Book Review on Easy Learning Calculus

1

The origin of book review

In May last year, the author got the book Easy to Learn Calculus presented by Zhang Zhongxing, editor (teacher) of Science Press. Curious about the title of the book, the author finished reading the book in one breath and found it really "easy" to read. Calculus is a compulsory course for science and engineering majors in universities, and such courses are also called "advanced mathematics". The so-called calculus, including differential calculus and integral calculus, has played a very good role in dealing with many practical problems. Therefore, it is very necessary for students majoring in science and engineering to learn calculus well.

There is no doubt that there are obviously various calculus textbooks and popular science books on the market. Then, the natural question is, why did Mr. Zhuo Yonghong, the author of this book, write such an "easy to learn" calculus textbook?

On this issue, the author has published such a view in the preface:

"The author deeply feels that many people's unsatisfactory results in calculus are often not due to poor talent or poor learning attitude, but because they have not grasped the core spirit of various topics of calculus and stayed in the operation of abstract symbols, so they can't get in."

Indeed, as the author said, at present, many calculus textbooks often focus on the simple listing of mathematical symbols and formulas, but fail to show some theorems in calculus to readers intuitively. For a long time, many people hate calculus, and even say that they don't want to see the derivative symbols of Newton and Leibniz all their lives.

However, it is not so easy to solve this problem, and the author of this book tries to alleviate this bad situation with his "easy to understand" explanation of calculus.

2

The characteristics of this book

A close look at this book reveals that it has the following characteristics:

(1) Introducing Calculus with the History of Mathematics

At present, many calculus textbooks mainly explain the mathematical results themselves, so most of them are introduced in the mode of "definition-theorem-example-exercise", which is difficult to arouse readers' interest in reading. One of the characteristics of this book is to introduce the history of mathematics alternately, and through the introduction of the history of mathematics, the effective integration of mathematics and history can be achieved. It is worth noting that at the beginning of each chapter of this book, there will be a famous saying by mathematicians or others. For example, in the second chapter of this book, "Differential calculus", there is a philosopher Voltaire's profound insight into calculus:

"Calculus is an art of accurately calculating and measuring unimaginable things."

Although in the eyes of many people, the famous sayings of mathematicians can't help them understand those seemingly boring mathematical formulas, it needs special attention that the opinions of these leading figures can often help them quickly understand the essential connotation of a subject. Of course, in this book, by putting famous sayings and sentences before each chapter, you can effectively lay the theme of this book (yes, you are reading calculus books! )。

In addition, the author also spent a lot of time in this book to elaborate some mathematical historical materials about calculus, such as the copyright dispute between Newton and Leibniz in history, the problem of the steepest descent line, and the story of L'H?pital and Bernoulli. Even though these are classic facts in calculus, the author does not stick to the routine and lists these old things in his own unique humorous language, which makes the author feel that there is an interesting math teacher teaching the history of calculus. In addition, a close reading of the author's words shows that the author has a strong Taiwan Province accent (for example, the middle text on page 174 in the book "In fact, these two spellings are equivalent in French and can be used!" ), so it can be understood as a history of calculus with the characteristics of Taiwan Province Province.

(2) Show the problem-solving ideas in detail.

The essence of calculus is differential calculus and integral calculus. Among them, the differential calculus involves the concepts of derivative and differentiability, and the mathematical theorems involved include Fermat's Last Theorem, Rolle's Theorem, Lagrange's Mean Value Theorem, Cauchy's Mean Value Theorem and so on. These mathematical theorems also help the majority of students studying calculus to understand a group of mathematicians abroad: Fermat, Lagrange and Cauchy. The theoretical part of integral mainly involves Riemann integral, which is different from Lebesgue integral in the course of real variable function in mathematics department.

The mastery of mathematical theory is mainly reflected in mathematical analysis, and the purpose of this book is to learn calculus easily, so it is natural to focus on mathematical calculation. For example, how to calculate the derivative of a function? How to calculate indefinite integral and definite integral of a function? How to calculate a double and triple integral? These are the key problems to be solved in calculus teaching, and theoretical proof is in a secondary position.

Another feature of this book is that the author shows the key ideas to solve the problem with his own popular language and way of thinking.

For example, when the author proves the function limit problem, he introduces step by step how to use mathematical skills to achieve the ultimate goal. For example, some exercises need molecular rationalization, while others need trigonometric inequality. For example, in the book Example 1.4. 12, the author wrote this passage:

"Next, use a trick to see clearly. This is called trigonometric inequality. "

Readers who don't know think they have strayed into the martial arts novels. What's wrong with it? In fact, the mathematical circle itself can also be regarded as a small river and lake, and the tricks used to do the problems here are also martial arts cheats in mathematics. The author once had an inappropriate view: "Mathematical skills are like gestures, and mathematical thinking is like internal work." . If used here, then the trigonometric inequality is really a move, just a simple gesture.

(3) Be good at visualizing mathematical concepts with charts.

