(1) The content of China University Students' Mathematics Competition (mathematics major) is the basic course teaching content of undergraduate mathematics major, that is, mathematical analysis accounts for 50%, advanced algebra accounts for 35%, and analytic geometry accounts for 65,438+05%. The specific contents are as follows: 1. Mathematical analysis Part I Set and function 65,438+0. Denseness of real number sets, rational and irrational numbers, bounds and supremum, existence theorem of supremum, closed interval set theorem, convergence point theorem, finite covering theorem. 2. Distance, neighborhood, convergence point, boundary, open set, closed set, bounded (unbounded) set, closed rectangular set theorem, convergence point theorem, finite covering theorem, base point sequence on. 3. Existence theorems of inverse function and elementary function and their related properties. 2. Limit and continuity 1. The limit of sequence, the basic properties of convergent sequences (limit uniqueness, boundedness, sign preservation, inequality). 2. Conditions, limits and applications of sequence convergence (Cauchy criterion, forced convergence, monotone bounded principle, the relationship between sequence convergence and its subsequence convergence). 3. Definition of limit of univariate function, basic properties of function limit (uniqueness, local boundedness, sign-preserving, inequality and forced convergence), resolution principle and Cauchy convergence criterion, two important limits and their applications, various calculation methods of limit of univariate function, comparison between infinitesimal and infinite size and order, meaning of O and O, concepts and basic properties of multiple limit and repeated limit of multivariate function. The relationship between double limit and repetition limit of binary function. 4. Continuity and discontinuity of functions, uniform continuity, local properties of continuous functions (locally bounded, sign-preserving), and properties of continuous functions on bounded closed sets (boundedness, maximum theorem, mean value theorem, uniform continuity). 3. Differential calculus of unary function 1. Derivative and its geometric meaning, the relationship between derivability and continuity, uniform continuity. Differential and its geometric significance, the relationship between differentiability and differentiability, the invariance of first-order differential form. 2. Basic theorems of differential calculus: Fermat theorem, Rolle theorem, Lagrange theorem, Cauchy theorem, Taylor formula (Peano remainder and Lagrange remainder). 3. Application of one-dimensional differential calculus: discrimination of monotonicity of function, extreme value, maximum value, minimum value, convex function and its application, convexity of curve, inflection point, asymptote, discussion of function image, Lobida's rule, approximate calculation. 4. Differential calculus of multivariate functions 1. The relationship between differentiability and the existence and continuity of partial derivatives, partial derivatives and total derivatives of composite functions, invariance of first-order differential forms, directional derivatives and gradients, higher-order partial derivatives and mixed partial derivatives have nothing to do with order, mean value theorem of binary functions and Taylor formula. 2. Existence theorem of implicit function, existence theorem of implicit function group, derivation method of implicit function (group), inverse function group, coordinate transformation. 3. Geometric application (tangent and normal of plane curve, tangent and normal of space curve, tangent and normal of surface). 4. Extreme value problem (necessary and sufficient condition), conditional extreme value and Lagrange multiplier method. V. Integral of unary function 1. Primitive function and indefinite integral, basic calculation methods of indefinite integral (direct integral method, method of substitution, partial integral), rational function integral: type, type. 2. Definite integral and its geometric meaning, integrable condition (necessary and sufficient condition:), integrable function. 3. Properties of definite integral (about interval additivity, inequality properties, absolute integrability, the first mean value theorem of definite integral), variable upper bound integral function, basic theorem of calculus, N-L formula and calculation of definite integral, and the second mean value theorem of definite integral. 4. Absolute convergence and conditional convergence, non-negative convergence criterion (comparison principle, Cauchy criterion), Abel criterion, Dirichlet criterion, the concept of generalized integral of unbounded function and its convergence criterion. 5. Differential element method, geometric application (area of plane figure, volume of known cross-sectional area function, arc length and arc differential of curve, volume of rotating body), and other applications. 6. Multivariate function integral 1. Double integral and its geometric significance, calculation of double integral (divided into repeated integral, polar coordinate transformation and general coordinate transformation). 2. Triple integral and calculation of triple integral (divided into repeated integral, cylindrical coordinate transformation and spherical coordinate transformation). 3. The application of double integral (volume, surface area, center of gravity, moment of inertia, etc. Interchangeability of operation sequence. Uniform convergence of generalized integrals with parameters and its discrimination method, continuity, differentiability and integrability of generalized integrals with parameters, interchangeability of operation sequences. 