When it comes to recommended books, in addition to the two classic schemes, there is actually a set: College Mathematics-Concepts, Methods and Skills. The first volume is advanced mathematics, and the second volume is linear algebra and probability statistics. It's from Tsinghua University. Very good. I borrowed it from the library, but I'm not sure if it's still there. Personally, I think it is enough to choose one of the three books, either Deng Ge's or Li Er's.
Mathematics for postgraduate entrance examination mainly examines: the basic concepts of comprehensive analysis, calculation ability and thinking method. And our usual semester exams basically only involve the first two parts.
Let's talk about the basic concepts first.
It is best to read the teaching materials before contacting the counseling books, and have a general understanding. It is best to combine the examination syllabus and be targeted. The outline for 2006 won't come out until the summer vacation. Let's look at it from 2005. Mathematics will not change every year like politics, and more than 90% of things will not change. Everyone should have Tongji version of advanced mathematics and Zhejiang University version of probability theory and mathematical statistics. As for line generation, the textbook of line generation used in our undergraduate study is Tongji version of linear algebra, but it is not recommended because this book is too abstract and dry. It is suggested to replace it with Advanced Algebra (Volume I) published by Peking University. When reading the textbook, you can skip the proof of all theorems. For example, in the first chapter, the seemingly dizzy "ε-δ" language is the work of colleagues in the Department of Mathematics. Don't worry. You only need to see an elementary function, and then use "method of substitution" to find its limit at a certain point. There are many things in the book that are written in detail. When reading, we should grasp the main contradiction and make a choice. Specifically, it is to focus on the exam. But because understanding the process also helps to remember the conclusion, if time permits, we can also have a general understanding of the proof of important theorems. Regardless of the process or not, there is only one ultimate goal: to remember formulas and theorems. Different from the college entrance examination, postgraduate mathematics needs to remember a lot of knowledge points, and it needs to be remembered as often as learning English words to deepen the impression.
What to do after remembering the knowledge points? Nature is used to solve problems. At this time, there is a noteworthy problem, that is, the conditions for the theorem and formula to be established, or take the above example, what are the conditions for the function to substitute the value of a certain point to find the limit? That is, this function is a continuous function. Although most of the functions we encounter are continuous, it is best not to take them for granted. There are many similar examples, and from my personal experience and the communication with classmates who reviewed together before, many people tend to ignore this link. Some properties of continuous function, such as maximum theorem and zero theorem, refer to the properties of continuous function on closed interval; In the chapter of mean value theorem, many theorems are established under the condition that a given function is continuous in a closed interval and differentiable in an open interval. Green's formula and Gaussian formula, which are widely used, are established under the condition that the region surrounded by the corresponding closed curve or closed surface does not contain singularities. When the obtained integral region is not closed, the method of filling straight lines or surfaces should be adopted. When there are singularities, a simple connected region should be transformed into a multi-connected region, so that the corresponding multi-connected region does not contain singularities, and the corresponding theorem can be applied. It is strongly recommended that you summarize yourself in the review process. Generally speaking, it is not difficult to remember knowledge points, but we must pay attention to mastering the applicable conditions corresponding to a certain knowledge point at the same time! Only by grasping these two aspects at the same time can the concept pass and the foundation be laid.
The next step is computing power.
The computing power mentioned here includes two aspects: speed and accuracy. I used to suffer from it when I was in high school. When a math paper is handed down, I can work out all the questions and have ideas, but when I do it, there are many loopholes. There are always mistakes, and time is naturally insufficient. In the final analysis, it's because I never practice. When I see a problem, I think about it first. If there are no obstacles in the method, I think there will be no problem. In fact, if I really do it, I may find it is not as simple as I thought. After entering the university, I often pay attention to practicing more while studying, because I started preparing for the postgraduate entrance examination earlier, so I have more time. I have done about 6,000 exercises in advanced mathematics alone, but not so much in linear algebra and probability statistics. Basically, they are all after-school exercises and after-school topics directed by Chen Wendeng. After all, high numbers are the most important part. My suggestion is: you don't need to do all the exercises after the book, because taking the high mathematics book as an example, the exercises after each chapter are divided into big questions and small questions. A big question may have several small questions, so these small questions are basically in one category and can be done selectively. Pay attention to different types of topics, and then review the reference books after Chen Wendeng or other exercises. The following summarizes some operations that I personally think are more important: finding limit, finding derivative, finding higher derivative, finding indefinite integral, finding dot product and cross product of vector, chain rule of derivative of compound function, elementary transformation of determinant or matrix, and multiplication of matrix. That's basically all. Be sure to practice until you are not familiar with it and basically make no mistakes. The speed of later operation is more important, because the whole paper needs to be completed in the sprint stage. At this time, it is necessary not only to allocate the time for each part of the topic, but also to ensure that the corresponding tasks are completed within the expected time, otherwise it will have an impact on personal emotions. Nine questions for the postgraduate entrance examination should be left for at least two hours. Personally, I think the better time allocation is: choose and fill in the questions for 45 minutes and solve the problems for 2 hours.
