《数学分析》考试大纲 - 河北教师教育网（Syllabus for examination of mathematical analysis - Hebei Teacher Education Network）

《数学分析》考试大纲 - 河北教师教育网（Syllabus for examination of mathematical analysis - Hebei Teacher Education Network） 《数学分析》考试大纲 - 河北教师教育网(Syllabus for examination of mathematical analysis - Hebei Teacher Education Network) Syllabus for examination of mathematical synthesis I. The Syllabus of mathematical analysis Textbook: mathematical analysis (Mathematics Department of East China Normal University) (Third Edition) First, the nature, purpose and requirements of the course: "" mathematical analysis "in mathematics the most important one of the basic courses, is the basis for subsequent learning of mathematics majors continue, theory method and its content is related to the analysis of mathematical rigor and logic for hundreds of years, and in various fields of modern mathematics are closely linked. Is engaged in mathematical theory and its application work necessary knowledge. You understand the basic concepts of mathematical analysis systematically, master the basic method to analyze the field, basically grasp the demonstration methods of mathematical analysis, with calculus skills and preliminary application ability and logical reasoning ability is good. Two, curriculum content and assessment requirements: The first chapter is the real number set and function (1) understanding the real field and its properties (2) master several main inequalities and applications. (3) mastery of the supremum, the supremum and the supremum. (4) grasp the function compound, basic elementary number, elementary function and some characteristics (monotonicity, periodicity, parity, boundedness, etc.). Example: P6: example 2; P7: Example 3; P17: case 2; Exercises: P9:6, 7; P20: 7; P22: 12,13. The second chapter is the limit of sequence (1) master the definition of sequence limit. (2) grasp some properties of convergence series (boundedness, number preserving, inequality preserving, etc.). (3) grasp the conditions of convergence of series (monotone boundedness principle, forcing convergence rule, Cauchy criterion, etc.). Example: P32: example 5; P36: example 2. Exercises: P34:4 (3) (4) (5), 9; P39:3 (1) (2); 11; P40 total exercises: 1, 3, 7, 8. The third chapter, function limit (1) proficiency in the use of epsilon delta language, describing the limits of various types of functions. (2) grasp the properties of function limit. (3) grasp the conditions of the existence of function limit (resolution principle, Cauchy criterion, left and right limit, monotone bounded). (4) apply the two important limits skillfully to find the limit of function. (5) firmly grasp the definition, property and order of infinitesimal (large). Example: P62: example 1, 2. Exercises: P55:1, 5; P59: 2; P67 total exercises: 2. The fourth chapter is the continuity of function (1) mastering the definition and equivalent definitions of a point of continuity. (2) grasp, break, order and classify. (3) understand the definition of continuity on the interval, and use the "left and right limit" approach to the limit. (4) grasp the continuity property of a point and its continuous property on the interval. (5) understand the continuity of elementary functions. Example: P77: example 3, 4; P79: case 9. P84: example 2. Exercises: P81:2, 6, 17, 19, P85, total exercises: 2, 7, 9. The fifth chapter is derivative and differential (1) master the definition of derivative, geometric and physical significance. The derivative is used to define the limit. (2) remember the rules of derivation and the formula of derivation. (3) all kinds of derivatives (complex, parametric, logarithmic and high order derivatives) are sought. (4) grasp the concept of differential. (5) deeply understand the relation of continuity, differentiability and differentiability. Example: P89: example 4, 5; P93: case 8. P109: example 6. Exercises: P94:4, 11, 12, 14, P117, total exercises: 4, 7. The sixth chapter is the differential mean value theorem and its application (1) firmly grasp the differential mean value theorem and Its Applications (including Rolle theorem, Lagrange theorem, Cauchy theorem, Taylor theorem). (2) use the law of law to limit. (3) judge the monotonic interval, the extreme value, the most value, the concavity and convexity and the inflection point of the function. Examples: P128: example 4; P130: cases 8; P131: 10 cases 12; P154 cases (not plotted) Exercises: P124:4, 7, 9, 13; P133: 8; P153: 5; 8; P159 total exercises: 6,12,13. The eighth chapter is indefinite integral (1) grasp the concepts of primitive function and indefinite integral. (2) remember the basic integral formula. (3) master the change element method and the partial integration method. (4) understand the integral steps of rational functions, and find the integrals that can be reduced to rational functions. Example: P185: example 6; P187: case 13, 14 Exercises: P188:1 (11) (12) (23), 2 (5) (6) The ninth chapter is definite integral (1) grasp the definition and properties of definite integral. The definite integral is used to define the limit of the sequence. (2) understanding integrable