The outline of non-mathematical advanced mathematics examination is as follows
I. Function, Limit and Continuity
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 graphics and elementary function.
4.？ Definition and properties of sequence limit and function limit, left limit and right limit of function.
5.？ The concepts of infinitesimal and infinitesimal and their relations, the properties of infinitesimal and the comparison of infinitesimal.
6.？ Four operations of limit, monotone boundedness criterion and pinch criterion of limit existence, two important limits.
7.？ The continuity of function (including left continuity and right continuity) and the types of function discontinuity.
8.？ Properties of continuous function and continuity of elementary function.
9.？ Properties of continuous functions on closed intervals (boundedness, maximum theorem, minimum theorem, intermediate value theorem).
Second, the differential calculus of unary function
1. Concepts of derivative and differential, geometric meaning and physical meaning of derivative, relationship between derivability and continuity of function, tangent and normal of plane curve.
2. Four operations of derivative, derivative and differential of basic elementary function, and 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. Robida's law and the limit of infinitive.
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. Arc differential, curvature and radius of curvature.
3. Integral calculus of unary function
1.？ The concepts of primitive function and indefinite integral.
2.？ Basic properties of indefinite integral and basic integral formula.
3.？ Concept and basic properties of definite integral, mean value theorem of definite integral, functions and derivatives of variable upper bound definite integral, Newton-Leibniz formula.
4. Substitution integration method for indefinite integral, definite integral and partial integral.
5.？ Rational expressions of rational functions, trigonometric functions and integrals of simple irrational functions.
6.？ Generalized integral
7.？ The 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 solid volume, work, gravity, pressure and function.
Four. ordinary differential equation
1.？ Basic concepts of ordinary differential equations: differential equations and their solutions, orders, general solutions, initial conditions and special solutions.
2.？ Differential equation of separable variables, homogeneous differential equation, first-order linear differential equation, Bernoulli equation, total differential equation.
3. Some differential equations and higher-order differential equations that can be solved by substitution of simple variables:
4. Properties of solutions of linear differential equations and structural theorems of solutions.
5.？ Second-order homogeneous linear differential equations with constant coefficients and 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
Verb (abbreviation of verb) Vector Algebra and Spatial Analytic Geometry
1.？ Concept of vector, linear operation of vector, quantitative product of vector, cross product and mixed product of vector.
2.？ The condition that two vectors are vertically parallel and the included angle between the two vectors.
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, angle between straight line and straight line, conditions of parallelism and verticality, distance from point to plane, distance from point to straight line.
6.？ Sphere, cylinder whose generatrix is parallel to the coordinate axis, equation of rotating surface with the axis of rotation as the coordinate axis, common quadratic equation and its graph.
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 bivariate function.
2.？ The concepts of limit and continuity of binary function and the properties of multivariate continuous function in bounded closed region.
3.？ Necessary and sufficient conditions for the existence of partial derivatives and total differentials of multivariate functions. 4. Derivation of multivariate composite function and implicit function. 5. Second-order partial derivative, directional derivative and gradient.
4.？ Tangent plane and normal of space curve, tangent plane and normal of surface.
5.？ Second-order Taylor formula of binary function
6.？ Extreme value and conditional extreme value of multivariate function, Lagrange multiplier method, maximum and minimum value of multivariate function and its simple application.
Seven, multivariate function integral calculus
1.？ The concepts and properties of double integral and triple integral, the calculation of double integral (rectangular coordinates, polar coordinates) and 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 condition that plane curve integral has nothing to do with path, and the original function of binary function is known.
4.？ The concept, properties and calculation of two kinds of surface integrals, and the relationship between the two kinds of surface integrals.
5.？ Concept and calculation of Gauss formula, Stokes formula, divergence and curl.
6.？ Application of multiple integral, curve integral and surface integral (area of plane figure, volume of three-dimensional figure, surface area, arc length, mass, center of mass, moment of inertia, gravity, work, flow, etc. )
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, discrimination of convergence of positive series, discrimination of staggered series and Leibniz.
3.？ Absolute convergence and conditional convergence of arbitrary series.
4.？ Convergence domain of function term series and the concept of sum function.
5.？ Power series and its convergence radius, convergence 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 sum function.
7.？ Power series expansion of elementary functions.
8.？ Fourier coefficients and Fourier series of functions, Dirichlet theorem, Fourier series of functions on [- 1, 1], sine series and cosine series of functions on [0, 1].
I. Function, Limit and Continuity
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 graphics and elementary function.
4.？ Definition and properties of sequence limit and function limit, left limit and right limit of function.
5.？ The concepts of infinitesimal and infinitesimal and their relations, the properties of infinitesimal and the comparison of infinitesimal.
