Advanced Mathematics: You don't know the law of setting questions and the types of common problems.
First, the law of high number proposition
1) focuses on the unique knowledge of numbers one and three. What is the unique knowledge of postgraduate mathematics? The big modules include spatial analytic geometry and multivariate integration (triple integration, curve integration and surface integration); The unique knowledge of number three includes economic application and sequence (as opposed to number two). For example, the real problem of 20 14, the tangent plane equation, Stokes formula and surface integral were tested; The third test is the sum expansion of marginal income and power series.
2) Examine the ability of candidates to analyze and solve problems by comprehensively applying what they have learned. To put it bluntly, it is an application problem. For example, the economic application of the number three mentioned above in the postgraduate entrance examination, and the number two got the center of mass. The former is the economic application of derivative, and the latter is the geometric application of definite integral.
3) The coverage of test sites is relatively complete. This reminds candidates not to take chances, not to ignore the secondary test sites, and to do a comprehensive review. This is not contradictory to getting to the point. Here, we can apply the basic principles of Marxist philosophy to the politics of postgraduate entrance examination: comprehensive review and dialectical unity of grasping the key points.
Second, frequently asked questions
? Vector Algebra and Spatial Analytic Geometry
1, understand the concept of vector and its representation.
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; Master unit vector, direction number and direction cosine, coordinate representation of vector and the method of vector operation with coordinate representation.
3, master the plane equation and straight line equation and their solutions, will use the relationship between plane and straight line to solve related problems.
4. Understand the concept of surface equation, understand the equation of common quadric surface and its figure, and find the equation of rotating surface with axis of rotation and cylinder with generatrix parallel to axis of rotation.
5. Understand the parametric equation and general equation of space curve; Understand the projection of space curve on the coordinate plane and find its equation.
? differential equation
1. Find the general solution or special solution of a typical first-order differential equation: This kind of problem first discriminates the type of the equation. Of course, some equations do not directly belong to the type we have learned. At this time, the common method is to switch x and y or make appropriate variable substitution to turn the original equation into the type we have learned;
2. Solve the reducible equation;
3. Find the special solution or general solution of homogeneous and nonhomogeneous equations with linear constant coefficients;
4. Establish and solve differential equations according to actual problems or given conditions;
? infinite series
1. Determine the convergence, absolute convergence and conditional convergence of series;
2. Find the convergence radius and convergence domain of power series;
3. Find the sum function of power series or the sum of several series;
4. Expand the function into a power series (including writing the convergence domain);
5. Expand a function into a Fourier series, or if a Fourier series has been given, determine its sum at a certain point (usually using Dirichlet's theorem);
? Integral of multivariate function
1. The calculation of double and triple integrals in various coordinates, and the exchange order of repeated integrals;
2. Calculation of the first kind of curve integral and surface integral;
3. Calculation of the second kind (coordinate) curve integral, Green formula, Stokes formula and their applications;
4. Calculation of the second kind (coordinate) surface integral, Gauss formula and its application;
5. Comprehensive calculation of gradient, divergence and curl;
6. Double integration, line and surface integration application; Find the area, volume, weight, center of gravity, gravity, variable force work, etc.
? multivariable differential calculus
1. Determine whether the binary function is continuous at one point, whether the partial derivative exists, whether it is differentiable, and whether the partial derivative is continuous;
2. Find the first and second partial derivatives of multivariate functions (especially those with abstract functions) and the first and second partial derivatives of implicit functions;
3. Find the directional derivatives and gradients of binary and ternary functions;
4. Find the tangent plane and normal direction of the surface and the tangent plane and normal direction of the space curve. This kind of problem is a comprehensive problem of multivariate function differential calculus, vector algebra and spatial analytic geometry, which should be reviewed together;
5. The application of extreme value or conditional extreme value of multivariate function in geometry, physics and economy;
6. Find the maximum and minimum values of the binary continuous function in the bounded plane region.
? Integral calculus of unary function
1. Calculate indefinite integral, definite integral and generalized integral;
2. Questions about variable upper bound integral: such as derivation, limit, etc.
3. On the proof of integral mean value theorem and integral properties;
Application of definite integral:
Calculate area, volume of rotating body, arc length of plane curve, area of rotating surface, pressure, gravity, variable force work, etc.
