Reprint:
https://www.cnblogs.com/xitingxie/p/5684254.html
Ability mathematics how to cultivate
Department of Mathematics Dr. anger answer!
I think we have such an experience: elementary school when you do not know what kind of junior high school math, high school, you would not believe what kind of university mathematics. The students, if you do not focus on math, I am afraid not know what he looks like modern mathematics. The following will explain how the motivation to learn mathematics from several aspects of mathematics, mathematical classification of the different disciplines of mathematics and how to cultivate practical ability. (In addition, welcome to watch how the Department of Mathematics reading experience? Experience the fun of mathematics, Department of Mathematics human characteristics and IQ is not enough how to do.)
================ enter the question of how to learn mathematics =============== ========
first, you need to understand
why you need to learn math, this is the first thing you need to think clearly problems. Mathematics and more sub-categories, each book has a lot of math theorems and conclusions, it takes a lot of time to study. The people's time is precious, limited, so you need to have a general objectives and plans, arrange your time.
1.1 Your goal is proficient in mathematics, studying mathematics, mathematical earn a living, you might aspire to master algebraic geometry, or want to be proficient in cutting-edge physics. Then you need to lay a solid modern algebra, geometry and analysis basis, you need to prepare a lot of time and effort, with unwavering determination. (Requirements: proficient in all three levels of higher mathematics)
1.2 Your goal is fluent in mathematics, solve problems, explore new weapons to master application areas, you might aspire to enter the field of computer vision, the field of economics or data mining. So, you need to lay a solid matrix theory, calculus, statistics and probability basis. (Requirements: proficient first grade mathematics)
1.3 Your goal is to understand the fun of mathematics, the mathematics of life as a great hobby. So, you need to lay a solid linear algebra, mathematical analysis, topology and basic probability and statistics, for you to experience the fun of learning mathematics is a more important goal. (Proficient mathematics first stage, second stage dip in Advanced Mathematics, trying to reach the third stage mathematics)
two, sufficient power to their
Mathematics need intelligence, but requires time and effort. The following phase we all understand a few facts:
1. Every useless things, or while useful, but less than what you learn quickly forget something else. Do not believe you recall your first year or the first day basic course, you still remember clearly?
2. Those who do not interest you (or do not feel fun) things, it's hard to adhere to finish it. A lot of people have this experience, a book, the first three chapters look at very carefully, gulping back on, ever more quickly, both boring useless anyway.
3. Primary Mathematics is the basis of high school math, middle school math is the basis for high school mathematics, high school mathematics is the foundation of the University of mathematics (and so can you).
Therefore, no matter what your goal is, do math with fun math, math or experience, to meet their own, there's a teenager from a dream. Learn and be happy, have learned to use, you will always be to maintain the momentum unabated two most important factors.
Third, what higher mathematics learning?
Well, take a look at the standard of Mathematics, University of Science and Technology tree:
a:
linear algebra (matrix theory), mathematical analysis, modern algebra (group ring domain), respectively, include a geometry, algebra and theoretical analysis. Do not forget there is probability theory (based on an analysis of the basic disciplines).
Class 2:
With these foundations, followed by base basis, abstract and promotion: measure theory (basis points, of course, is the basis of probability theory), topology (about collections, space, an extremely important basic disciplines of geometry ), functional analysis (promotion linear algebra), complex function (promotion analysis), ordinary differential equations and partial differential equations (promotion analysis), mathematical statistics and stochastic processes (promotion of probability theory), differential geometry (analysis and the combination of geometry).
Then, some small fresh and Applied science: numerical analysis (algorithm), cryptography, graphics, information theory, time series, graph theory, and so on.
Three:
Beyond graduate program is often algebra, geometry, and with the analysis to: differential manifold, algebraic geometry, like stochastic dynamics.
The third tier of the tree, and elementary, middle and high school mathematics is very similar to the one school not proficient, the next layer of hieroglyphics.
