60 1- syllabus of advanced mathematics examination
I. Basic requirements
1, function, limit and continuity
Understanding the concept of function will establish the functional relationship of application problems; Understand the boundedness, monotonicity, periodicity and parity of functions; Understand the concepts of compound function and piecewise function, and understand the concepts of inverse function and implicit function; Grasp the nature of basic elementary function and understand the concept of elementary function; Understand the concept of limit, the concept of left and right limit of function and the relationship between the existence of function limit and left and right limit; Master the nature of limit and four algorithms; Master two criteria of limit existence, and use them to find the limit, and master the method of finding the limit by using two important limits; Understand the concepts of infinitesimal and infinitesimal, master the comparison method of infinitesimal, and find the limit with equivalent infinitesimal; Understanding the concept of function continuity will distinguish the types of function discontinuity; Understand the properties of continuous functions and the continuity of elementary functions, and understand the properties of continuous functions on closed intervals and apply them.
2. Differential calculus of unary function
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, and understand the relationship between function derivability and continuity; Master the four arithmetic rules 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; If you understand the concept of higher derivative, you will find the higher derivative of simple function. Can find the derivative of piecewise function, derivative of implicit function, function determined by parameter equation, inverse function. Understand and apply Rolle theorem, Lagrange mean value theorem and Taylor theorem, and understand and apply Cauchy mean value theorem; Master L'H?pital's law to find the limit of indeterminate form; 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 application; I will use the derivative to judge the concavity and convexity of the function graph, and I will find the inflection point and the horizontal, vertical and oblique asymptotes of the function graph.
3. Integral calculus of unary function
Understand the concepts of original function, indefinite integral and definite integral; 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; Can find the integral of rational function, rational formula of trigonometric function, simple unreasonable function; Understand the function of the upper limit of integral, find its derivative and master Newton-Leibniz formula; Understand the concept of generalized integral, and can calculate generalized integral; Through definite integral, we can master the representation and calculation of some geometric physical quantities (the area of plane figure, the arc length of plane curve, the volume and lateral area of rotating body, the area of parallel section, the volume, work, gravity, pressure, center of mass, centroid of solid, etc.). ).
4. Vector Algebra and Spatial Analytic Geometry
Master the operation of vectors and understand the conditions of vertical and parallel vectors; Understand the coordinate expressions of unit vector, direction number, direction cosine and vector, and master the method of vector operation with coordinate expressions. Will find the angles between planes, planes and straight lines, straight lines and straight lines, and will use the relationship between planes and straight lines to solve related problems; Will find the distance from point to straight line and point to plane; Understand the concepts of surface equation and space curve equation; Knowing the equation of quadric surface and its figure, we can find out the equation of simple cylinder and rotating surface. 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.
5. Differential calculus of multivariate functions
Understand the concept of multivariate function; Understand the concepts of limit and continuity of binary functions and the properties of continuous functions in bounded closed regions; Understanding the concepts of partial derivative and total differential of multivariate functions will help us to 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. Understand the concepts of directional derivative and gradient, and master their calculation methods; Master the solution of the first and second partial derivatives of multivariate composite functions; Knowing the existence theorem of implicit function, we will find the partial derivative of multivariate implicit function; Understand the concepts of tangent and normal plane of space curve and tangent and normal plane of surface, and solve their equations; Understand the second-order Taylor formula of binary function; 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, and use Lagrange multiplier method to find the extreme value, conditional extreme value, maximum value and minimum value of simple multivariate function to solve some simple application problems.
6, multivariate function integral calculus
Understand the concepts of double integral and triple integral, and understand the properties of double integral; Master the calculation method of double integral (rectangular coordinates and polar coordinates), and can calculate triple integral (rectangular coordinates, cylindrical coordinates and spherical coordinates); Understand the concepts, properties and relationships of two kinds of curve integrals; Master the calculation methods of two kinds of curve integrals; Master Green's formula and use the condition that the plane curve integral has nothing to do with the path to find the original function of the total differential of binary function; 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; The concepts of dissolution and rotation are introduced and calculated. We will use multiple integrals, curve integrals and surface integrals to find some geometric physical quantities.
7. Infinite series
Understand the concepts of convergence and divergence and sum of convergent constant series, and master the basic properties of series and the necessary conditions for convergence; Master the conditions of convergence and divergence of geometric series and p series; Master the comparison discrimination method and ratio discrimination method of positive series convergence, and use the root value discrimination method; Master Leibniz discriminant method of staggered series; Understand the concepts of absolute convergence and conditional convergence of arbitrary series and the relationship between absolute convergence and convergence; Understand the convergence domain of function term series and the concept of function; Understand the concept of convergence radius of power series, and master the solution of convergence radius, convergence interval and convergence domain of power series; Knowing the 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 some series. Understand the necessary and sufficient conditions for the function to expand into Taylor series; If you master Maclaurin expansions of functions, you will use them to indirectly expand some simple functions into power series. Understand the concept of Fourier series and Dirichlet convergence theorem, expand the function into Fourier series, sine series and cosine series, and write the expression of Fourier series and function.