Finally, the thinking method of comprehensive analysis.
Because of the wide coverage of mathematics knowledge points in postgraduate entrance examination, the coverage that a paper can examine is limited, which will naturally improve the comprehensive requirements. Imagine a topic that only involves derivatives or a topic that combines derivatives, extremum and spatial analytic geometry. Which is easier to use as an examination question? For example, there is a topic in Chen Wendeng's exam-oriented practice, which is to find a point on the ellipsoid to maximize the geometric volume between the tangent plane passing through the point and the coordinate plane. This is a good comprehensive topic. For another example, as a bridge between multiple integrals and curve (surface) integrals, Green's formula, Gaussian formula or Stokes formula are tested almost every year for the simple reason that a topic can cover two knowledge points, which is the best for the proposer.
There are also some mathematical thinking methods: classified discussion, combination of numbers and shapes, differential element analysis and so on. Because function plays an important role in higher mathematics, it is necessary to be familiar with the properties of some commonly used functions. Speaking of this, it is best to combine numbers and shapes to facilitate analysis, and it is not limited to rectangular coordinates. Some curves in polar coordinates should also be mastered, such as star lines and logarithmic spirals. If the object is expanded to the spatial coordinate system, there are various surfaces of revolution, cylindrical surfaces, conical surfaces and so on. You need to write their cylindrical coordinates or spherical coordinate equations. When it comes to the use of symmetry, the combination of numbers and shapes is helpful for analysis. On the discussion of classification, linear algebra is widely used, especially when it comes to the topic of linear equations, and it is often necessary to discuss the values of unknown parameters. Trace element analysis is the most important thinking method in college mathematics, which is not only used in mathematics, but also used in many subsequent courses. Specific ideas can refer to the application of definite integral. There are many specific examples in the book, so I won't explain them in detail, because they are so useful, so I personally feel that I must master them skillfully. There are also some mathematical thinking methods: classified discussion, combination of numbers and shapes, differential element analysis and so on. Because function plays an important role in higher mathematics, it is necessary to be familiar with the properties of some commonly used functions. Speaking of this, it is best to combine numbers and shapes to facilitate analysis, and it is not limited to rectangular coordinates. Some curves in polar coordinates should also be mastered, such as star lines and logarithmic spirals. If the object is expanded to the spatial coordinate system, there are various surfaces of revolution, cylindrical surfaces, conical surfaces and so on. You need to write their cylindrical coordinates or spherical coordinate equations. When it comes to the use of symmetry, the combination of numbers and shapes is helpful for analysis. On the discussion of classification, linear algebra is widely used, especially when it comes to the topic of linear equations, and it is often necessary to discuss the values of unknown parameters. Trace element analysis is the most important thinking method in college mathematics, which is not only used in mathematics, but also used in many subsequent courses. Specific ideas can refer to the application of definite integral. There are many specific examples in the book, so I won't explain them in detail, because they are so useful, so I personally feel that I must master them skillfully. The application problem in the postgraduate entrance examination is a modeling process from the actual problem to the mathematical model, and then the mathematical model is solved, so how to establish it? Generally, differential element method is used for analysis, such as finding area, volume, arc length, variable force work, flow rate, etc., which are fundamentally related. Sometimes combined with the extreme value problem, it is divided into two parts: the extreme value of univariate function and the extreme value of multivariate function. Multivariate functions have conditional extremum and unconditional extremum. I did a simulation and thought it was good. First, I give a random variable and find the estimated value of its parameters. First of all, I demand impartiality. In fact, this gives a restrictive condition, and then I ask for optimization. At this time, it becomes a multivariate extreme value problem, and it is a conditional extreme value. This topic combines the content of probability theory and high number.
Having said that, we are all talking about what is important and what is to be mastered, so naturally there are some corresponding parts, which I call "marginal content". These contents are basically multiple-choice questions or fill-in-the-blank questions every few years, and big questions will definitely not be involved. I summarize myself as follows: asymptote, higher derivatives of order 3 and above, area of revolving surface, Fourier series, Taylor formula of binary function, Euler equation, Vandermonde determinant, two-dimensional normal distribution, large number theorem, central limit theorem, Chebyshev inequality, interval estimation, hypothesis test, all these things are written in the outline and can be understood. As for the two contents of spatial analytic geometry and inequality, the postgraduate entrance examination will generally not be directly involved. It is generally required to be mastered as a tool, that is, to be examined as part of other topics. I haven't seen a big topic specifically for spatial analytic geometry (such as finding the equation of common perpendicular) and proving inequality. Again, because there are too many contents, in order to avoid being upset too early, we should concentrate on breaking through important and more points in the first review, and then solve the marginal contents. There is no need to have pressure when facing it.