Fourth, how to learn
4.1 do question the amount of
Qianwanqianwan do not do crazy question. Strategy game played against the students know, made a few low-level soldiers on the line to save money in order to win the senior soldier in the late, low attack not only low-level soldiers, yet fun magic, the meaning of their existence is to give you the ability to stay up late. So many courses listed above, you spend the first five years to finish Jimmy Norwich six mathematical analysis problem sets, you're 30 years old, the secondary school curriculum has not started back yet. So, do some after-school exercise, to help you review, thinking, maintain brain operation on the line, we must continue to learn backwards. If you do not understand fully learn, to return to do the exercises to help sort things out themselves.
4.2 understand the idea of
mathematical essence not do the number of questions, but rather to grasp ideas. Each branch of mathematics has its own main line of thinking and methodology, different branches are also ways of thinking and learn from each other for comparison. Pay attention to it, imitate it, trivial knowledge strung on a necklace, you will master a lesson. I thought not read a textbook can easily understand, you have to read several books to understand some applications can appreciate. Two examples:
the main line of the calculus there are so few: Recognizing that micro and macro are linked, differential to portray how things change, enlarge it to show you the details, but integral to characterize the overall nature of things; differential and integral a phenomenon sometimes described in different ways, that you may not be easily found in the mathematical analysis in the book, but if you learn physics, you will find Maxwell's equations also have equivalent differential form and integral form; integral transformation be able to establish a link between different spaces, establishing contact for space and space boundaries, which is Stokes theorem: that this formula at the latest in differentiable manifold you can find it all.
Matrix is a linear transformation of space in the abstract, linear algebra course this is all about is to study how to express, simplification, classification space linear transformation operator; SVD decomposition not only have a very wide range of appearance in the applied sciences, but also you understand matrix a powerful tool; matrix is a linear operator on a finite-dimensional space, the understanding of "space" not only allow you to re-recognize matrix, more functional analysis of learning off to a good start.
4.3 circuitous progressive learning, compared to learning
a lot of time, read a book, perhaps because of a jump somewhere thinking about the future you will no longer keep up. A knack for learning mathematics is that you also get several of the internationally renowned materials, contrast with each other to see, or read a book and then look at another of the same subject, already familiar with the contents jump in the past, if you do not understand , or doing exercises to stop and think, or do not understand it to the back of a retreat, advancing from the section can understand, when you see more, you will find something appears in many places, on the understanding of it deepened. Two examples:
exterior differential this thing, and some domestic mathematical analysis may not be introduced in the book, I first met in Peng Jiagui of "differential geometry", I feel this is a convenient clever tool; then read Zhuoli Qi "mathematical analysis" and Rudin of the "principle of mathematical analysis", have talked about this thing, seen in the West outside the differential is a foundation of knowledge. You want to read it, it may be appreciated that first matrix, determinant understand exactly drawn in spatial volume multiple of the transformation matrix, which is a linear form. Finally, when you read differentiable manifold, the differential is found outside the tool Stokes theorem on manifolds available.
Point-set topology this thing, do not use the application. But deep down whenever you want to learn this discipline must be mastered, because it provides accurate characterization of such open sets, tight set, continuous, complete the basic concepts of mathematics. Later learn functional analysis, differential manifold, these concepts you will not move an inch. First of all you want to read Mans Chris enduring masterpiece "topology", followed by reading other books written by foreigners, more or less contact with some related concepts, your understanding will deepen, such as read Rudin's "Functional Analysis "start is to introduce linear topological space, knowledge in front of you will be able to spend.
4.4 to establish contacts from different disciplines
to see things in a lot of places with you to deepen understanding of it, and slowly will also be able to appreciate the subtleties of this thing, and finally you will find all the basic disciplines are intertwined, and in follow-up applications to help each other, they really truly appreciate the very basic, very useful. This is a fun way to experience mathematics.