conditions and Integrable classes. (3) deeply understand the basic theorem of calculus and apply it skillfully. (4) skilled calculation of definite points. Examples: P216: cases 1, 2 cases; P225: cases 1, cases 4. Exercise: P207:2; P229: 2,3,5,7; P237 exercises: 2,3. The tenth chapter is the application of definite integral (1) skilled in calculating the area of various plane figures. (2) the volume of a rotating body or the area of a known cross section. (3) the definite integral is used to calculate the length, curvature, and the lateral area of the rotator. (4) some physical problems (pressure, force work, static moment, center of gravity, etc.) are solved by infinitesimal method. Example: P239: example 1; P244: case 2; P249 case 1, 2; P254: case 1.. Exercise: P242:1; P255: 1 (1) last of two or three volumes The twelfth chapter, number series (1) grasp the definition and properties of convergence and divergence of several series. (2) mastering the criterion of convergence and divergence of positive series. (3) Master condition, absolute convergence and staggered series Leibniz method. Examples: P2: cases 2; P8: cases 2; P11 cases 7. Exercises: P5:3, 4, 5, P16: 4, 5, 6, 7, P25, 2,3.: The thirteenth chapter is function series and function series (1) understand the relation between function column and function item level, and grasp the uniform convergence definition of function column and function item series. (2) the criterion of uniform convergence of function series and function series. (3) understand the limit function of function column, the sum of function series and the property of function. Example: P30: example 3. Exercises: P35:1 (1) (2) (4), 2, 4, 8 (2); P41: 7. The fourteenth chapter is power series (1) the convergent domain, the radius of convergence and the method of sum function of skilled power series. (2) understanding the properties of power series. (3) to understand the method of power series expansion for general differentiable functions of any order. Strongly remember the Maclaurin expansions of six basic elementary functions. (4) the power series expansion of some elementary functions is obtained by the indirect method. Example: P47: example 4. Exercises: P51:2,8; P61 total exercises: 3 (2) (3) The sixteenth chapter is the limit and continuity of multivariate functions (1) understanding some concepts of plane set of points. (2) grasp the definition and nature of the double limit of two variables function. (3) master the two limit and master the relation between the double limit and the two limit. (4) grasp the definition and property of two yuan continuous function. (5) to understand the relation of two variables to the continuity and continuity of the two variables. Example: P95: example 3; P98: example 8. The seventeenth chapter is differential calculus of multivariate functions (1) mastering the meaning of differentiable and partial guidance. (2) grasp the relation between the two variables, differentiable, partial, continuous and partial derivative functions. (3) accounting calculates various types of partial derivatives, accounting for total differential. (4) the tangent plane of the curved surface of space, the normal line. The normal plane and tangent of a space curve. (5) the Taylor expansion and unconditional extreme value for the two variables function. Examples: P110: cases 5; P115: cases 6; P120: cases 2. Exercises: P117:4,5,6,7,11,12; P61 total exercises: 3 (2) (3) The twentieth chapter is curvilinear integral (1) master the calculation method of the integral of the first curve. (2) master the calculation method of integral of type second curve. Example: P200: example 2; P205: case 1,2,3; P120: case 2. Exercise: P201:1 (1) (2) (4); P208: 1 (4) (5) The twentieth chapter is the multiple integral (1) understanding the double integral and the definition and properties of the three integral. (2) master the permutation of double integral and the method of variable substitution. (3) to understand the order of the three integral integrals, the substitution of spheres, columns, and generalized spherical coordinates is used to compute the three integral. (4) the Green curve formula is used to calculate the integral of type second. (5) application of multiple integrals: the surface area, the moment of inertia, the center of gravity coordinates, etc.. Examples: P221: cases 2, 3; P226: cases 1,2; P240: cases 3; 4; P248: cases 3. Exercise: P231:1 (1) (2); P242: 2 (1) (3) The twenty-second chapter is surface integral (1) master the calculation method of surface integral of the first type. (2) master the calculation method of surface integral of type second. (3) skilled use of Gauss formula to calculate surface integral of type second. Example: P291: example 1. Exercise: P295:1. Two, advanced algebra Teaching materials: Advanced Algebra (Department of geometry and algebra, Department of mathematics, Peking University) First, the nature, purpose and requirements of the course: "One of the important basic courses of Higher Algebra" is the Mathematics Department of Hebei Normal University, the main task is to enable students to obtain the knowledge