Advanced mathematics

Chapter 1: Function, Limit and Continuity

Examination contents: the concept and expression of function, boundedness, monotonicity, periodicity and parity of function, the properties of basic elementary functions of inverse function, piecewise function and implicit function, and the establishment of functional relationship of graphic elementary function.

Definitions and properties of sequence limit and function limit, left limit and right limit of function, concepts and relationships of infinitesimal and infinitesimal, properties of infinitesimal and four operational limits of infinitesimal, two important limits: monotone bounded criterion and pinch criterion;

The concept of continuous function, the type of function discontinuity, the continuity of elementary function, and the properties of continuous function on closed interval;

1. Understand the concept of function, master the representation of function, and establish the function relationship in 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 and right limits of function, and the relationship between the existence of function limit and left and right limits. 6. Master the nature of limit and four algorithms.

7. Master two criteria for the existence of limit, and use them to find the limit, and master the method of using two important limits to find the limit. 8. Understand the concepts of infinitesimal and infinitesimal, master the comparison method of infinitesimal, and find the limit with equivalent infinitesimal. 9. Understand the concept of function continuity (including left continuity and right continuity) and distinguish the types of function discontinuity points. 10. Understand the properties of continuous functions and elementary functions.

Chapter 2: Differential calculus of unary function.

Examination content: the geometric meaning of derivative and differential concepts and the relationship between derivability and continuity of physical meaning function; Four operations of tangent, normal derivative and differential of plane curve: derivative of function determined by parametric equation, compound function, inverse function, implicit function and differential method, first-order differential invariant differential mean value theorem of L'H?pital's law function monotonicity discriminates extreme value function graph concavity, inflection point and asymptote function graph describes function maximum and minimum arc differential curvature concept curvature circle curvature radius.

Examination 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 first-order differential form, we can find the differential of function.

3. If you understand the concept of higher derivative, you will find the higher derivative of simple function.

4. We can find the derivative of piecewise function, implicit function, function determined by parametric equation and inverse function.

5. Understand and apply Rolle theorem, Lagrange mean value theorem, Taylor theorem, and Cauchy mean value theorem.

6. Master the method of finding the limit of infinitive with L'H?pital's law.

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, master the method of finding maximum and minimum value of function and its simple application.

8. Will judge the concavity and convexity of the function graph by derivative (note: in the interval (a, b), let the function f(x) have the second derivative. At that time, the graph of f(x) was 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, circle of curvature and radius of curvature, and calculate curvature and radius of curvature.

Chapter 3: Integral of unary function.

Examination contents: the concepts of original function and indefinite integral, the basic properties of indefinite integral, the concept and basic properties of definite integral formula, the mean value theorem of definite integral, Newton-Leibniz formula for expressing and calculating the upper limit of centroid integral with definite integral and its derivatives, the substitution integration method of indefinite integral and definite integral, the rational formula of partial rational function integral, and the application of integral generalized abnormal (generalized) integral definite integral of trigonometric function and simple irrational function.

Examination requirements:

1. Understand the concepts of original function and indefinite integral and definite integral.

2. Master the basic formula of indefinite integral, the properties of indefinite integral and definite integral and the mean value theorem of definite integral, and master the integration methods of method of substitution and integration by parts.

3. Can find the integral of rational function, rational formula of trigonometric function and simple unreasonable function.

4. Understand the function of the upper limit of integral, find its derivative and master Newton-Leibniz formula.

5. Knowing the concept of generalized improper integral, we can calculate the generalized improper integral.

6. Master the expression and calculation of some geometric physical quantities (the area of a plane figure, the arc length of a plane curve, the volume and lateral area of a rotating body, the area of a parallel section, the volume, work, gravity, pressure, center of mass, centroid, etc. of a known solid. ) and definite integral to find the average value of the function.

Chapter 4: Vector Algebra and Spatial Analytic Geometry.