When I first read this book, it was hard not to be attracted by the exquisite geometric images made by the author. In calculus, formal symbolic operation is inevitably boring, and few people are willing to deal with mathematical formulas all the time. In fact, if you know the journal articles of biology, it is not difficult to see that their articles are all "looking at pictures and talking". In fact, mathematics should be like this. It is said that when discussing academic problems among mathematicians, they often draw a picture first, and then make up the corresponding mathematical description according to the picture.

In this book, the amazing graphics are naturally two-dimensional or three-dimensional geometric images. Especially when we encounter the problems of double integral and triple integral, if there are more intuitive geometric images to help us understand the problems, then we will achieve the goal of simplifying the complex. In fact, there are too many illustrations in this book, not to mention the effect of intuitively understanding mathematical concepts.

Carefully typesetting books with latex

Soon after I got this book, I was satisfied with its format. In addition to being surprised by the author's typesetting skills, Mr. Zhang Zhongxing, the editor of this book, told the author: "The author, Mr. Zhuo Yonghong, is a tex typesetting master and the second powerful figure she has known so far." In addition, Mr. Zhang added: "Apart from not being able to draw pictures quickly, it is basically the typesetting speed of the teacher with lecture notes."

Although I have never met Mr. Zhuo Yonghong, by reading the layout of this book and Mr. Zhang Zhongxing's description, I conclude that what I said is definitely true (after all, mathematicians are rigorous! )。

This book is really skillful in LaTeX typesetting and can be compared with other LaTex players. A remarkable fact is that the author has inserted many arrows in the process of solving problems in this book, which is unique in the arrangement of various definitions, theorems and properties. The author believes that the typesetting of many domestic mathematics books will inevitably draw lessons from this book.

three

Write it at the end

Regarding the book Calculus of Yi Xue, the features listed are only some of the advantages in the book, and other advantages need to be discovered by readers themselves. One of the shortcomings of this book is naturally that it does not continue to introduce the theory of surface integral and curve integral. This book is divided into twelve chapters. However, chapter 12 only introduces double integrals and triple integrals, so in my opinion, this is not enough for students who want to learn calculus.

The author also studies mathematics, so it is not appropriate to write a book to evaluate math teachers. So, here is the end of a sentence that Chen Yue, a teacher in the Department of Mathematics of Shanghai Normal University, once warned the author:

"When reading a book, imagine yourself as the author himself. If you write, can you write? Why do you want to write like this? "

four

editorial comment/note

Mr Zhu Xiao is too modest. Mr. Zhu Xiao is a graduate student in the School of Mathematical Sciences of Tongji University. Thank him for his book review of Calculus Little Book after his study and research. The detailed book information of Easy Calculus is as follows, dedicated to you who like calculus and study ~

brief Introduction of the content

Learn calculus easily.

Author: Zhuo Yonghong

Easy and interesting calculus reading.

Suitable for audience: people who are interested in calculus and want to know about it, liberal arts students who want to improve their mathematical literacy, students who find calculus difficult to learn in class preparation, and other readers who want to know about calculus.

This is a book that teaches readers how to get started with calculus easily, and it is also an easy and simple book suitable for self-study. Easy to learn calculus language, easy and humorous. Through a large number of appropriate and concrete graphics and images, this paper introduces the origin of various concepts of calculus as vividly as possible, perfectly connects middle school mathematics with advanced mathematics, interspersed with mathematical history, and restores the context of mathematical thought. There are also common interesting talks about the symbols of higher mathematics, which deepen readers' learning impression and understand the ins and outs of the development of calculus. The author summarizes many years' teaching experience in calculus, uses simple and easy-to-understand language as much as possible, summarizes learning methods and practical rules, points out common mistakes and students' blind spots, provides detailed problem-solving skills, and intersperses one problem with multiple solutions to broaden the horizons, helping readers to grasp the specific knowledge points of calculus from a higher angle easily and happily, so that readers can have a clearer understanding of calculus. This book introduces China's ancient mathematics and ancient mathematics thoughts in particular, so that readers can appreciate the contribution of ancient philosophers in China to mathematics while introducing calculus easily.

Table of contents of this book

catalogue

Chapter 1 Limit and continuity

1. 1 the origin of calculus 1

Limit of 1.2 sequence 5

1.3 continuous functions and limits of functions 16

1.4 Strict definition of limit 30

1.4. 1 Definition of limit 30

1.4.2 is proved by limit definition 35.