6.？ Four operations of limit, monotone boundedness criterion and pinch criterion of limit existence, two important limits.
7.？ The continuity of function (including left continuity and right continuity) and the types of function discontinuity.
8.？ Properties of continuous function and continuity of elementary function.
9.？ Properties of continuous functions on closed intervals (boundedness, maximum theorem, minimum theorem, intermediate value theorem).
Second, the differential calculus of unary function
1. Concepts of derivative and differential, geometric meaning and physical meaning of derivative, relationship between derivability and continuity of function, tangent and normal of plane curve.
2. Four operations of derivative, derivative and differential of basic elementary function, and 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. Robida's law and the limit of infinitive.
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. Arc differential, curvature and radius of curvature.
3. Integral calculus of unary function
1.？ The concepts of primitive function and indefinite integral.
2.？ Basic properties of indefinite integral and basic integral formula.
3.？ Concept and basic properties of definite integral, mean value theorem of definite integral, functions and derivatives of variable upper bound definite integral, Newton-Leibniz formula.
4. Substitution integration method for indefinite integral, definite integral and partial integral.
5.？ Rational expressions of rational functions, trigonometric functions and integrals of simple irrational functions.
6.？ Generalized integral
7.？ The 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 solid volume, work, gravity, pressure and function.
Four. ordinary differential equation
1.？ Basic concepts of ordinary differential equations: differential equations and their solutions, orders, general solutions, initial conditions and special solutions.
2.？ Differential equation of separable variables, homogeneous differential equation, first-order linear differential equation, Bernoulli equation, total differential equation.
3. Some differential equations and higher-order differential equations that can be solved by substitution of simple variables:
4. Properties of solutions of linear differential equations and structural theorems of solutions.
5.？ Second-order homogeneous linear differential equations with constant coefficients and 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
Verb (abbreviation of verb) Vector Algebra and Spatial Analytic Geometry
1.？ Concept of vector, linear operation of vector, quantitative product of vector, cross product and mixed product of vector.
2.？ The condition that two vectors are vertically parallel and the included angle between the two vectors.
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, angle between straight line and straight line, conditions of parallelism and verticality, distance from point to plane, distance from point to straight line.
6.？ Sphere, cylinder whose generatrix is parallel to the coordinate axis, equation of rotating surface with the axis of rotation as the coordinate axis, common quadratic equation and its graph.
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 bivariate function.
2.？ The concepts of limit and continuity of binary function and the properties of multivariate continuous function in bounded closed region.
3.？ Necessary and sufficient conditions for the existence of partial derivatives and total differentials of multivariate functions. 4. Derivation of multivariate composite function and implicit function. 5. Second-order partial derivative, directional derivative and gradient.
4.？ Tangent plane and normal of space curve, tangent plane and normal of surface.
5.？ Second-order Taylor formula of binary function
6.？ Extreme value and conditional extreme value of multivariate function, Lagrange multiplier method, maximum and minimum value of multivariate function and its simple application.
Seven, multivariate function integral calculus
1.？ The concepts and properties of double integral and triple integral, the calculation of double integral (rectangular coordinates, polar coordinates) and 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 condition that plane curve integral has nothing to do with path, and the original function of binary function is known.
4.？ The concept, properties and calculation of two kinds of surface integrals, and the relationship between the two kinds of surface integrals.
5.？ Concept and calculation of Gauss formula, Stokes formula, divergence and curl.
6.？ Application of multiple integral, curve integral and surface integral (area of plane figure, volume of three-dimensional figure, surface area, arc length, mass, center of mass, moment of inertia, gravity, work, flow, etc. )
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, discrimination of convergence of positive series, discrimination of staggered series and Leibniz.
3.？ Absolute convergence and conditional convergence of arbitrary series.
4.？ Convergence domain of function term series and the concept of sum function.
5.？ Power series and its convergence radius, convergence 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 sum function.
7.？ Power series expansion of elementary functions.
8.？ Fourier coefficients and Fourier series of functions, Dirichlet theorem, Fourier series of functions on [- 1, 1], sine series and cosine series of functions on [0, 1].
The outline of the national college students' mathematics competition is as follows:
Mathematical analysis accounts for 50%, advanced algebra accounts for 35%, and analytic geometry accounts for 15%. Details are as follows:
ⅰ. Mathematical analysis part?
1. Settings and functions?
1. Denseness of real number set, rational number and irrational number, boundary and supremum of real number set, existence theorem of supremum, closed interval set theorem, aggregation point theorem, finite covering theorem.
2. Distance, neighborhood, gathering point, boundary point, boundary, open set, closed set, bounded (unbounded) set, closed rectangular sleeve theorem, gathering point theorem, finite covering theorem, basic point sequence on and the generalization of the above concepts and theorems.