5. The concept, basic properties and calculation of the first kind of curve integral and surface integral. 6. The concept, properties and calculation of the second kind of curve integral: Green's formula, the condition that the plane curve integral is independent of the path. 7. The concepts, properties and calculation of the edge of surface and the second kind of surface integral, and the relationship between the two kinds of line integral and the two kinds of area fraction. 7. Infinite series 1. Sequence and its convergence and divergence, sum of series, Cauchy criterion, necessary conditions for convergence, basic properties of convergent series. Necessary and sufficient conditions, comparison principle, ratio discrimination, root discrimination and their limit forms of convergence of positive series; Leibniz discriminant of staggered series: absolute convergence, conditional convergence, Abel discriminant, Dirichlet discriminant of general term series. 2. Uniform convergence of function series and function series, Cauchy criterion, uniform convergence discrimination (M- discrimination, Abel discrimination, Dirichlet discrimination), uniform convergence function series, properties and applications of function series. 3. The concept of power series, Abel theorem, convergence radius and interval, uniform convergence of power series, item-by-item integrability and differentiability of power series and their applications, the relationship between the coefficient of power series and its sum function, power series expansion of functions, Taylor series and Kraulin series. 4. Trigonometric series of Fourier series, orthogonality of trigonometric function system, Fourier series expansion of 2 and 2 periodic functions, Puwell inequality, Riemann-Leberg theorem, convergence theorem of Fourier series of piecewise smooth functions. Ⅱ. Advanced Algebra Part I: Polynomial 1. The concept of number field and unary polynomial 2. Polynomial divisibility, divisibility with remainder, greatest common factor, alternate divisibility of 3. Coprime, irreducible polynomials, multiple factorials and multiple roots 4. Polynomial function, remainder theorem, roots and properties of polynomial 5. Basic theorem of algebra, complex coefficient factorization and real coefficient polynomial 6. Factorization of polynomials with rational coefficients, eisenstein discriminant method, rational roots of polynomials in rational number field. 7. Multivariate polynomials and symmetric polynomials, Vieta theorem. 2. Determinant 1. Definition of determinant of order n. 2. The properties of determinant of order n.. 3. Calculation of determinant. 4. The determinant is expanded by rows (columns). 5. Laplace expansion theorem. 6. Cramer's Law. 3. Linear equations 1. Gauss. General solution of linear equations. 2. the operation of n-dimensional vectors and vector groups. 3. Linear combination of vectors, linear correlation and linear independence, and equivalence of two vector groups. 4. Maximum independent group and rank of vector group. 5. The relationship among row rank, column rank, rank and its subforms of a matrix. 6. The discriminant theorem of linear equations and the structure of solutions of linear equations. 7. Solution space and its dimension. Matrix 1. The concept of matrix, its operations (such as addition, multiplication, multiplication, transposition, etc. ) and its operating rules. 2. The relationship between determinant of matrix product and rank of matrix product and rank of its factors. 3. Conditions for matrix inversion, adjoint matrix and matrix reversibility. 4. Block matrix and its operation and properties. 5. elementary. Equivalent canonical form of matrix. 6. Block elementary matrix and block elementary transformation. 5. Bilinear function and quadratic form 1. Bilinear function and dual space 2. Quadratic form and its matrix representation. 3. The standard form of quadratic form, the matching method of transforming quadratic form into standard form, the elementary transformation method and the orthogonal transformation method. 4. Uniqueness of quadratic canonical form in complex number field and real number field. Semi-positive definite, semi-positive definite quadratic form and positive definite and semi-positive definite matrix VI. Linear space 1. Definition and simple properties of linear space II. Dimensions, foundations and coordinates. 3. Base transformation and coordinate transformation. 4. Linear subspace. 5. Intersection of subspaces, dimension formula and direct sum of subspaces. 7. Linear transformation 1. Definition of linear transformation, operation of linear transformation, matrix of linear transformation. 2. Eigenvalues and eigenvectors, diagonalizable linear transformations. 3. Similar matrices and similar invariants. Hamilton-Kelly theorem. 4. Range, kernel and invariant subspace of linear transformation. 8. Jordan canonical form 1. Matrix. 2. Conditions of similarity between determinant factor, invariant factor, elementary factor and matrix. 3. Jordan canonical form. 9. Euclidean space 1. Length of inner product and euclidean space and vector. Schmidt orthogonalization method. 3. Isomorphism of Euclidean space. 4. Orthogonal transformation, orthogonal complement of subspace. 5. Symmetric transformation, canonical form of real symmetric matrix. 6. Principal axis theorem, transforming real quadratic form or real symmetric matrix into standard form by orthogonal transformation. 