalues and eigenvectors of matrices.
(2) Understand the concept and properties of matrix similarity and the necessary and sufficient conditions for matrix similarity diagonalization, and transform the matrix into a similar diagonal matrix.
(3) Understand the properties of eigenvalues and eigenvectors of real symmetric matrices.
Sixth, quadratic form
1, exam content
Quadratic form and its matrix; Rank inertia theorem representing contract transformation and quadratic form of contract matrix: standard form and standard form of quadratic form; Using orthogonal transformation and matching method to convert quadratic form into standard form; Quadratic form and positive definiteness of its matrix.
2. Examination requirements
(1), understand the concept of quadratic form, express quadratic form in matrix form, and understand the concepts of contract transformation and contract matrix.
(2) Understand the concepts of rank of quadratic form, canonical form and canonical form of quadratic form, and inertia theorem. We will transform quadratic form into standard form through orthogonal transformation and configuration.
(3) Understand the concepts of positive definite quadratic form and positive definite matrix, and master their discrimination methods.
References:
Baidu Encyclopedia-Two Outline of Postgraduate Mathematics