Examination content:

The quantitative product of the linear operation vector of the concept vector of the vector and the mixed product of the cross product vector are the conditions that the two vectors are vertically parallel. The coordinate expression of two vector included angle vector and its operation unit vector direction number and direction cosine surface equation and space curve equation are the concepts of plane equation, straight line equation, plane to plane, plane to straight line, straight line to straight line included angle and parallelism. The distance between vertical condition point and plane and the distance between point and straight line are commonly used parametric equations of spherical cylindrical surface of revolution and its graphic space curve and projection curve equation of general equation space curve on coordinate plane.

Examination requirements:

1. Understand the spatial rectangular coordinate system and the concept and representation of vectors.

2. Master the operation of vectors (linear operation, quantitative product, cross product, mixed product) and understand the conditions for two vectors to be vertically parallel.

3. Understand the coordinate expressions of unit vector, direction number, direction cosine and vector, and master the method of vector operation with coordinate expressions.

4. Principal plane equation and straight line equation and their solutions.

5. Will find the included angle between plane, plane and straight line, straight line and straight line, and will use the relationship between plane and straight line (parallel, vertical, intersecting, etc.). ) to solve related problems.

6. You can find the distance from a point to a straight line and the distance from a point to a plane.

7. Understand the concepts of surface equation and space curve equation.

8. Understand the equation of quadric surface and its figure, and find the simple equation of cylinder and revolution surface.

9. Understand the parametric equation and general equation of space curve. Understand the projection of space curve on the coordinate plane, and find the equation of projection curve.

Chapter 5: Differential calculus of multivariate functions.

Examination content:

Concept of multivariate function, geometric meaning of bivariate function, concept of limit and continuity of bivariate function, properties of multivariate continuous function in bounded closed region, necessary and sufficient conditions for existence of partial derivative and total differential of multivariate function, derivation method of implicit function, second-order partial derivative direction derivative of multivariate composite function and tangent of gradient space curve, second-order Taylor formula of tangent and normal of plane surface, maximum and conditional extreme value of multivariate function and its simple application.

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 functions and the properties of continuous functions in bounded closed regions.

3. By understanding the concepts of partial derivative and total differential of multivariate functions, we can find the total differential, understand the necessary and sufficient conditions for the existence of the total differential, and understand the invariance of the total differential form.

4. Understand the concepts of directional derivative and gradient, and master their calculation methods.

5. Master the solution of the first and second partial derivatives of multivariate composite functions.

6. Knowing the existence theorem of implicit function, we can find the partial derivative of multivariate implicit function.

7. Understand the concepts of tangent and normal plane of space curve and tangent and normal plane of surface, and work out their equations.

8. Understand the second-order Taylor formula of binary function.

9. 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 some simple application problems.

Chapter 6: Integration of multivariate functions.

Examination content:

The concepts, properties, calculation and application of double integral and triple integral; The concept, properties and calculation of two kinds of curve integrals: Green's formula; The condition that the plane curve integral is independent of the path; Original function of binary function; The concept, properties and calculation of two kinds of surface integrals: Gaussian formula; Stokes formula; The concepts of divergence and curl; And the calculation of curve integral and surface integral.

Examination requirements:

1. Understand the concept, properties and mean value theorem of double integral.

2. Master the calculation method of double integrals (rectangular coordinates and polar coordinates), and be able to calculate triple integrals (rectangular coordinates, cylindrical coordinates and spherical coordinates).

3. Understand the concepts, properties and relationships of two kinds of curve integrals.

4. Master the calculation methods of two kinds of curve integrals.

5. Master Green's formula and use the conditions of plane curve integral and path elements to find the original function of total differential of binary function.

6. Understand the concepts, properties and relations of two kinds of surface integrals, master the calculation methods of two kinds of surface integrals, master the method of calculating surface integrals with Gaussian formula, and calculate curve integrals with Stokes formula.

7. The concepts of dissolution and rotation are introduced and calculated.

8. We can use multiple integrals, curve integrals and surface integrals to find some geometric physical quantities (area, volume, surface area, arc length, mass, center of mass, moment of inertia, gravity, work and flow of plane figures, etc.). ).

