Exploring the power of calculus in the forest of thorns in Tongji

Document directory

• 2.1 variables and functions
• 2.2 functions and Calculus
• 2.3 limit and Calculus
• 3.1 micro-Learning
• 3.2 points
• 5.1 law of opposites
• 5.2 examples of corresponding points

Now we have started a round of postgraduate mathematics review, using the sixth edition of Tongji higher mathematics. For postgraduate entrance exams, it is a good teaching material, and all the knowledge points are clearly listed. But if you really want to learn calculus and grasp its knowledge context, then it is the next option. In this book, the branches and leaves derived from each knowledge point are too complicated and complicated to be overwhelmed by the main context of calculus. Therefore, taking the opportunity of postgraduate entrance exam, I plan to read some relevant materials and try to find the main ins and outs of calculus in Tongji teaching materials. "Advanced Mathematics (I)" mainly discusses one-dimensional calculus, while "Advanced Mathematics (ii)" mainly discusses binary calculus. This article focuses on the previous book, that is, the one-dimensional calculus part.

1. Framework of One-element Calculus

First, we will give a framework map of one-dimensional calculus, which corresponds to the Tongji edition of Advanced Mathematics (I ). Let's just give a simple explanation and then describe it one by one.

　　①The limit of functions and functions is the foundation of calculus;

　　②Calculus consists of three parts:Differentiation,PointsAnd points out that differentiation and integration are a pair of ContradictionsBasic theorem of calculus (Newton-Leibniz Formula);

　　③The opposite of the contradiction between the differential and the integral. A theorem or formula in the differential should also have the corresponding theorem or formula in the integral. And vice versa, that is, they correspond to each other. Therefore, you can see the two-way arrow above, indicating the relationship between some scattered knowledge points.

The above figure is the framework of the knowledge structure in the book of advanced mathematics. to grasp this figure, we have grasped the essence of one-dimensional calculus.

2. Foundation of calculus-functions and limits

No matter which Calculus Teaching Material is opened, it is indispensable.FunctionAndLimit. For example, the calculus we used in class (Department of mathematics, Huazhong University of Science and Technology) will start with two chapters (Chapter 1: functions, Chapter 2: Limits and continuity of functions) to introduce them; tongji "Advanced Mathematics" compresses the content into a chapter, that is, "Chapter 1: Functions and limits ".

The first question comes with it: in modern university calculus textbooks, why are functions and limits all started?

2.1 variables and functions

Since its birth, humans have never stopped trying to understand the world we are in. As early as the original period, our ancestors began to use the concept of "Number" to represent the number of objects, and the lines engraved on the rock wall to represent the trajectory of the movements of the celestial body.

However, everything in the objective world is moving and changing from particle to universe. Constants can only reflect the external performance of a change process at a certain moment. Using constants clearly cannot reveal the internal laws of changes. The development of productivity in the 17th century has promoted the development of natural science and technology. Not only has the existing mathematical achievements been further consolidated, enriched, and expanded, but also due to practical needs, people began to study Moving Objects and the amount of changes, so as to obtain the concept of variables. This is a turning point in the development of mathematics. After studying the general nature of variables and their dependencies, people come up with the concept of functions. The extension of mathematical objects makes mathematics enter a new period.

The prime energy equation of Einstein is a classic example of using functions to express the relationship between variables.

A function represents the essence of the internal connection of an objective object and a deep image of the real world.

2.2 functions and Calculus

Functions provide powerful tools for describing the changing world, but humans are not satisfied with this. We know that changes and changes are often inextricably linked. One change is often the cause of another change. Therefore, we often need to explore the variation rules of another unknown element through the variation rules of a known element, such:

　　The function expression of displacement is known, and the function expression of instantaneous velocity is obtained. In turn, the function expression of velocity is known, and the function expression of displacement is obtained.　　

If you have learned calculus, you can see at a glance that the former is the process of differentiation, while the latter is the process of its inverse operation-integral. (If you are a freshman, don't worry if you don't understand this sentence. We will explain in detail the internal meanings of "differentiation" and "points .)