Properties of 1.5 continuous function 40

1.6 natural index and natural logarithm 45

1.6. 1 natural index 45

1.6.2 natural logarithm 48

1.6.3 using the definition of e to solve the limit 49

Interesting talk 1.6.4 e 52

1.7 equivalent infinitesimal substitution 56

1.7. 1 motivation introduction 56

1.7.2 Infinitely small gradation 57

1.7.3 Equivalent infinitesimal substitution 58

1.8 asymptote 63

1.8. 1 horizontal asymptote 64

1.8.2 vertical asymptote 66

1.8.3 Oblique asymptote 67

Chapter II Differential calculus

2. 1 derivative definition 73

2.2 Properties of Derivative and Derivative Functions of Power Function 80

2.3 Derivative function of trigonometric function and logarithmic function 9 1

2.4 Higher derivative 96

2.5 Chain rule 99

2.6 Unilateral derivatives 103

2.7 Derivation of implicit function 1 1 1

2.8 Derivation of Inverse Function 1 17

2.9 logarithmic derivative method 122

2. 10 parameter derivation 125

2. 1 1 difference 13 1

Chapter III Application of Differential calculus 135

3. 1 Tangents and normals 135

3.2 Variable interest rate problem 140

3.3 Monotonicity and concavity of functions 143

3.3. 1 monotonicity of function 143

3.3.2 concavity and convexity of function 147

3.4 Extreme Value Problem 153

3.4. 1 First-order verification method 155

3.4.2 Second-order verification method 157

3.5 Drawing function diagram 160

3.6 Differential Mean Value Theorem 165

3.7 L'H?pital Law 170

3.7. Introduction to the use of1Lobida Law 170

3.7.2 Discussion on Abuse of L'H?pital Rule 176

Chapter IV Integral calculus 18 1

4. Definition of1integer 18 1

4.2 Basic Properties of Integer 19 1

4.3 Basic Theorem of Calculus 196

4.3. 1 Basic Theorem of Calculus Part I 196

4.3.2 Basic Theorem of Calculus Part II 200

4.4 indefinite integral 202

4.5 Areas between Curves 206

The fifth chapter integral skills 2 1 1

5. 1 partial integral 2 1 1

5.2 Variable substitution 2 17

5.2. 1 First alternative method 2 17

The second alternative method 223

5.3 Triangle Substitution 225

5.4 Integral of Rational Function: Partial Fraction Method 232

5.5 Integral of Trigonometric Function 243

5.5. 1 power of trigonometric function 243

5.5.2 Rational formula 252 containing sin(x) and cos(x)

5.5.3 Ingeniously exchange RMB 254.

5.6 improper integral 256

5.6. 1 improper integral of the first kind (unbounded integral range) 256

5.6.2 improper integral of the second kind (unbounded function) 259

5.6.3 Convergence and divergence of improper integral 26 1

5.7 Integral Skills Miscellaneous Talk 265

Chapter VI Application of Integral calculus 276

6. 1 curve arc length 276

6.2 Find Volume 283

6.3 Volume of Rotating Body 287

6.3. 1 disk method 287

6.3.2 Shelling method 29 1

6.4 Surface Area of Rotating Body 295

Chapter VII Special Functions 299

7. 1 hyperbolic function 299

7. 1. 1 definition of hyperbolic function 299

7. 1.2 Basic Formula of Hyperbolic Function 302

7. 1.3 Derivative function of hyperbolic function 306

7. 1.4 inverse hyperbolic function 306

Derivative function of 7.65438+inverse hyperbolic function 308 of 0.5

7. 1.6 Application of Hyperbolic Function in Calculus 309 for Freshmen

7.2 Gamma Function 3 10

Chapter 8 Infinite Series 3 13

8. Convergence and divergence of1infinite series 3 13

8.2 Integral Convergence Method 32 1

8.3 Comparison and Collection Methods 326

8.4 ratio convergence method and root convergence method 33 1

8.5 staggered series convergence method 335

8.6 Conditional Convergence and Absolute Convergence 34 1

8.7 Power Series 349

Chapter 9 Taylor spread 356

9. 1 Taylor expansion: polynomial approximation function 356

9. 1. 1 Taylor expansion 356

9. 1.2 indirect expansion method 360

9.2 Application of Polynomial Approximation 368

9.3 Taylor Theorem and Remainder 373

9.4 Sum function of power series 38 1

Chapter 65438 +00 polar coordinates 390

Introduction to 10. 1 polar coordinate 390

Common curves in 10.2 polar coordinate 399

10.3 polar coordinates of area 402

10.4 polar coordinates of arc length 409

Chapter 1 1 Differential calculus of multivariate functions 4 13

1 1. 1 introduction to multivariate functions 4 13

1 1.2 limit of multivariate function 4 16

1 1.3 partial derivative 422

1 1.4 Total difference 429

1 1.4. 1 popular and imprecise discussions/kloc-0 /5666.000000000805

Theoretical discussion on 1 1.4.2 3 1

Chain rule of 1 1.5 multivariate function 434

Derivation of the implicit function of 1 1.6 multivariate function566661

1 1.7 gradient, directional derivative and tangent plane 443

1 1.7. 1 the definition of gradient 443

1 1.7.2 directional derivative 443

1 1.7.3 section 449

Extreme value problem of multivariate function 1 1.8 450

1 1.9 conditional extremum: Lagrange multiplier method 456

Chapter 12 Multiple Integrals 466

12. 1 double integral 466

12.2 triple integral 480

Method of replacing 488 with 12.3 multiple integral

12.4 polar coordinates replace 499

12.5 cylindrical coordinate replacement 504

12.6 spherical coordinate substitution 508

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