Chapter 7: Infinite Series

Examination content:

The basic properties and necessary conditions of convergence of the concept series of the convergence of geometric series and P series of constant term series and its convergence, and the basic properties of the concept power series and its convergence radius of staggered series and Leibniz theorem and function, and the sum function of power series in convergence interval (refers to open interval) and convergence region; Solution of simple power series sum function: Fourier coefficient of power series expansion of elementary function and Dirichlet theorem of Fourier series: sine series and cosine series of Fourier series function on [-l, l].

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. To master the comparison method and ratio method of positive series convergence, the root value method will be used.

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 convergence domain of function term series and the concept of function.

7. Understand the concept of convergence radius of power series and master the solution of convergence radius, convergence interval and convergence domain of power series.

8. Knowing some 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 several terms of some series.

9. Understand the necessary and sufficient conditions for the function to expand into Taylor series.

10. Master maclaurin expansions of,, and, and use them to indirectly expand some simple functions into power series.

1 1. Knowing the concept of Fourier series and Dirichlet's convergence theorem, we will expand the functions defined on the ground into Fourier series, expand the functions defined on the ground into sine series and cosine series, and write the expression of the sum of Fourier series.

Chapter 8: Ordinary differential equations

Examination content:

Basic concept of ordinary differential equation separable variable differential equation homogeneous differential equation Bernoulli equation All differential equations can be solved by simple variable substitution. Some differential equations can be simplified to properties and structure theorems of solutions of higher-order linear differential equations. Second-order homogeneous linear differential equations with constant coefficients are higher than some second-order homogeneous linear differential equations with constant coefficients. Simple application of second-order nonhomogeneous linear differential equations Euler differential equations.

Examination requirements:

1. Understand the concept of differential equation and its order, solution, general solution, initial condition and special solution. (Knowledge points before adjustment: Understand the concepts of differential equations and their solutions, orders, general solutions, initial conditions, special solutions, etc. )

2. Master the solutions of differential equations with separable variables and first-order linear differential equations.

3. Homogeneous differential equations, Bernoulli equations and total differential equations can be solved, and some differential equations can be replaced by simple variables.

4. The following equations will be solved by order reduction method. 5. Understand the properties and structure of solutions of linear differential equations.

6. 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.

7. 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.

8. Euler equation can be solved.

9. Can use differential equations to solve some simple application problems.

linear algebra

Chapter 1: Determinant

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.

Chapter 2: Matrix

Examination content:

Concept of matrix, linear operation of matrix, multiplication of matrix, concept and properties of transposed inverse matrix of determinant matrix, necessary and sufficient condition of matrix reversibility, elementary transformation of adjoint matrix, rank matrix equivalent block matrix of elementary matrix and its operation.

Examination requirements:

1. Understand the concepts and properties of matrix, identity matrix, quantitative matrix, diagonal matrix, triangular matrix, symmetric matrix and antisymmetric 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 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.

Chapter 3: Vector

Examination content:

The linear combination of concept vectors of vectors and the linear representation of linear correlation of vector groups have nothing to do with the maximum linearity of linear independent vector groups. The orthogonal normalization method of vector space and related concepts between the rank of vector groups and the rank of matrix. N-dimensional vector space base transformation and coordinate transformation transformation matrix vector inner product linear independent vector groups orthogonal base orthogonal matrix specification and its properties.

Examination requirements:

1. Understand the concepts of n-dimensional vectors, linear combinations of vectors and linear representations.

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. Understand the concepts of maximal linearly independent group and rank 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 concepts of N-dimensional star space, subspace, basement, dimension and coordinate.

6. Understand the formulas of base transformation and coordinate transformation, and find the transformation matrix.

7. Understand the concept of inner product and master the Schmidt method of orthogonal normalization of linear independent vector groups.

8. Understand the concepts and properties of standard orthogonal bases and orthogonal matrices.

Chapter four: linear equations.