Comprehensive examination questions.
Vector Algebra and Spatial Analytic Geometry
Calculation problem:
1. Find the scalar product, cross product and mixed product of vectors;
2. Find the linear equation and plane equation;
3. Determine the parallel and vertical relationship between the plane and the straight line, and find the included angle;
4. Establish the equation of revolution surface;
Topics related to the application of differential calculus of multivariate functions in geometry or linear algebra.
? Differential calculus of univariate function
1. Find the derivative and differential of a given function (including higher-order derivative), the derivative of implicit function and the function determined by parameter equation, especially the discussion on the differentiability of piecewise function and function with absolute value;
2. Use Robida's law to find the limit of infinitive;
3. Discuss the extreme value of function and the root of equation, and prove the inequality of function;
4. Prove related propositions by using Rolle's theorem, Lagrange's mean value theorem, Cauchy's mean value theorem and Taylor's mean value theorem, such as "Prove that at least one in the open interval satisfies ……", and prove that such problems often need to construct auxiliary functions;
5. The application of maximum and minimum in geometry, physics and economy. To solve this kind of problems, it is mainly to determine the objective function and constraint conditions, and to determine the discussion interval;
6. Use derivative to study the behavior of function and describe the graph of function, and find the asymptote of curve.
? Features, limitations and links
1. Find the composite function of piecewise function;
2. Find the limit or known limit and determine the constant in the original formula;
3. Discuss the continuity of function and judge the type of discontinuity;
4. Comparison of infinitesimal orders;
5. Discuss the number of zeros of continuous function in a given interval, or judge whether the equation has real roots in a given interval.
This chapter is assessed by multiple-choice questions, fill-in-the-blank questions, or as part of a big question. The key to review is to have an essential understanding of these concepts, and on this basis, find exercises to strengthen them.
Third, how to judge that you have mastered the knowledge points?
You can choose a real question for postgraduate entrance examination, which may be more difficult and comprehensive, but the decomposed test sites are all within the test sites specified in the outline, indicating that postgraduate entrance examination mathematics focuses on the foundation.
So can you easily cope with the exam by laying a good foundation? Not enough, we need to summarize the methods on this basis. For example, many candidates have a headache in proving the mean value theorem. After mastering the basic contents (properties of continuous functions on closed intervals, Fermat's lemma, Rolle's theorem, Lagrange's theorem, Cauchy's theorem) (the contents of the theorem can be completely expressed, and the theorem itself will be proved), you may have no clue and don't know which direction to think.
There is a process from understanding to application of knowledge: understanding does not mean being able to use it, and there is still a directional problem in application-where is the application? At this time, the value of the real question appears: the real question is a good material. Through the analysis and summary of real questions over the years, we can have an intuitive understanding of the specific application of real questions and a comprehensive understanding of propositional thinking of real questions.
In other words, you can know which topic to think about from which direction by "summarizing the questions and methods" for the real questions of mathematics in the postgraduate entrance examination. Taking the proof of the mean value theorem as an example, if the summary is in place, the following effects can be achieved: when you get a problem of this kind, you can generally think from the conditions to see whether the formula to be proved contains one mean value or two mean values. If it is one, let's see if it contains a derivative. If the derivative is involved, Rolle's theorem is given priority, otherwise, the properties of continuous functions on closed intervals (mainly two theorems-the intermediate value theorem and the existence theorem of zeros) are considered. If the formula to be proved contains two median values, consider Lagrange theorem and Cauchy theorem.
After this dry goods, is everyone's grasp of high numbers greatly deepened? Find the right methods and skills, mathematics is not difficult for you!