"Calculus" 01-The Mathematical Dragon Slayer Knife

When it comes to calculus, everyone is unfamiliar, and people who go to college are basically the ones who have been on it, and countless people are hanging on that tall tree every semester ... Liberal arts students in the "University of Mathematics" in the look of its face, engineering students in the "higher mathematics" and it met, and science students in the "mathematical analysis" see its true colors. But to tell the truth, compared with other abstract mathematics disciplines, calculus is very straightforward, it is also the object of the study of our intuitive can feel. People feel difficult, in fact, there is no form of "problem-style" learning habits, and just blindly try to understand and remember.

In my blog, I have always emphasized that any mathematical subject is: (1) Abstract the object to be discussed and strictly define it, (2) analyze the basic problems to be discussed, and solve the problem from the definition, (3) Extend the object, method and conclusion to a more general scenario. This is a mathematical study, the basic quality of research, put aside the formula theorem to discuss the basic issues, you will find that the general textbook will not appear a lot of difficult theories, but only some basic problems of basic conclusions.

Well, we say back to calculus, strictly speaking it is divided into differential (differentiation) and integration (integration), they are originally irrelevant, but since Newton, Leibniz to hand them, the differential, The points will grow old and stay together. Integral thought appears very early, it is in fact the whole as a myriad of pieces together. As early as in ancient Greece, the almighty Archimedes used integral ideas to solve a lot of the problem of area, volume, his tombstone on the ball and column, is to commemorate his proud conclusion. But in order to avoid the concept of "infinity" and "limit", after drawing conclusions, Archimedes used other methods to demonstrate.

In the Renaissance, the maturity of algebra, the development of analytic and functional, has been put forward and studied a lot of practical problems, the idea of differentiation is already well known methods. Later, we all know that Newton, Leibniz almost simultaneously found the relationship between calculus, from this subject is considered to have been produced. As for the cattle and the two people that caused the British to break with the continental war, you can find its details on the Internet, read it is quite interesting. This controversy of priority, in the history of mathematics is not uncommon, some mathematical ideas to the shell out of time, there are many people also found it is not surprising. We are now in the use of Leibniz's symbolic system, which is more accurate to express the idea of calculus.

Newton (1643-1727) Leibniz (1646-1716)

Newton and Leibniz completed the groundbreaking ceremony, the next 100 years, the development of calculus almost hoodwink, it is not only rich in content, covering a wide range, but also in various fields have been important applications. The discovery of calculus opened the prologue to modern mathematics, which was called "The Age of Heroes" for 100 of years, because there were vast tracts of territory that were excavated and more treasures were revealed. In history, a large number of outstanding mathematicians, Euler, Bernoulli family, the French school, and countless other Daniel built a grand building of calculus.

But after the passion is always an inexplicable emptiness, because people to that ghostly "infinity" is unbearable, must give it a strict definition. As a result, the crazy development of calculus is accompanied by the "second mathematical crisis" together, until the beginning of the 19th century "limit" theory people only put this uneasy heart down. This is back to the previous "set theory" and "real number system" of the course, it is with these two obstacles, I was slow to pick up the book of calculus. As for the history of calculus concepts and ideas, the most detailed and thorough reference [10], learning history is equally important to mathematics.

After solving the basic problems, calculus also derived a lot of independent branches: differential equations, vector analysis, complex analysis, and so on, these disciplines are often referred to as analytical mathematics (Analysis ), it is the largest branch of mathematics. In the time of modern mathematics fusion, mathematical analysis plays an amazing role in other irrelevant branches. Of course, we are still a little bit, this blog I only intend to learn the basic elements of calculus, including only one dollar calculus, multivariate calculus, Series theory three pieces. The general calculus and mathematical analysis textbooks all take these content as the main body, but also will introduce some extensibility knowledge, those content is I divides into other branches, here will not introduce.

It is becoming increasingly felt that it is almost impossible to do everything. Originally want to refer to more information, so that the content of the blog more comprehensive, more high, but the energy is not enough. Often refer to the one or two book after the beginning of writing, that there is no way, can only say that later back to slowly and diligently. Knowledge and thought is not a cluster, and the heart is still to the ground, the list of these reference books is to give themselves a reminder.

"Pre-order Discipline" real-number system, analytic geometry

Resources

[1] Calculus tutorial (one ~ three) "(8th), Fihkingorts, 2006

The curriculum of The Godfather class of calculus, although the age is relatively long, but its classic status still can not shake, almost all of the domestic textbooks are based on it, the content is comprehensive, large space. A large number of examples and applications, theory and application of the combination of close.

[2] Mathematical analysis Tutorials (top and bottom), Delhusza, Song, 2000

This is my undergraduate teaching materials used, by the teacher, although not famous, but personal highly respected, but also the main reference book of this blog. The textbook is rigorous, well-organized and easy to demonstrate, covering the basic and important conclusions in mathematical analysis, and extending the introduction of some content. Textbooks selected a lot of examples and exercises, the difficulty distribution is reasonable, it is worth careful consideration and study.

[3] Mathematical analysis (i ~ III), Wu Shengjian, 2009

[4] Mathematical analysis (i ~ III), Xu Forest, 2005

[5] Mathematical Analysis Course (i ~ iii), Chang, 2003

[6] Mathematical analysis (upper and lower), European sunshine, 2007

[7] Mathematical analysis (upper and lower), Chen Jisiu, 2003

These sets are also domestic relatively excellent textbooks, can choose according to their own preferences.

[8] Lecture notes on mathematical analysis (i ~ III), Chen Tianquan, 2010

This textbook is more high-level content, a lot of space extended to follow the subject, you can choose advanced reading.

[9] Demidovich Mathematical analysis problem Solving (1~6) (4th), Fetinghui, 2012

The Soviets wrote the problem sets, you know: The number is huge. It is also the most famous mathematical analysis of a problem set, to the analysis of scientific research must be done, the teacher told us Takeshi done once. For the ordinary students, it is best to pick some practice practiced hand good, and do the best not to see the answer to the question. A part of the problem is very difficult, difficult to out of the book people will not.

[10] The History of calculus concept (the calculus and Tts conceptual development), b.b. boyer,2007

Very old but very classical calculus of thought history, the book more emphasis on the evolution of mathematical thought. Content has a certain depth, rather than a story book, it is necessary to first have a disciplinary basis, to improve the understanding of mathematics is very helpful.

"Calculus" 01-The Mathematical Dragon Slayer Knife