In other words, calculus is actually an operation of a function: input is a function, and output is also a function. Through the differential or integral operation, we can easily calculate the function expression of an unknown variable from the function expression of a known variable.

With the concept of a function, differentiation and integration make sense. In the university calculus textbooks, the function section is put at the very beginning to lay a solid foundation for the emergence of calculus.

2.3 limit and Calculus

So what is the relationship between the extreme and calculus?

That's because we use the Limit Concept when defining differentiation and integration. Believe it? Go to p79 (derivative definition) and P226 (definition of points), and you will find that the limit plays a key role in it. One sentence: To describe the differential and integral buildings, you must use the limit as the foundation.

3. Micro-learning and point-based learning-once unknown

In the era before Newton and laveniz, micro-learning and integral points are actually two independent development disciplines, which were born to solve different problems respectively. I think it would be better to study calculus along this historical process. Next, we will first introduce the sub-learning and the integral points respectively, and then use the Newton-laveniz formula to unify them.

3.1 micro-Learning

The most important core concept in micro-learning is derivative and differentiation.

Birth of Derivative

How is the concept of derivative obtained? Here is a very simple physical example: the function expression of displacement is used to calculate the instantaneous speed.

Currently, when a particle moves along a straight line, the function expression of known particle displacement is S (T). We need to study the speed of the particle movement accordingly. The distance between the time point T0 and the time point T is S (t)-S (T0), so we can use the formula:

　　　　

It indicates the average speed of a particle in the interval [t0, T]. It is called the "average speed" of a particle in the interval [T, t0 ", it is obviously a function with continuous variable t as the independent variable. However, most movements are non-even (that is, the distance traveled during the same time interval is not equal), so it is not possible to accurately express the instantaneous dynamics at t0. However, when we infinitely shorten the time interval [t0, T], that is, when T infinitely approaches T0, the average speed and the trend of the limit V (t0) it expresses the concept of "Instantaneous Velocity" of a particle at moment t0. The mathematical expression is as follows:

　　　　

The above limit represents the instantaneous change rate of T, shift S (t) relative to time at a certain time point.

There are more than one such example of "Change Rate". In the field of natural science and engineering technology, there are still many concepts that involve "Change Rate ", for example, the current intensity I (the charge of a cross-section of the wire per unit time), the angular velocity w (the angle of rotation within the unit time), and so on. It can be seen that this problem is quite common, as long as you study the change rate of a variable a relative to BWe need to use the above limit to express this change rate..

Since the limit is so sharp, mathematicians give it a rather memorable short name --DerivativeAnd provides a strict definition:

　　　　

In a word:The change rate of variable Y relative to the independent variable X in the Function Y = f (x) gave birth to the concept of derivative.

From derivative to differential

The concept of differentiation is closely related to derivatives. From the above discussion, we know that in order to solve the problem of "Change Rate", the birth of the concept of derivative is catalytic. So what kind of problems have led to the birth of differentiation?

In many specific problemsIt is often necessary to calculate the number of changes to the function.:

　　　　

For example, to find the volume of the shell with the inner radius R and thickness, we know that the formula of the ball volume is. When the radius increases from R + to R +, the ball size increases.

　　　　

The function here is very simple, so it is not too difficult to calculate the expression (three-way expansion, and then subtract ). However, some functions are very complex, so it is not so easy to calculate. For example, how can you find existing functions?

Hey, mathematicians have also considered this problem. If some things are hard to work, they can use the "Approximation! It's okay to use a relatively simple formula to approximate things that cannot be directly computed! (Although it is more or less likely to cause errors, it doesn't matter if the error is very small and approaches 0)

Now, the concept of "differentiation" is coming soon!

Let's proceed from the definition of the derivative:

　　　　

If order:

　　　　

Obviously, that is, when ,. Deformation of the preceding formula:

　　　　

That is to say, when the independent variable X is a change volume, the change volume obtained by Function Y is composed of two parts: one is (linear) and the other is (nonlinear ).