7. unitary space. Three. Analytic geometry part I. Vector and coordinate 1. Vector. Decomposition and geometric operation of vectors. 2. The concept of coordinate system, the coordinates between vectors and points, and the algebraic operation of vectors. 3. The projection of the vector on the axis and its properties, direction cosine and vector included angle. 4. The definition, geometric meaning, operational properties, calculation methods and applications of the quantity product, cross product and mixed product of vectors. 5. Solve some geometric and trigonometric problems with vectors. 2. Trajectory and equation 1. Definition of surface equation: general equation, parametric equation (mutual transformation between vector and coordinate) and their relationship. 2. The general form of space curve equation, parameter equation and their relationship. 3. The general method of establishing space surface and curve equation, simple surface and curve equation with vector. 4. Standard equation and general equation of spherical surface, cylindrical equation with bus parallel to coordinate axis. 3. The straight line between plane and space 1. Various forms of plane equation and straight line equation, and the meaning of each letter in the equation. 2. Starting from determining the geometric conditions of the plane and the straight line, select the appropriate method to establish the equations of the plane and the straight line. 3. According to the equation of plane and straight line and the coordinates of points, determine the positional relationship between plane and plane, straight line and straight line, and plane and straight line. Find the common perpendicular equation of two straight lines in different planes. Quadratic surface 1. Define cylindrical surface, conical surface and surface of revolution, and find the equations of cylindrical surface, conical surface and surface of revolution. 2. The standard equations and main properties of ellipsoid, hyperboloid and paraboloid. The standard equation of quadric surface is established according to different situations. 3. The straightness of hyperboloid and hyperbolic paraboloid and the solution of straight generatrix of hyperboloid and hyperbolic paraboloid. 4. According to the given line family, the ruled surface equation represented by it is obtained. Find the trajectory of moving straight line and moving curve. 5. General theory of conic 1. The direction, center and asymptote of the conic. 2. Tangents of conic, normal points and singular points of conic. 3. Quadric diameter, yoke direction and yoke diameter. 4. Principal axis and direction of conic. Characteristic equation, characteristic root. 5. Simplify the quadratic curve equation and draw the position diagram of the curve in the coordinate system. (2) The content of Chinese college students' mathematics competition (non-mathematics majors) is the teaching content of advanced mathematics courses for undergraduate science and engineering majors. The details are as follows: 1. function, limit, continuous 1. The concept and representation of function, and the establishment of functional relationship of simple application problems. 2. Properties of functions: boundedness, monotonicity, periodicity and parity. 3. The properties of compound function, inverse function, piecewise function and implicit function, basic elementary function and its graph and elementary function. 4. Limit of series and function. Left and right limits of functions. 5. The concepts of infinitesimal and infinitesimal and their relationship, the nature of infinitesimal and the comparison of infinitesimal. 6. Four operations of limit, monotone bounded criterion and pinching criterion of limit existence, two important limits. 7. Continuity of function (including left and right continuity) and types of function discontinuity. 8. The nature of continuous function and the continuity of elementary function. 9. Properties of continuous functions on closed intervals (boundedness, maximum theorem, mean value theorem). 2. Differential calculus of unary function 1. The concepts of derivative and differential, the geometric meaning and physical meaning of derivative, the relationship between differentiability and continuity of function, the tangent and normal of plane curve. 2. Four operations of derivative and differential, invariance of first-order differential form. 3. Differential methods of complex variable functions, inverse functions, implicit functions and functions determined by parametric equations. 4. The concept of higher derivative, the second derivative of piecewise function and the n derivative of some simple functions. 5. Differential mean value theorem, including Rolle theorem, Lagrange mean value theorem, Cauchy mean value theorem and Taylor theorem. 6. Lobida's law and the limit of indefinite form. 7. Extreme value of function, monotonicity of function, concavity and convexity of function graph, inflection point and asymptote (horizontal, vertical and oblique asymptote), and description of function graph. 8. Maximum and minimum values of functions and their simple applications. 9. Integral of unary function 1. The concepts of primitive function and indefinite integral. 2. Basic properties and basic integral formulas. 3. The concept and basic properties of definite integral, the mean value theorem of definite integral, the function determined by variable upper bound definite integral and its derivative, Newton-Leibniz formula. 