3. The concepts of function, mapping and transformation and their geometric significance, the concepts of implicit function, inverse function and inverse transformation, the existence theorem of inverse function, elementary function and its related properties.
Second, limit and continuity?
The limit of 1. sequence, the basic properties of convergent sequences (limit uniqueness, boundedness, sign-preserving, inequality properties).
2. Conditions of sequence convergence (Cauchy criterion, forced convergence, monotone bounded principle, the relationship between sequence convergence and its subsequence convergence), important limits and their applications.
3. Definition of limit of univariate function, basic properties of function limit (uniqueness, local boundedness, sign preservation, 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 of sum order between infinitesimal and infinitesimal, meaning of O and O, concepts and basic properties of multiple limits and repeated limits of multivariate function, and binary function.
4. Continuity and discontinuity of functions, uniform continuity, local properties of continuous functions (locally bounded and sign-preserving), and properties of continuous functions on bounded closed sets (boundedness, maximum value theorem, mean value theorem, uniform continuity).
Third, the differential calculus of unary function?
1. Derivative and its geometric meaning, the relationship between differentiability and continuity, various calculation methods of derivative, differential and its geometric meaning, the relationship between differentiability and differentiability, and 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, concavity and convexity of curve, inflection point, asymptote, discussion of function image, Lobida's rule, approximate calculation.
Fourth, differential calculus of multivariate functions?
1. Partial derivative, total derivative and its geometric significance, the relationship between differentiability and existence and continuity of partial derivative, partial derivative and total derivative of composite function, invariance of first-order differential form, directional derivative and gradient, higher-order partial derivative, mixed partial derivative and order independence, mean value theorem of binary function 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.
Verb (verb's abbreviation) Integral calculus of unary function?
1. Primitive function and indefinite integral, basic calculation method of indefinite integral (direct integral method, method of substitution, partial integral), rational function integral (triangular rational type, radical type).
2. Definite integral and its geometric meaning, integrable conditions (necessary and sufficient conditions) and integrable function classes.
3. The properties of definite integral (about interval additivity, inequality, absolute integrability, first mean value theorem of definite integral), variable upper bound integral function, basic theorem of calculus, N-L formula and calculation of definite integral, second mean value theorem of definite integral.
4. Generalized integral on infinite interval, Canchy convergence criterion, absolute convergence and conditional convergence, convergence judgment method of infinite interval when f(x) is nonnegative (comparison principle, Cauchy judgment method), Abel judgment method, Dirichlet judgment method, unbounded function generalized integral concept and its convergence judgment method.
5. Infinitesimal 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.
Six, multivariate function integral?
1. Double integral and its geometric meaning, calculation of double integral (divided into repeated integral, polar coordinate transformation and general coordinate transformation).
2. Triple integral, triple integral calculation (divided into repeated integral, cylindrical coordinate and spherical coordinate transformation).
3. Application of multiple integrals (volume, surface area, center of gravity, moment of inertia, etc.). )?
4. Normal integrals with parameters and their continuity, differentiability and integrability, interchangeability of operation order, uniform convergence of generalized integrals with parameters and its discrimination method, continuity, differentiability and integrability of generalized integrals with parameters, interchangeability of operation order.
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 edge integral and the second kind of surface integral, Auger formula and Stoke formula, and the relationship between two kinds of line integral and two kinds of area fraction.
Seven, infinite series?
1. series?
Series 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 of general term series, Dirichlet discriminant.
2. Function term series?
Uniform convergence of function sequence and function term series, Cauchy criterion, uniform convergence discrimination (M- discrimination, Abel discrimination, Dirichlet discrimination), uniform convergence function sequence and function term series and their applications.
3. Power series?
Concept of power series, Abel theorem, convergence radius and interval, uniform convergence of power series, item-by-item integrability, differentiability and its application, relationship between coefficient of power series and its sum function, power series expansion of function, Taylor series and Kraulin series.
4. Fourier series?
Trigonometric 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?
First, polynomial?
The concepts of 1. number field and unary polynomial?
2. Polynomial divisibility, divisibility with remainder, divisibility with greatest common factor and divisibility with phase?
3. coprime, irreducible polynomials, multiple factors and multiple roots.
4. Polynomial function, remainder theorem, roots and properties of polynomials.
5. Basic theorem of algebra, factorization of complex coefficients, polynomial of real coefficients.
6. Primitive polynomial, Gauss lemma, factorization of rational coefficient polynomial, eisenstein discriminant method, rational root of polynomial in rational number field. 7. Multivariate polynomials and symmetric polynomials, Vieta theorem.
Second, determinant?
The definition of 1 Determinant.