In this case, it is a volume that tends to be zero faster. That is to say, when it is very small, the proportion of the first part is much larger than that of the second part. Therefore, the second part can be ignored. In general:When it is very small, the non-linear part tends to be 0 faster, so we can use the linear part for approximation.

Well, we can name this linear part --DifferentiationAnd use the following symbol:

　　　　

In particular, if y = x is used, there is (this formula indicates that the increment of the independent variable is equal to the differential of the independent variable). Therefore, the above formula can be rewritten:

　　　　

In this way, we have introduced the concept of differentiation from the concept of derivative, and described the concept of differentiation.When the variable X changes DX enough hours, the dy of the function value y can be approximate using f' (x) dx.You might think that's amazing. It is not an approximation. But take a closer look. when X is a specific value, f' (X) is a constant! That is to say, Dy is proportional to DX! Furthermore, in the range [x, x +] (small enough), Y = f (x) can be considered as a straight line segment! The smaller the value, the more accurate the approximation! In this way, we can use a linear method to solve non-linear problems, which is quite convenient!

If you are not quite familiar with the above, you can see the following figure, which is extremely classic and illustrates the geometric meaning of the differential:

　　

Both P and M points are on the curve, and their X coordinates are X and x +, respectively..

If the tangent om of the curve is N after P points and the tangent slope is the derivative F' (x), then DY = f' (x) dx indicates the line segment on. In this case, we can use a straight line segment Pn to represent the curve segment PM.

Overview of micro-Learning

Micro-learning is a research functionLocal Features. Derivative and differentiation are the two most important concepts in micro-segmentation.

We can find the derivative F' (x) of the function, which represents the change rate of the dependent variable Y relative to the independent variable X at a certain point of the function (in geometric terms, this is the slope of the tangent of the vertex). Further, we have derived the concept of differentiation, which can approximate a complex function to a linear function in a local range, and the slope is the derivative of the vertex. This linear approximation simplifies local research on complex functions and provides strong support for subsequent research.

The increment infinitely approaches zero, the cut line infinitely approaches the tangent, and the curve infinitely approaches the straight line, so as to solve the non-linear problem through direct generation and linearity. This is the essence of the micro-segmentation theory.

3.2 points

Current points, including fixed points and indefinite points. Although there is only one word "no" between them, they were not compatible before the Newton-laveniz formula came into being! In general calculus textbooks, the indefinite points are usually arranged between the differential points and the fixed points (flip the book, isn't it ?), However, in this case, it is easy to confuse the logic of the students and give them the illusion that "determining points and indefinite points are something. I think the better way is: first talk about "differentiation" and "definite points" separately, then introduce "Newton-laveniz" to establish a connection for them, and then introduce indefinite points.

Starting from Area

To introduce the concept of points, let's take a simple classic example: finding the area enclosed by the X axis within the range of [0, 2] is reflected in the figure as follows:

　　

OK. Now we need to specify the OAC area of the image. What should I do? Of course, if you know calculus, you can do it in a moment. But now I am not allowed to use it. How can I solve this problem?

Think back to the idea of finding the area of the circle: divide the circle along the radius into small triangles, and find out the sum of the area of these triangles to approximate the area of the circle. The more detailed the split, the closer the approximate result is to the actual result. In other words, can we also use this "subdivision" idea to try to solve the parabolic area problem? As follows:

　　

For example, divide the range [0, 1] into N equal points, n-1 rectangles, and add up their area:

　　　　

When N is larger, the number of rectangles is more, and the smaller the white part is, the closer it is to the actual area of the irregular image Oca. At that time, it is equal to 1/3, that is, the image OCA area.