8. Ordinary differential equations
Understand the concepts of differential equation and its order, solution, general solution, initial condition and special solution; Master the solution of differential equations with separable variables and first-order linear differential equations; Can solve homogeneous differential equations, Bernoulli equations and total differential equations, and can replace some differential equations with simple variables; Differential equations can be solved by order reduction method; Understand the properties and structure of solutions of linear differential equations; Master the solution of second-order homogeneous linear differential equations with constant coefficients, and can solve some homogeneous linear differential equations with constant coefficients higher than the second order; Can use polynomial, exponential function, sine function, cosine function and their sum and product to solve second-order non-homogeneous linear differential equations with constant coefficients; Able to solve Euler equation; Differential equations can be used to solve some simple application problems.
Second, the scope of the examination
1, function, limit and continuity
Concept of function, boundedness, monotonicity, periodicity and parity of function, composite function, inverse function, piecewise function and implicit function, basic elementary function; Definition and properties of sequence limit and function limit, left limit and right limit of function, concept and relationship between infinitesimal and infinitesimal, properties and comparison of infinitesimal, four operational limits of limit, two discriminant methods, monotone bounded discriminant method and pinch discriminant method, two important limits; The concept of function continuity, the types of function discontinuity points, the continuity of elementary functions, and the properties of continuous functions on closed intervals.
2. Differential calculus of unary function
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, four operations of derivative and differential, derivative of basic elementary function, differential method of functions determined by compound function, inverse function, implicit function and parameter equation, higher derivative, invariance of first-order differential form, differential mean value theorem, Lobida's law, monotonicity of function.
3. Integral calculus of unary function
Concepts of primitive function and indefinite integral, basic properties of indefinite integral, basic integral formula, concept and basic properties of definite integral, mean value theorem of definite integral, function of upper limit of integral and its derivative, Newton-Leibniz formula, substitution integral method and partial integral of indefinite integral and definite integral, rational and simple irrational integral of rational function and trigonometric function, abnormal (generalized) integral and application of definite integral.
4. Vector Algebra and Spatial Analytic Geometry
Concept of vector, linear operation of vector, scalar product and cross product of vector, mixed product of vector, condition of two vectors being vertically parallel, included angle of two vectors, coordinate representation and operation of vector, unit vector, cosine of direction number and direction, concepts of surface equation and space curve equation, plane equation, straight line equation, included angle between plane and plane, parallel and vertical condition, distance from point to plane, and point to line. Sphere, cylinder, revolution surface, common quadratic equation and its graph, parametric equation and general equation of space curve, projection curve equation of space curve on coordinate plane.
5. Differential calculus of multivariate functions
The concept of multivariate function, the concept of limit and continuity of multivariate function, the properties of continuous function of multivariate function in bounded closed region, partial derivative and total differential of multivariate function, necessary and sufficient conditions for the existence of total differential, derivative method of multivariate composite function and implicit function, second-order partial derivative, directional derivative and gradient, tangent and normal plane of space curve, tangent plane and normal line of surface, second-order Taylor formula of multivariate function, extreme value and conditional extreme value of multivariate function, and maximum value of multivariate function.
6, multivariate function integral calculus
The concepts, properties, calculation and application of double integral and triple integral, the concepts, properties and calculation of two kinds of curve integral, the relationship between two kinds of curve integral, Green's formula, the condition that plane curve integral has nothing to do with path, the original function of binary function total differential, the concepts, properties and calculation of two kinds of surface integral, the relationship between two kinds of surface integral, Gauss formula and Stokes formula.
7. Infinite series
Concept of convergence and divergence of constant series, concept of convergence series sum, basic properties and necessary conditions of series convergence, geometric series and p- series and their convergence, judgment of convergence of positive series, staggered series and Leibniz theorem, absolute convergence and conditional convergence of arbitrary series, concept of convergence domain and function of function series, power series and its convergence radius, convergence interval and convergence domain, sum function of power series, basic properties of power series in its convergence interval, Solution of simple power series and function, power series expansion of elementary function, Fourier coefficient and Fourier series of function, Dirichlet theorem, function expansion into Fourier series, sine series and cosine series.
8. Ordinary differential equations
Basic concepts of ordinary differential equations, differential equations with separable variables, homogeneous differential equations, first-order Euler linear differential equations, Bernoulli equations, total differential equations, some differential equations that can be solved by substitution of simple variables, low-order higher-order differential equations, properties and structure theorems of solutions of linear differential equations, second-order homogeneous linear differential equations with constant coefficients, some homogeneous linear differential equations with constant coefficients higher than second-order, and simple second-order homogeneous linear differential equations with constant coefficients.
Third, the main reference books
Advanced Mathematics (6th Edition), Higher Education Press, Department of Mathematics, Tongji University.