Liu Shibo

Mathematical analysis is the most important course in the department of mathematics. Many subsequent courses are based on it, such as ordinary differential equations, partial differential equations, complex variable functions, real variable functions and functional analysis. These all belong to the category of analytical mathematics. In addition, topology, as a branch of geometry, mainly studies the invariance of topological space under continuous mapping, which is the generalization of continuous function studied in mathematical analysis. Differential geometry is the most important department in mathematical research today. It is developed in the process of applying calculus to geometry, so it is also inseparable from the theory and method of mathematical analysis. Therefore, it is very important to learn mathematical analysis well in order to successfully complete the undergraduate study of mathematics department. Furthermore, mastering the theory and method of mathematical analysis is the key first step for any young student who is interested in mathematical research.

There is no shortcut to learn mathematical analysis well. So are other courses. If there was such a shortcut, the teacher would have told everyone in the class. In this case, don't worry too much, just work hard? If you just do it recklessly, it will not have a good result and you will be very tired. I have met many classmates whose books are all broken books, and the pages are full of notes or experiences in the process of reading. Looks like they're still trying. But the questions he asked me were very simple, and some of them were even clearly explained in the book. I turned the book over to him, and it dawned on him. I think this is because although he spent a lot of time, he didn't seriously think about the following aspects. So he didn't have a deep impression on the basic content.

Next, I want to talk about my views on the study of mathematical analysis. When it comes to learning mathematics, many people think of doing more problems. However, I think the most important thing is to study the textbook carefully and make every definition and theorem clear. On this basis, doing some exercises properly will get twice the result with half the effort. If you don't understand the basic concepts and the theorems you have learned, you will inevitably encounter many difficulties and even lose confidence. This is an undesirable learning method.

First of all, we should thoroughly understand every definition we come into contact with. Mathematical definitions are abstracted from many concrete examples. Although these definitions are abstract of concrete examples, they are natural. We should think more in our study and grasp the connotation of each definition through concrete examples. The definition of mathematics often has various conditions. We should think carefully about these conditions and understand their functions. Sometimes it is necessary to distinguish different concepts through positive and negative examples. Only in this way can we really master it and use it flexibly in reasoning.

Secondly, every time we learn a theorem, we must clarify the meaning of this theorem from the connotation, that is, what it says. This can often be grasped by combining geometric intuition. Then there are the conditions required to study the theorem. We can understand the function of these conditions by studying the proof of theorems, and we can also find out why the conclusion is wrong when a certain condition is not true by counterexamples. Through such positive and negative thinking, we will have a better understanding of this theorem. I have met many students in the department of mathematics. They said that "F is bounded because it is a continuous function on the closed set F". The reason why we make such a mistake is that we have not mastered the theorem that continuous functions on bounded closed sets must be bounded.

In addition, the proof of the theorem is also worth studying. By studying the proof of the theorem, we can deepen our understanding of this theorem. Moreover, in the process of theorem proving, we can also learn various basic argumentation methods of this subject. After being familiar with these methods, we can naturally apply them to the problems we face. The proof of some theorems is very beautiful, which fully shows the beauty of mathematics. We should also appreciate this beauty in the process of learning, which is conducive to improving our mathematics literacy. Of course, some theorems are difficult to prove. In order not to hurt self-confidence, you can skip it first and come back to study later. In fact, some theorems are important in themselves, but their proof is not necessarily important. In this regard, we can look at the description of arc-length quadratic variational formula on page iv of "Words for Readers" in front of "Preliminary Riemannian Geometry" published by Mr. Wu Hongxi of Peking University Publishing House. This whole "information for readers" is very enlightening to people who study mathematics.

In addition, after learning a theorem, a very important aspect is how to apply it to various problems. This is even more important than the proof of the theorem itself. Imagine that if you derive some interesting conclusions from a theorem, you will definitely find this theorem wonderful. Many theorems in mathematical analysis have intuitive geometric significance. Many proof questions are easy to understand if they are geometrically intuitive. Such geometric intuition often inspires us to find a solution to the problem.

We can also look at the theorem we have learned from a global perspective and see how it relates to other theorems in mathematical analysis. For example, why is this theorem needed? Imagine what the whole mathematical analysis theory would look like if there were no theorems about the properties of continuous functions on closed intervals. Linking various definitions and theorems to form an organic network in your mind, you can use knowledge more flexibly when solving problems.

After firmly grasping the definition and theorem. Do some exercises to deepen your understanding. The exercises at the end of each section of a good textbook are the application of the knowledge learned in this section. Doing these exercises will help you master this section better. The process of doing exercises is also a kind of training for yourself, which is another purpose of doing exercises. Just as long-term physical exercise will gradually strengthen the body, the ability to persist in doing exercises, analyzing problems and solving problems will gradually improve.

The practice of mathematical analysis is more flexible. We often have the experience of being helpless in the face of problems. This is a normal phenomenon. Don't lose heart. This is because we are inexperienced. Now there are many reference books such as problem-solving guides. There are many typical examples in these books, some of which are very enlightening. You can look at some according to your own situation. However, every good exercise is very precious. If you encounter a problem and are eager to see the answer without in-depth thinking, then although you know the solution to this problem or even this kind of problem, you have lost an opportunity to think independently. You wasted the question. So be sure to think about the exercises independently (or discuss them with several students). If we really can't do it, we will look at the answer and seriously sum up the reasons for our failure. For the examples in the reference book, if time permits, you should try them yourself first. If you don't try to do the examples in the reference book, you must choose to do some exercises in it.

Finally, the content of mathematical analysis is very rich, which is closely related to many subsequent courses. In the study of subsequent courses, we sometimes come back to look at the relevant contents in mathematical analysis and sort out the relationship between them. This is beneficial to better master mathematical analysis and follow-up courses.

The above is a teacher's article. The landlord should be able to meet the requirements with a slight change.

Although I found it online, it was carefully selected. I hope I can help you O(∩_∩)O~