There are still some confusing places. For example, in a univariate function, derivable must be continuous and derivable is equivalent to derivable, but in a multivariate function, even if partial derivative exists, it is not necessarily derivable, and the condition is strengthened to be continuous. Some concepts in linear algebra, such as equivalence (same bias), similarity and contraction, are they related? For example, is equivalence necessarily similar, is similarity necessarily a contract, and vice versa? These must be clarified, not a little knowledge. I said it's best to master the principle if you don't recite it. Personally, I think the combination of the two. You can master the principle and you can master it. It is really impossible to master it in a short time and then memorize it. Formulas and theorems have been mentioned before, but there is another content in the basic concept: definition. In the process of learning, I take definition as the starting point to master the principle. For the above example, what is equivalence? What is similarity? What is a contract? Strictly expressing these statements in mathematical language is the definition, and then analyzing the relationship between them. There will be some multiple-choice questions in postgraduate mathematics. This kind of question is to pick up those parts that are easy to confuse, and it is pervasive. You can turn over the real questions over the years.
Finally, I will talk about my views on this year's examination questions in combination with the real questions in 2005, that is, this paper I did in the examination room. I won't write the title. I can compare it with the original question. They should all be out by now. Let's talk about my views on knowledge points. Generally speaking, this year's math problem once again verified the saying that "the postgraduate entrance examination focuses on the foundation", and there is nothing strange about it. I mentioned a saying "1: 2: 7" before, and 1 is a difficult problem, 2 is a simple problem, and 7 is a medium problem. In recent years, almost all the test questions are structured according to this ratio.
Fill in the blanks for the first asymptote. There were Fourier series in 2003 and Euler equation in 2004. The edge content is generally a small problem, and the asymptote is easy to find, but don't confuse it. The function given in this question has two asymptotes, and the oblique asymptote is necessary. Of course, later I heard that someone wrote both of them. In short, let's look at the problem carefully. The second problem is to solve differential equations. Both sides of the equation are transformed into first-order linear differential equations, but the non-homogeneous ones should use the constant variation method, and be careful not to make mistakes in operation. The third way to find directional derivative is to mention that there are many concepts in the multivariate integral part, such as directional derivative, gradient, flux, divergence, circulation and curl. We need to know their relationship, directional derivative and gradient, flux and divergence, circulation and curl. The directional derivative is a number and the gradient is a vector. In this problem, the gradient is obtained first, and then the directional derivative is obtained. The fourth problem is the direct application of Gaussian formula. According to the given equation, the integral area is directly determined. Pay attention to whether the area is closed, it must be outside. The inside will add a negative sign before the whole result. These are the details. If the topic changes slightly, you will suffer if you are not careful. The fifth problem is to find determinant. Because it is an abstract determinant, we must make good use of the relationship between the known quantity and the unknown quantity. This is why we say that we should master the elementary transformation of determinant skillfully. If the relationship between them is written in the form of matrix, it will be more clear at a glance, and it will be much easier to solve it with "the product determinant is equal to the product of determinant" Therefore, the postgraduate entrance examination questions are generally not limited to one knowledge point, but usually cross-chapter. The last question is to find the probability of a probability. First, discuss it by classification, and then get it with the formula of total probability.