2. What are 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.
Third, linear equations?
1. gauss elimination, elementary transformation of linear equations, general solution of linear equations.
2.n-dimensional vector operation and vector group.
3. Linear combination of vectors, linear correlation has nothing to do with linearity, and the two vector groups are equivalent.
4. Maximal independent group of vector group and rank of vector group.
5. The row rank, column rank and rank of a matrix, and the relationship between the rank of a matrix and its sub-formulas.
6. The discriminant theorem of linear equations and the structure of solutions of linear equations.
7. What is the basic solution system, solution space and dimension of homogeneous linear equations?
Fourth, the matrix?
The concept of 1. matrix, its operations (addition, number multiplication, multiplication, transposition, etc. ) and its operating rules.
2. The relationship between determinant of matrix product, rank of matrix product and rank of its factors.
3. Conditions of inverse, adjoint and invertibility of matrices.
4. Block matrix and its operation and properties.
5. Elementary matrix, elementary transformation and equivalent canonical form of matrix.
6. Block elementary matrix and block elementary transformation.
5. Bilinear function and quadratic form?
1. Bilinear function, dual space?
2. Quadratic form and its matrix representation.
3. The canonical form of quadratic form, the matching method of transforming quadratic form into canonical form, the elementary transformation method and the orthogonal transformation method.
4. Uniqueness and inertia theorem of quadratic canonical form in complex number field and real number field.
5. Positive definite, semi-positive definite, positive definite quadratic form and positive definite and semi-positive definite matrix?
Sixth, linear space?
The definition and simple properties of 1. linear space.
2. Size, foundation and coordinates.
3. Base transformation and coordinate transformation.
4. Linear subspace.
5. Intersection of subspaces, dimension formula and direct sum of subspaces.
Seven, linear transformation?
Definition, operation and matrix of 1. linear transformation.
2. Eigenvalues and eigenvectors, diagonalizable linear transformations.
3. Similarity matrix, similarity invariants, Hamilton-Kelly theorem.
4. Range, kernel and invariant subspace of linear transformation.
Eight, if it is a standard shape?
1. matrix.
2. Conditions of similarity between determinant factor, invariant factor, elementary factor and matrix.
3. If it is a standard shape.
9. Euclidean space?
1. Inner product and Euclidean space, length, included angle and orthogonality of vectors, metric matrix.
2. Standard orthogonal basis, orthogonal matrix and Schmidt orthogonalization method.
3. Isomorphism of Euclidean space.
4. Orthogonal transformation and 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.
Ⅲ. Analytic geometry?
First, vectors and coordinates?
Definition and representation of 1. vector, linear operation of vector, decomposition of vector, geometric operation.
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, the direction cosine and included angle of the vector.
4. Definition, geometric meaning, operational properties, calculation methods and applications of vector product, cross product and mixed product.
5. Solve some geometric and trigonometric problems with vectors.
Second, trajectory and equation?
1. Definition of surface equation: general equation, parametric equation (mutual transformation between vector formula and coordinate formula) and their relationship.
2. The general form of space curve equation, the form of parameter equation and their relationship.
3. The general method of establishing space surface and curve equation, and the application of vector to establish simple surface and curve equation.
4. Standard equation and general equation of spherical surface, and cylindrical equation with bus parallel to coordinate axis.
Three, plane and space straight line?
1. Various forms of plane equation and straight line equation, and the meanings of related letters in the equation.
2. According to the geometric conditions of plane and straight line, the equations of plane and straight line are established by appropriate methods.
3. According to the equation of plane and straight line, determine the positional relationship between plane and plane, straight line and straight line, and plane and straight line.
4. According to the equations of plane and straight line and the coordinates of points, judge the positional relationship between points, planes and straight lines, and calculate the distance and intersection angle between them. Find the common perpendicular equation of two straight lines in different planes.
Quadratic surface?
1. Define cylinder, cone and surface of revolution, and find the equations of cylinder, cone and surface of revolution.
2. According to different situations, the standard equations and main properties of ellipsoid, hyperboloid and paraboloid, and the standard equations of quadric are established.
3. The straightness of hyperboloid and hyperbolic paraboloid and the solution of straight generatrix of hyperboloid and hyperbolic paraboloid.
4. According to the given straight line family, the ruled surface equation expressed by it is solved, and the trajectory problem of moving straight line and moving curve is solved.
5. General theory of conic?
Asymptotic direction, center and asymptote of 1. conic.
2. Tangents, normal points and singular points of conic.
3. Diameter of cone, direction of yoke and diameter of yoke.
4. Principal axis, principal direction, characteristic equation and characteristic root of conic curve.
5. Simplify the quadratic curve equation and draw the position of the curve in the coordinate system.