Basic knowledge of advanced mathematics 1, function, limit, continuity
This paper focuses on the calculation of limit, the unknown parameters in the original formula of known limit, the discussion of function continuity, the judgment of discontinuous point type, the comparison of infinitesimal order, the discussion of the number of zeros of continuous function in a given interval, and the judgment of whether the equation has real roots in a given interval.
2. Integral calculus of unary function
This paper mainly introduces the calculation of derivative and limit of indefinite integral, definite integral, generalized integral and variable upper bound function, and proves the integral property, geometric application and physical application of definite integral by using the mean value theorem of integral.
3. Differential calculus of unary function
The definition of derivative and differential, calculation of derivative and differential of function (including derivative of implicit function), limit of infinitive, extreme value and maximum value of function, number of roots of equation, proof of functional inequality, relevant proof of mean value theorem, practical application in physics and economy, and solution of curve asymptote are emphatically introduced.
4. Vector Algebra and Spatial Analytic Geometry (1)
This paper mainly examines the operation of vectors, plane equations and straight line equations and their solutions, the angles between planes, planes and straight lines, and straight lines and straight lines, and makes use of the relationship between planes and straight lines (parallel, vertical, intersecting, etc.). ) solve related problems, etc. This part is generally not investigated separately, but is mainly used as the basis of curve integral and surface integral.
5. Differential calculus of multivariate functions
This paper focuses on the limit existence, continuity, existence, differentiability and continuity of partial derivatives of multivariate functions, the solution of the first and second partial derivatives of multivariate functions and implicit functions, and the conditional extremum and unconditional extremum. In addition, the number one also requires mastering the directional derivative, gradient, tangent plane and normal plane of curve, tangent plane and normal plane of surface.
6, multivariate function integral calculus
The calculation, repeated integration and the change of integration order of double integration in rectangular coordinates and polar coordinates are emphatically introduced. In addition, the number one also requires mastering the calculation of triple integral, two kinds of curve integral and two kinds of surface integral, Green formula, Gaussian formula and Stokes formula.
7. Infinite series (No.1 and No.3)
This paper mainly discusses the basic properties of positive series and the discrimination of convergence and divergence, the discrimination of absolute convergence and conditional convergence of general series, the solution of convergence radius, convergence domain and sum function of power series, and the expansion of power series at specific points.
8. Ordinary differential equations and difference equations
This paper mainly introduces the general solution or special solution of the first-order differential equation, the special solution or general solution of the second-order linear homogeneous and inhomogeneous equations with constant coefficients, and the establishment and solution of differential equations. In addition, the number three examines the basic concept of difference equation and the solution method of a linear equation with constant coefficients. Number one also needs Bernoulli equation, Euler formula and so on.
Advanced mathematics postgraduate knowledge 1. The contents of advanced mathematics examination include: function, limit and continuity.
Examination requirements
1, understand the concept of function
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 concept of left and right limit of function and the relationship between the existence of function limit and left and right limit.
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. Understanding the concept of function continuity (including left continuity and right continuity) will distinguish the types of function discontinuity points.
10. Understand the properties of continuous function and continuity of elementary function, understand the properties of continuous function on closed interval (boundedness, maximum theorem, mean value theorem), and apply these properties.
Second, the differential calculus of unary function
Examination requirements
1, understand the concepts of derivative and differential, understand the relationship between derivative and differential, and understand the relationship between derivability and continuity of functions.
2. Mastering the four algorithms of derivative and the derivative rule of compound function, mastering the derivative formula of basic elementary function, and understanding the four algorithms of differential and the invariance of first-order differential form, we can get the differential of function.
If you understand the concept of higher derivative, you will find the higher derivative of a simple function.
4. The derivative of piecewise function, implicit function, function determined by parametric equation and inverse function can be obtained.
5. Understand and apply Rolle theorem, Lagrange mean value theorem, Taylor theorem, and Cauchy mean value theorem.
6. Master the method of using L'H?pital's law to find the limit of indefinite form.
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, and master the method of finding maximum and minimum value of function and its application.