system of mathematics thinking method and the basic theory of polynomial, determinant, linear equations, matrix theory, the two type, linear space, linear transformation in Euclidean space, etc.. On the one hand, for subsequent courses (such as algebraic number theory, discrete mathematics, calculation method, differential equations, functional analysis) to provide some necessary basic theory and knowledge; on the other hand is to improve the thinking ability of students to develop students' intelligence, strengthen the "Three Basics" (basic knowledge, basic theory, basic the theory of cultivating students' creative ability) and other important role. The main part of this course (the lectures and discussion, class exercises, homework tutoring, etc.), so that students of the polynomial theory and linear algebra "analytic theory", and "geometric theory" and its thought to have deep understanding and understanding, which is the basic concept and demonstration methods to help students correct understanding of Higher Algebra and improve the ability to analyze and solve problems. Two, curriculum content and assessment requirements: Chapter 1 polynomial The definition of section 1 number field, and will determine whether a system is the algebraic number field. The definition of a polynomial of the 2 number field P polynomial multiplication, the number of a polynomial ring concept. The operation and operation law of polynomial. The definition, divisible by 3, property division and divisibility. Section 4 of two (or more) the greatest common divisor of polynomials, coprime concepts and properties. The greatest common factor of two polynomials by means of the division of the opposite phase. Section 5 is about the definition and properties of polynomial; factorization and uniqueness theorem; standard factorization of a polynomial. The definition of section 6 square. The concept of section 7 polynomial remainder theorem and properties of polynomial roots. The relation between polynomial and polynomial function. The fundamental theorem of algebra 8. Polynomial decomposition theorem and standard decomposition formula of complex (real) coefficients. The relationship between decomposition and decomposition of integer polynomial 9 the rational coefficient polynomial. The definition of primitive polynomial, Gauss's lemma, the property of rational root of integral coefficient polynomial and Eisenstein discriminant method. Examples: P2: examples 1; P14: examples; P32: cases 1; cases 2; Exercises: P4446:1, 2, 3, 5, 6, 7, 8, 11, 12, 14, 15, 19, 24, 25, 27, 28. Second chapter determinant The introduction of section 1 introduces the background of n determinant. The definition of the 2 arrangement, reverse, reverse, parity arrangement; relationship between the parity arrangement and exchange. The definition of section 3 level determinant, can calculate some special determinant by definition. The basic properties of section 4 determinant. Section 5 matrix, matrix determinant, matrix elementary transformation concept, some simple calculation using determinant determinant. More than 6 sub elements, algebraic concepts. A formula for the expansion of a determinant by a row (column). "Triangle method", "recursive reduced order method", "mathematical induction" and other skills to calculate the determinant. 7 Clem (Cramer) rules. Exercises: P97101:8, 10, 11, 13, 16, 17, 18, 19. The third chapter, linear equations Section 1 general linear equations, elementary transformation concepts and properties, the augmented matrix of linear equations, the general solution of linear equations. The definition of the 2 dimensional vector and vector equal; vector; concept dimension vector space. Section 3 linear combination, linear correlation, linear independent definition and properties. The definition of equivalence of two vector groups and the theorem of equivalent property. The maximal independent group of vectors and the definition of rank, and a maximal independent group of vectors. The definition of row rank, 4 rank, rank of matrix columns. The relation between the rank of a matrix and its child. Section 5 linear equations with discriminant theorem of solution. The basic concept of 6 homogeneous linear equations, The solution of fundamental solutions and the structural theorem of solutions of linear equations. Solving all solutions of general linear equations. Exercises: P155158:2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 23, 24. Fourth chapter matrix Section 1 matrix concept background. Section 2 matrix addition, multiplication, multiplication, transpose operations and operational rules. Section 3 determinant theorem of matrix product, relationship between the rank of matrix product and its factor of rank. Section 4 of invertible matrix, inverse matrix, adjoint matrix concept, a necessary and sufficient condition for a matrix of N reversible and seek a matrix with the formula method of the inverse matrix. Section 5 block matrix, operations and properties of addition and multiplication of the partitioned matrix. The relationship between the 6 elementary matrix, elementary transformation concept