Another example

For example, if the velocity expression V (t) is known and the distance taken by an object within the interval [, T] is obtained, we can also use the above method to deal with it:

Divide the interval [, T] into N parts and insert the shard. When the interval is very small, it is reasonable to imagine that the speed v = f (t) cannot change much during this interval, therefore, we can regard the speed as constant, and we can regard the T = speed v = f () at any time point as the speed at each time point, therefore, the distance traveled by an object within the time interval is approximately equal:

　　　　

The distance taken during the entire time interval is:

　　　　

The elders of the N short periods are (think, why ?), Obviously, the closer the interval [, T] is divided, the more accurate the above approximation is. If the range [, T] is infinitely subdivided, that is, the upper limit is equal to the required distance:

　　　　

Definition of points

Use this"First segment, then accumulate, and then take the limit"There are still many problems to solve. The above problems can all come down to a limit such as computing. Since this limit is so widely used, we give it a name --Set pointsAnd summarize its general definition (p225-P226 in the book has provided a detailed definition, so only the final definition is provided here ):

　　　　

From the formula, we can see that the definite integral is actually an operation, and its operator number is an operation for the product function f (x )."First segment, then accumulate, and then take the limit". The result of the integral calculation is the final result of getting the limit. PointsGeometric meaningThat is, the area under the f (x) curve. PointsPractical SignificanceIs usedEvaluate the cumulative effect of a variable F (x) over another variable X(For example, area S indicates the cumulative effect of line L on line m; distance s indicates the cumulative effect of speed V on time t; power W is the cumulative effect of force F on displacement S ).

This is the integral point, that is, the core content of the integral point before the emergence of the Newton-laveniz formula.

4. Newton-laveniz formula-the marriage of differential and integral

In this way, the differential sister and the integral brother took their respective steps. They did not achieve good deeds between them until they met Newton and laveniz, but also achieved a good story in the field of mathematics.

This is the greatness of Newton and laveniz. They pointed out that differentiation and integral are inverse operations, so that the problem of determining integral can be converted into the problem of finding the difference of the original function, this greatly simplifies the process of solving points.

However, before introducing the Newton-laveniz formula, let's make a small comparison between the development of differentiation and integral ~

	Micro-Learning	Points
Core Content	Derivative and Differentiation	Set points
Geometric meaning	Tangent of a curve at a certain point	Area under the curve
Practical Significance	Change Rate of variable A to variable B	Cumulative effect of variable A on variable B
Application Example	Known displacement S (t) for speed expression V (t)	Displacement expression s (T) at a known speed V (t)

How is it so difficult to calculate 4.1 credits?

Before we introduced the fixed points, we used to calculate the area of the image enclosed by the X axis in the range of [0, 1. We usedDefinitionTo calculate the points, cut the irregular graph into small rectangles, accumulate and take the limit, and the final result is obtained. However, it is very simple. If I give you a function that allows you to calculate the area in the range [0, 1], if you still use the definite points definition method, the following statement is displayed:

　　　　

You should have cried ~ Really difficult to calculate! Although the example I gave is a little extreme, it illustrates a very difficult problem:It is difficult, difficult, and difficult to use the definition method to calculate the points of complex functions!

What should I do? Newton-laveniz "debut "!

4.2 troublesome solution: Newton-laveniz Formula

This formula is provided directly, and the proof section is omitted (for proof, see p239 ):

　　If f' (x) = f (x) exists in the range [a, B], then:

　　　　

The problem described in this formula is:The calculation of the definite points of a function f (x) can be converted into the difference operation of its original function f (x )..

With this strong formula, let's take a look at this problem. That is to say, we only need to find the original function, and the problem will naturally be solved! So how can we get to the original function of f (x) early?

This requires the help of the export public table. On the p116 page, we find that the first export formula is:

　　　　

Then, we only need to find the function after the derivation. Let's repeat it and make it. Then, the original function is obviously! According to Newton-laveniz's formula, there are:

　　　　

Is it super simple? No longer need to hate and bother defining!

In other words, is this progress? Points are so important, but their definition method is so troublesome, while Newton and laveniz point out this formula, which greatly simplifies the calculation process of points, and greatly promotes the development of Science (especially physics ), in this way, productivity is promoted, and the pace of social advancement is accelerated. It is taken for granted that they are great!