8. Derivative will be used to judge the concavity and convexity of the function graph (Note: in the interval, let the function have the second derivative. When, the figure is concave; When the graph is convex, the inflection point and horizontal, vertical and oblique asymptotes of the function graph will be found, and the function graph will be portrayed.
9. Understand the concepts of curvature, circle of curvature and radius of curvature, and calculate curvature and radius of curvature.
3. Integral calculus of unary function
Examination requirements
1, understand the concept of original function, and understand the concepts of indefinite integral and definite integral.
2. Master the basic formula of indefinite integral, the properties of indefinite integral and definite integral, the mean value theorem of definite integral, and the integration methods of substitution integral and partial integral.
3. Can find the integral of rational function, rational formula of trigonometric function and simple unreasonable function.
4. If you understand the function of the upper limit of integral, you will find its derivative and master Newton-Leibniz formula.
5. Understand the concept of generalized integral and be able to calculate generalized 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.
4. Vector Algebra and Spatial Analytic Geometry
Examination requirements
1, understand the spatial rectangular coordinate system, and understand the concept and representation of vectors.
2. Master vector operations (linear operation, scalar product, cross product, mixed product) and understand the conditions for two vectors to be vertical and 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 plane and plane, plane and straight line, straight line and straight line angle, will use the relationship between plane and straight line (parallel, vertical, intersection, etc. ) to solve related problems.
6, will find the distance from point to line and point to plane.
7. Understand the concepts of surface equation and space curve equation.
8. Knowing the equation of quadric surface and its graph, we can find out the equation of simple cylindrical surface and revolving surface.
9. Understand the parametric equation and general equation of spatial curve, understand the projection of spatial curve on the coordinate plane, and find the equation of projection curve.
Verb (abbreviation of verb) Differential calculus of multivariate functions
Examination requirements
1, understand the concept of multivariate function and understand the geometric meaning of binary function.
2. Understand the concepts of limit and continuity of binary functions and the properties of continuous functions in bounded closed regions.
3. Understand the concepts of partial derivative and total differential of multivariate function, you will find total differential, understand the necessary and sufficient conditions for the existence of total differential, and understand the invariance of 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 multivariate function extremum and conditional extremum, and solve some simple application problems.
Six, multivariate function integral calculus
Examination requirements
1, understand the concepts, properties and mean value theorem of double integral.
2, master the calculation method of double integral (rectangular coordinates, polar coordinates), can calculate triple integral (rectangular coordinates, cylindrical coordinates, 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 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.
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. Some geometric and physical quantities (area, volume, surface area, arc length, mass, centroid, centroid, moment of inertia, gravity, work and flow, etc.). ) can be obtained by using multiple integral, curve integral and surface integral.
Seven, infinite series
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 conditions of geometric series and convergence and divergence of series.
3. Master the comparison discrimination method and ratio discrimination method of positive series convergence, and use the root value discrimination method.
4. Master the Leibniz discriminant method of staggered series.
5. Understand the concepts of absolute convergence and conditional convergence of arbitrary series.
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. I will find the sum function of some power series in the convergence interval, and I will find the sum of some series from this.
9. Understand the necessary and sufficient conditions for the function to expand into Taylor series.
10, master maclaurin expansion 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 expand the functions defined on the ground into Fourier series, and expand the functions defined on the ground into sine series and cosine series, and write the expressions of Fourier series and functions.
Eight, ordinary differential equations
Examination requirements
1. Understand differential equations and their concepts such as order, solution, general solution, initial condition and special solution.
2. Master the solutions of differential equations with separable variables and first-order linear differential equations.
3, can solve homogeneous differential equations, Bernoulli equations and total differential equations, and can solve some differential equations with simple variables.
4. The following differential equation 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, will solve the free term is polynomial, exponential function, sine function, cosine function and their sum and product of second-order constant coefficient non-homogeneous linear differential equation.
8, can solve Euler equation.
9, can use differential equations to solve some simple application problems.