and the necessary and sufficient conditions for a matrix of the equivalent standard form and invertible matrix; inverse matrix is a square matrix with the method of elementary transformation. The relationship between the 7 block multiplication of elementary transformation and Generalized Elementary Matrices, and block matrix inverse. Exercises: P197202:1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 23, 24. The fifth chapter, the two type Section 1 of two type and non degenerate linear substitution; corresponding relation matrix for two times and two times of that type and symmetric matrix; contract concept and properties of matrix. The standard form of section 2 two type, two type method to the standard form (with elementary transformation method). The uniqueness of the normal form 3 complex domain and real domain of the two type; inertia theorem. The concept of 4 positive definite, positive semidefinite, negative definite quadratic and two positive definite, positive semi definite matrix; positive definite quadratic and two equivalent conditions of positive semidefinite quadratic two. Exercises: P232234:1, 2, 3, 4, 5, 7, 9, 11, 13, 14. The sixth chapter is linear space Section 1 mapping, injective and surjective (mapping on), the concept of mapping, inverse mapping etc.. The definition and properties of 2 linear space; to determine whether an algebra system is linear space. Section 3 linear combination, linear representation, linear correlation, linear independence concept; concept and properties of n-dimensional linear space. The relationship between 4 base transformation and coordinate transformation. The definition of section 5 subspaces and discriminant theorem; definition and equivalent conditions of vector group generated subspace. The definition and properties of section 6 and the intersection of subspaces and the dimension formula. The concept of section 7 subspace and straight and the necessary and sufficient conditions for a direct sum. The definition, properties and two finite dimensional space is isomorphic to the necessary and sufficient conditions of section 8 of the isomorphism of linear spaces. Exercises: P267270:3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 20, 21. The seventh chapter is about linear transformation The definition and properties of 1 Linear transformation. Operation and operation rules of section 2 of linear transformation, linear polynomial transform. Section 3 linear transformation and matrix; matrix concept of matrix similarity and linear transformation in different medium under similar properties. Section 4 characteristic value and characteristic vector matrix, characteristic polynomial and the concept of nature; seek a matrix of the characteristic value and characteristic vector; Relations between similar matrices and their characteristic polynomials and Hamilton Kailai theorem. Necessary and sufficient conditions for a linear transformation matrix, 5 dimensional linear space in a group of diagonal. The concept of domain, kernel, rank, zero, section 6 linear transformation matrix; range of linear transformation and the corresponding rank relations and linear transformation and the relationship between rank zero. The definition of section 7 of the invariant subspace; to determine whether A- is a subspace subspace; invariant subspace and the relationship between the linear transformation matrix; the spatial V according to eigenvalue decomposition into straight and expression invariant subspace. The definition of section 8 of the Jordan canonical form. Section 9 the concept of minimal polynomial; a minimal polynomial matrix similar to a diagonal matrix and the. Exercises: P320325:1, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 23. The eighth chapter is lambda matrix Some basic concepts are introduced, some simple conclusions are made, and the proof of the theorem is not required. The ninth chapter, Euclidean space The definition and properties of section 1 in Euclidean space; vector length, angle, and two orthogonal vectors concept and basic properties of metric matrix, to enable students to master the relationship and difference between the various concepts. The concept of 2 orthogonal vectors, the standard orthogonal basis, Schmidt orthogonalization process, the vector of a linearly independent set of orthogonal unit. Section 3 defines two Euclidean space isomorphism. The meaning of isomorphism between two Euclidean spaces and the relation between isomorphism and dimension of space. The concept and several equivalence relations 4 orthogonal transform, orthogonal transformation and vector length, standard orthogonal basis, the relationship between the orthogonal matrix. The concept of section 5 of two subspace orthogonal, and the relationship between orthogonal and straight, each sub space and Euclidean space are only orthogonal complement properties. Section 6 any symmetric matrix can be orthogonal similar to a diagonal matrix method for orthogonal array. With orthogonal transformation, the two order is standard type. Exercises: P393396:1, 2, 4, 5, 6, 7, 8, 10, 11, 17, 18. We are not good children!