4.3 credits

After talking about the differential, definite points, and the Newton-laveniz formula connecting them, let's take a look at what the indefinite points are.

As we have said before, to evaluate the original function of f (x), we only need to compare the derivative formula of p116-p117 and then reverse derivation.

However, it seems awkward to always push the data in this way. As a result, mathematicians may wish to sort out this inverse process of derivation and give it a name --Indefinite pointsAnd obtained the integral table of the P188-P189, but also with some integral skills, such as change element method, Division integral method.

　　An indefinite point is the inverse operation of the derivation. It is completely irrelevant to the definite point before the Newton-laveniz formula is obtained! (Because you do not want to do this before the formula is used ). Only after this formula is used, the arithmetic rule of indefinite points is incorporated into the range of integral points, which greatly facilitates the calculation of definite points.

At this point, the main framework of one-dimensional calculus has been established: calculus, which consists of the microlearning, the integral point, and the Newton-laveniz formula linking the two. The core content of micro-learning is derivative and differentiation, and the core content of the integral point is fixed points and indefinite points.

5. There are still some residues. What should I do?

OK. The above discussion has covered the first chapter of "Advanced Mathematics" Tongji edition (previous ~ Chapter 6: Chapter 7 "Ordinary Differential Equations" is nothing more than an application of calculus in solving differential equations. Okay. Now I have finished speaking. What else can I do?

Although we have built a calculus framework, some fragmented things, such as some theorems, formulas, and calculation rules, are too scattered. The next task is to establish a connection for them and assign them a framework. In the following discussion, some people may feel a little mysterious and not rigorous, but it is very helpful for us to have a deep understanding of differentiation and points.

5.1 law of opposites

Makesi (max) is estimated to be one of the most annoying people. Chinese children's shoes are almost inevitably suffering from the tragic fate of his round: Political and high-definition political examinations in the senior high school entrance exam, politics is also required for postgraduate entrance exams. It is estimated that when you hear the name "Max", your mind will be reflected in a way that is boring and boring ~ Haha, I hate Ma, too. But then again, some of his things are indeed valuable, otherwise they will not gain such a high international reputation. We hate him, it may be a bias caused by incorrect school education.

In Max's theory, the dialectical materialism is a very important part.Unity of Opposites"Is a law that he attaches great importance to and is called" the fundamental law of the development of things ". Next, let's take a few paragraphs on the page "Overview of basic principles of Marxism" in my sophomore year. It can be used directly without explanation, because I can't really explain the stuff in the philosophical realm...

"The Law of opposites provides a fundamental way for people to understand the world and transform the world --Contradiction Analysis. The comparison and unification reflect the two basic attributes of the contradiction.UnifiedIt refers to the nature and trend of the interdependence and interconnectivity between the two sides of the conflict.OppositionIt refers to the nature and trend of mutual exclusion and separation between two conflicting posts ."

While differential and integral are the two sides of the conflict, and it is the Newton-laveniz formula that points out this contradiction. (refer to the fifth lecture of calculus by, I don't know, Xuan Ba ~). They are both opposite (one is the form of differentiation, and the other is the form of integral) and unified (their research object is a function, the difference studies the local characteristics of the function, credits study the overall features of Functions ). In principle,A Theorem or formula in the differentiation should also have corresponding theorem or formula in the integral; and vice versa, that is, they correspond to each other.

From a conflicting point of view, let's take a look at some of the corresponding points in the differential and integral points.

5.2 example algorithm of corresponding Vertex

For differential operations, there are several operational rules in section 2 "function derivation rules:

①AndThe Derivation rule:

　　　　

②ProductThe Derivation rule:

　　　　

③Composite FunctionsThe Derivation rule:

If, There are:

For the integral operation, chapter 4 "Indefinite Integral" introduces the three most important integral rules:

④Points of rational functions:

　　　　

⑤Division integration method:

　　　　

⑥Change element method:

Set:

We can see that: ① can launch ④, ② can launch ⑤, and ③ can launch ⑥. That is to say, ① corresponds to ④, ② and ⑤, and ③ corresponds to ⑥ one by one.