4. Substitution integral method of indefinite integral and definite integral and integration by parts. 5. Rational formula of trigonometric function and integral of simple irrational function. 6. Generalized integration. 7. Application of definite integral: the area of plane figure, the arc length of plane curve, the volume of rotating body, lateral area and the area of parallel section are the average values of known three-dimensional volume, work, gravity, pressure and function. 4. Ordinary differential equation 1. Basic concepts of ordinary differential equations: differential equations and their solutions, orders and connections. Initial conditions and special solutions, etc. 2. Differential equations with separable variables, homogeneous differential equations, first-order linear differential equations, Bernoulli equations and total differential equations. 3. Some differential equations that can be solved by simple variable substitution, and higher-order differential equations that can be reduced: 4. Properties of solutions of linear differential equations and structural theorems of solutions. 5. Second-order homogeneous linear differential equations with constant coefficients, some homogeneous linear differential equations with constant coefficients higher than the second order. 6. Simple second-order non-homogeneous linear differential equation with constant coefficients: the free term is polynomial, exponential function, sine function, cosine function and their sum and product 7. Euler equation 8. Simple application of differential equation 5. Vector algebra and spatial analytic geometry 1. Concept of vector, linear operation of vector, quantitative product of vector and mixed product of cross product and vector 2. The condition that two vectors are vertical and parallel, and the included angle between the two vectors is 3. Coordinate representation of vector and its operation, unit vector, direction number, direction cosine. 4. The concepts of surface equation and space curve equation, plane equation and straight line equation. 5. Angle between plane and plane, angle between plane and straight line, conditions of parallelism and verticality, distance from point to plane and distance from point to straight line. 6. Equation of spherical surface, cylinder whose generatrix is parallel to the coordinate axis, and rotation surface with the rotation axis as the coordinate axis. Quadratic equation in common use and its graphs. 7. Parametric equation and general equation of space curve, projection curve equation of space curve on coordinate plane. 6. Differential calculus of multivariate functions 1. The concept of multivariate function and the geometric meaning of binary function. 2. The concepts of limit and continuity of binary functions, and the properties of multivariate continuous functions on bounded closed fields. 3. The necessary and sufficient conditions for the existence of partial derivatives and total differential of multivariate functions. 4. Derivation of multivariate composite function and implicit function. 5. Second-order partial derivative, directional derivative and gradient. 6. The tangent plane and normal plane of a space curve, and the tangent plane and normal of a surface. 7. The second-order Taylor formula of binary function. 8. Extreme value and conditional extreme value of multivariate function, Lagrange multiplier method, maximum and minimum value of multivariate function and its simple application. 7. Multivariate function integral 1. The concepts and properties of double integral and triple integral, the calculation of double integral (rectangular coordinates, polar coordinates) and the calculation of triple integral (rectangular coordinates, cylindrical coordinates, spherical coordinates). 2. The concept, properties and calculation of two kinds of curve integrals, and the relationship between the two kinds of curve integrals. 3. Green's formula, the conditional plane curve integral has nothing to do with the path, and it is known that the binary function is fully differentiated to find the original function. 4. The concept, properties and calculation of two kinds of surface integrals, and the relationship between the two kinds of surface integrals. 5. The concepts and calculation of Gauss formula, Stokes formula, divergence and curl. 6. The application and process of multiple integral, curve integral and surface integral (area of plane figure, volume, surface area, arc length and quality of three-dimensional figure). ) eight. Infinite series 1. Convergence and divergence of constant series, sum of convergent series, basic properties of series and necessary conditions for convergence. 2. Geometric series and P series and their convergence, and the discrimination of convergence of positive series, staggered series and Leibniz series. Discrimination method. 3. Absolute convergence and conditional convergence of arbitrary series. 4. The concept of convergence domain and the sum function of function series. 5. Power series and its convergence radius, convergence interval (open interval), convergence domain and function. 6. The basic properties of power series in its convergence interval (continuity of sum function, item-by-item derivation, item-by-item integration), and the solution of simple power series and function. 7. Power series expansion of elementary functions. 8. Extended reading of Fourier coefficients and Fourier series of functions, Dirichlet theorem, Fourier series of functions on [- 1, l], sine series and cosine series of functions on [0, l];