Examination content:

Cramer's Law of Linear Equations Necessary and Sufficient Conditions for Homogeneous Linear Equations to Have Non-zero Solutions Necessary and Sufficient Conditions for Non-homogeneous Linear Equations to Have Solutions Properties and Structures of Linear Equations; Basic solution system of homogeneous linear equations and general solution of nonhomogeneous linear equations in general solution space

Examination requirements

The length can be determined by 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, general solution and solution space 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. Master the method of solving linear equations with elementary line transformation.

Chapter 5: Eigenvalues and eigenvectors of matrices.

Examination content:

Concepts of eigenvalues and eigenvectors of matrices, property similarity transformation, concepts of similar matrices and necessary and sufficient conditions for similar diagonalization of property matrices, eigenvalues, eigenvectors and similar diagonal matrices of real symmetric matrices of similar diagonal matrices.

Examination requirements:

1. Understand the concepts and properties of eigenvalues and eigenvectors of a matrix, and you will find the eigenvalues and eigenvectors of the matrix.

2. Understand the concept and properties of similar matrix and the necessary and sufficient conditions for matrix similarity diagonalization, and master the method of transforming matrix into similar diagonal matrix.

3. Master the properties of eigenvalues and eigenvectors of real symmetric matrices.

Chapter six: quadratic form

Examination content:

Quadratic form and its matrix represent contract transformation and rank inertia theorem of quadratic form of contract matrix. The canonical form and canonical form of quadratic form are transformed into canonical quadratic form and the positive definiteness of its matrix by orthogonal transformation and matching method.

Examination requirements:

1. Master quadratic form and its matrix representation, understand the concept of quadratic form rank, understand the concepts of contract change and contract matrix, and understand the concepts of standard form and standard form of quadratic form and inertia theorem.

2. Master the method of transforming quadratic form into standard form by orthogonal transformation, and can transform quadratic form into standard form by matching method.

3. Understand the concepts of positive definite quadratic form and positive definite matrix, and master their discrimination methods.

Probability and Statistics

Chapter 1: Random Events and Probability

Examination content:

The relationship between random events and sample space events and the basic properties of concept probability of complete operation event group probability; Classical probability, geometric probability, basic formula of conditional probability;

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.

Chapter 2: Random variables and their distribution.

Examination content:

Concept and properties of distribution function of random variables Probability distribution of discrete random variables Probability density of continuous random variables Distribution of common random variables Distribution of random variable functions

Examination requirements:

1. Understand the concept of random variables. Understand the distribution function.

The probability of events related to random variables will be calculated.

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. Understand the conclusion and application conditions of Poisson theorem, and use Poisson distribution to approximately represent binomial distribution.

4. Understand the concept and probability density of continuous random variables, and master uniform distribution, normal distribution and exponential distribution.

And its application, in which the parameter is λ (λ >; The probability density of exponential distribution of 0) is

5. Find the distribution of random variable function.

Chapter 3: Multidimensional random variables and their distribution.

Examination content:

Multi-dimensional random variables and their distributions Probability distribution, edge distribution and conditional distribution of two-dimensional discrete random variables Probability density, marginal probability density and conditional density of two-dimensional continuous random variables Independence and irrelevance of two or more simple functions.

Examination requirements:

1. Understand the concept of multidimensional random variables, understand the concept and properties of multidimensional random variable distribution, and understand the probability distribution, edge distribution and conditional distribution of two-dimensional discrete random variables; Understand the probability density, edge density and conditional density of two-dimensional continuous random variables, and find the probability of related events of two-dimensional random variables.

2. Understand the concepts of independence and irrelevance of random variables, and master the conditions of mutual independence of random variables.

3. Grasp the two-dimensional uniform distribution, understand the probability density of the two-dimensional normal distribution, and understand the probability meaning of the parameters.

4. Will find the distribution of simple functions of two random variables, and will find the distribution of simple functions of multiple independent random variables.