Public table

The score table of P95 and p188 correspond to each other.

Mean Value Theorem

The differential mean theorem of p129 corresponds to the Integral Mean Theorem of p233.

In short,Recognizing that differentiation and integration are the opposite of contradictions helps sort out some of their theorems and formulas and generalize them into the framework.

6. Calculus Overview

// Todo

7. Conclusion

The original intention of writing this article is mainly because I want to take a postgraduate entrance exam and have the opportunity to contact calculus again two years later. Although my freshman calculus scored 98 points, I learned a lot consciously... after all, "exams" and "Learning" are essentially different activities. The former is score-oriented, and the latter is knowledge-oriented ). Therefore, the basic course of calculus is also a good choice. Now, I have put all the duties of the class committee and the club away. I have nothing but to do with my postgraduate entrance exam. Why don't I calm down and repeat calculus? Then, the learning experience is written into a log, which is conducive to the organization of personal thinking, and facilitates future review, and can provide some thin help to other students. This kind of self-interest is nothing more than that.

Among the many calculus textbooks I have read, the most beneficial thing is Professor Yan's concise calculus. I think it is one of the best domestic calculus textbooks. Here, I would like to borrow a comment on excellence:

Professor Yu is one of my most admired masters! He wrote this concise calculus, which broke through the structural framework of the traditional high-number teaching materials, grasped the main contradiction of calculus, and wrote down the content of calculus from a higher level, the content of this book is wonderful and fascinating. It can be said that "calculus" is "active! This book is a rare excellent teaching material for beginners and those who have studied advanced mathematics and want to improve their mathematical literacy! What is the stream of the Tongji version? Compared with this book, it is simply not a realm! If we work with the introduction to advanced mathematics of the University of Science and Technology and the lecture on calculus of Professor Yu, we will learn from each other for reference. That is a wonderful feeling! Professor Yan wrote in his essay: "I have no degree, not an academician, but an ordinary old scholar ". However, his mathematical literacy and level are no less than any academician. There are not many masters with high levels of Ethics who can read the masterpiece with respect in Chinese Mathematical materials. Such masters include Hua lugeng, Chen xiyong, Li Shangzhi, Zhang zhusheng, and Shi jihuai, zeng kencheng. Only this digit.

After reading this book, I also learned the fact that the University of Science and Technology of China is one of the best schools in China for teaching basic courses.

It took a week to write this log intermittently. It is easy to say, but when writing, you will find that it is really not easy. First of all, a blog post is "the content is King", so you need to ensure the quality of the content, refer to different Calculus Teaching Materials, and straighten out the logic framework, in addition, it will be constantly modified during the writing process. To write this kind of mathematics blog, you need to use formulas and images, but you cannot use the keyboard to knock it out, so you have to use a variety of tools: use the formula editor to knock out all the formula code, use the function drawing tool to draw each function image; finally, you have to consider the blog user experience, therefore, some modifications are made with a small amount of CSS and JS Code.

Writing a logic blog is like building a software product. First, you need to build a framework, and then iterate and debug continuously until you finally get a satisfactory article. This kind of experience tells me that it is very difficult to accomplish anything overnight. We should constantly refine our work with peace of mind, and constantly innovate and change in order to make good things, it will not fall to the point of "never waste the rivers and the ages. Why can't we make iPhone or other products in China? Why can't we make a great job like Warcraft or Diablo 2? It may all be defeated by the word "impetuous.

Hey, my IQ and EQ are very limited. Although I have done my best, there must be some improper or even wrong information. Where there are typos, where the statements are not fluent or incomplete, and which image has an error. Or even further, you can't understand what you don't understand, even if you criticize and correct it! Looking forward to your feedback!