4.5 focus on applied science
Nothing applications can stimulate your desire for new knowledge and new tools. Find some interest in applied science textbook, read, broaden their horizons, to accumulate resources for their own future. The following combination of their professional (computer vision) and love to talk about some of the best professional books:
learn calculus, you can read without pressure, "Feynman Lectures on Physics Volume I" for power, heat, light, space and time mystery; learn partial differential equations, you can read without pressure, "the second volume Feynman lectures on Physics" to understand the mysteries of electricity; learn Matrix theory, you can buy a "computer vision multi-view geometry" for imaging mysteries programming three-dimensional reconstruction image sequence; students learn probability theory should be heard and frequency of Bayesian school, two schools who pulled the battlefield in machine learning, the achievements of two classic book "Pattern Recognition and machine learning" and "the Elements of Statistical learning", read them, I was the mathematical basis for the field of machine learning to provide fruitful results and insights deeply impressed; read "Ray Tracing from the Ground up" he has written a ray tracer to render the real scene, its foundation is a little bit of calculus and matrix ......
advanced mathematics Application is too much, if you like programming, automation, robotics, computer vision, pattern recognition, data mining, graphic images, information theory and cryptography ...... everywhere large number of models for you to play, and only needs a little bit of higher mathematics. In these areas, you may be able to find more interesting than math book, and easier to find a job target.
4.6 find interesting books to read
Mathematician wrote the book sometimes is more rigid, but there are always some textbooks, their authors have a strong desire to want to show you, "This thing is actually very interesting," "that way this thing is totally not what you think" and so on, they succeeded; some authors, a thing they like to use the application, and different things in different areas in a particular area of focus show you. This book will give you plenty of fun reading. A typical representative of the domestic publishing of a "Turing Mathematical Statistics Series", which is a really great book, such as "Linear Algebra should learn this" "complex analysis: visualization methods," "differential equations, dynamical systems and chaos Introduction "I personally think is a must-read classic textbook of mathematics, very, very interesting.
Five more books, read good books
if only one sentence how to cultivate mathematical ability, it is this one: read more books, read good books. So this step I would like to say a few words out alone.
Surely they are very proficient and skilled application of elementary school mathematics. Want to understand algebraic geometry, or take a step back, want to understand the basis of information theory, you have to pick a few good basic teaching materials, preferably written by foreigners, so as to master the elementary school mathematics master it. Do not just read a find three different authors of the book, see the comparison, literally line by line to see. In some places certainly do not understand, write it down, maybe somewhere in another book on this stuff comes from another angle.
If you later have to learn later, each basic theorems are seeing now, the future will be used.
Each foundation of this book, you give up today, tomorrow, but also obediently to start over.
To read scriptures like, like cross-comparison of the similarities and differences of different reading materials content.
5.1 recommended textbooks (in fact, I've read that good book):
First stage:
"Linear Algebra should learn this"
Zhuoli Qi "mathematical analysis (two)" (read it in English, it is easy to know there. Friends say this is still not too simple, that you can look at domestic materials, and then go back and look at this)
, Fudan University, "probability theory"
of the second stage:
Mount Chris "topology"
Some Volume Turing Books
Curls Terry gold "Introduction to Algebra"
"The essence of Statistical Learning Theory" Vapnik
Rudin "principle of mathematical analysis"
Rudin "functional analysis"
Gamelin "complex analysis"
Peng Jiagui "Differential Geometry"
Cover "information theory"
Third level:
"differential and Riemannian popular"
"modern geometry, methods and applications" three volumes of
5.2 to read some science textbooks
"What is mathematics"
, "elementary mathematics at the high point of view"
, "Bach, Escher, Gödel"
"E story "
. 5.3 read all areas of the funniest, liveliest and most make you grow knowledge, the most important application, the most understandable writing materials and books
," Feynman lectures on Physics "three
" Chaos and fractals: Science Xinjiang community, "
" differential equations, dynamical systems and chaos Introduction, "
" complex analysis: visualization methods "
Finally want to say that mathematics is a bottomless pit, will consume your precious youth. You may know nothing about the inspirational get to know modern mathematics, but many will be deterred halfway, while the rest of the time was not well versed in the other sciences. And even if you are proficient in pure mathematics, not a few good articles is not easy to find a job.
My advice is to open up the process of reading vision mathematics, pure and applied mathematics disciplines to see, to find interesting, widely used to find good work (for money) direction and then go on to become a Mengzha your career. Such as mathematics solid, strong programming ability of people promising.