Choosing the first one is also a classified discussion. According to the different range of independent variables, we can get the function expression on the corresponding interval, and then judge whether the point is derivable or not. Similar topics are found in the after-class exercises of advanced mathematics, but I actually chose the wrong one, and I was very depressed afterwards. Therefore, to maintain a high degree of concentration in the examination room requires a lot of simulated sprint exercises to support it. The second question is the statement mentioned above. If you remember this conclusion, you can choose directly, but most people won't remember it so clearly. Generally, only the last two items can be quickly excluded. So which is right, A or B? Don't forget that the original function is solved by an arbitrary integer constant c, which requires odd function to cross the origin, so any value of the constant in option B can't guarantee that the original function must cross the origin, so it is not necessarily odd function, thus excluding the strong interference term. The third way requires the second-order partial derivative. Because it is a compound function, so the calculation needs to be extremely careful, as long as there is no mistake, you can get the answer. Fourth, the new test site in 2005, the existence theorem of implicit function. What I want to mention here is that the new test center is generally required to take the exam every year. Fortunately, mathematics generally changes one or two knowledge points every year. Just pay attention to it when it comes out this year. The fifth problem is the eigenvalues and eigenvectors in line generation. Note that the eigenvectors corresponding to different eigenvalues must be linearly independent, so it is easy to use this conclusion. The rest is a typical problem, and it is deduced that one set of vectors is linearly independent and the other set of vectors is linearly independent, and there is a certain relationship between the two sets of vectors. Such exercises can be found everywhere in books. The sixth method involves the elementary transformation of matrices. In fact, in the chapter of elementary transformation, it is said that the elementary transformation of a matrix is equivalent to multiplying a corresponding elementary matrix, and all the sentences in the topic are translated into mathematical language, and the rest is mathematical transformation. Question 7: Test two-dimensional random variables. In fact, it is enough to make full use of some of their properties, that is, pay attention to the independence of use. The last problem is the sampling distribution and its deformation commonly used in mathematical statistics. If you remember it, it's easy. You just need to analyze the options one by one. The questioner really has ulterior motives, and sets the correct item at the last item ... Of course, if you can see the right one at a glance, you don't have to count the others. The sampling distribution mentioned in the sixth chapter of probability theory and mathematical statistics is difficult to remember, and it is easy to be confused and forgotten. You can only strengthen your memory by looking at it more.
Then it is to solve the problem.
The first method is to find the double integral, but the surfaces involved are not single. The integrand function needs to be split according to the integral area. In fact, it is an idea of classified discussion. The key is not to be scared by the rounding function at first. After a calm analysis, it is not difficult, but it is a bit strange in form.
The second method is to find the convergence domain first and then sum the functions. The first part is simple, but the second part is difficult. When summing functions, the method of derivative by integral item by item is troublesome, which requires good knowledge of integral, otherwise, even if you know how to do it, you may not be able to complete it smoothly. By the way, the series expansions of five commonly used functions must be memorized, such as proportional series, exponential function, two trigonometric functions and binomial expansion. Don't forget the corresponding convergence domain.
The third way can be regarded as an application problem, which is relatively simple, and the result can be obtained directly by using Newton-Lai formula and distributed integral.
The fourth method is to prove the mean value theorem. The most effective way to solve this kind of problem is to use the "original function method", that is, the equation that shilling requires to prove is a new function, try to find out the original function of this new function, see if it meets the conditions of some mean value theorems (generally), and then smoothly apply the theorems. The starting point is to construct a suitable function, which also needs to be accumulated when reviewing. There's another question. There are two or three questions. Pay attention to the conclusion of the previous question, and the difficulty of the latter question will be reduced.
The fifth question is the most difficult one for me personally. I'll basically throw my score here. The relevant knowledge points are Green's formula and differential equation. The first question is to prove the conclusion. If you have seen (roughly remember) the proof process of Green's formula, you will have a clue. You can draw a conclusion by filling the closed curve, and pay attention to the coordination of the curve direction. Then a differential equation is obtained by using Green's formula, which can be solved, but the solution process is very troublesome. Finally, I saw the unknown function through observation and then pieced it together. I guess this is where I lost points.
Then there are two problems of linear algebra. The first involves many knowledge points, from eigenvalue to quadratic form, but it is very simple and the calculation is not very annoying. The only thing to pay attention to is not to forget unitization after finding the feature vector, and there is nothing else to say. The second question is very novel and the only one I haven't seen before the exam. We use the idea of classified discussion to discuss the values of unknown parameters, because the ranks of matrices are different, and the solutions of linear equations have different forms. If we know this common conclusion: if AB=0, then R (a)+R (b)
Finally, probability theory and mathematical statistics. The first is the distribution function and probability density of two-dimensional random variables. If the meaning of random variable function is clear, this model is not difficult to establish according to known conditions. Let's go back to the principle. The things in probability theory are abstract, but if we think more about it, it may be easier to grasp in practical sense. What is a random variable? Fundamentally speaking, it is a function, but the independent variable is not the usual number, but some events, and the function value is the corresponding occurrence probability of these events. I don't advocate memorizing formulas when finding the distribution of random variables of functions, but suggest that I derive mathematical expressions from the statements and definitions of random variables. The second test of digital characteristics, of course, also includes the samples in mathematical statistics, that is, the independence between samples. Note that some features of digital features require independence between random variables, while others do not. In short, it is best to distinguish these attributes and classify them accurately. For example, a linear combination of two normal distributions or a normal distribution? There is nothing wrong with roughness, but this conclusion is wrong, because there must be two independent normal distributions to have this property.