Math Learning (turn to know)

How to develop the ability of mathematics
Dr. Fury of the Department of Mathematics!

I think we all have this experience: elementary School When you do not know what is the Junior high school mathematics, high School when you can not imagine College mathematics is what. and college students, if you don't focus on math, I'm afraid I don't know what modern mathematics looks like. The following will explain how to learn mathematics from the motivation of learning Mathematics, the classification of different disciplines of mathematics and how to develop mathematical ability. (Also, how do you feel about reading in the Math department?) Experience the fun of maths, the characteristics of the math department and the lack of IQ. ）

================ into the subject ======== How to learn maths ===============
First, recognize your needs
Why you need to learn math, this is the first thing you need to think clearly. There are many theorems and conclusions in the sub-classification of mathematics, and in every mathematics book, it takes a lot of time to study. And people's time is precious, limited, so you need to have a general goal and plan, reasonable arrangement of time.
1.1 Your goal is to be proficient in mathematics, to study mathematics, to earn a living in mathematics, you may aspire to master algebraic geometry, or to be proficient in frontier physics. Then you need to lay a solid foundation of modern algebra, geometry, and analysis, and you need to prepare a lot of time and energy, with unwavering determination. (Requirements: Proficient in all three levels of advanced mathematics)
1.2 Your goal is to be able to skillfully apply advanced mathematics, solve problems, and master weapons in exploring new applications, and you may aspire to enter the field of computer vision, economics, or data mining. Well, you need to lay down a solid matrix, calculus, and Probability statistics Foundation. (Requirements: Proficiency in the first level of advanced mathematics)
1.3 Your goal is to understand the fun of mathematics, to learn mathematics as a big hobby in life. So, you need to lay a solid foundation of linear algebra, mathematical analysis, topology, and probability statistics, and for you, the pleasure of learning mathematics is a more important goal. (Proficient in the first level of advanced mathematics, swim in the second level of higher mathematics, try to contact the third level of higher mathematics)

Second, give yourself enough power
It takes time and energy to learn maths and need intelligence. Here are a few facts that have been deeply felt by everyone:
1. Anything that is not used, or although useful, but you can not use the things, learn quickly forget quickly. Do not believe you remember your freshman or the basic course, you remember clearly?
2. If you are not interested (or do not feel fun), it is difficult for you to persist in accomplishing it. A lot of people have this experience, a book, the first three chapters look very carefully, behind the swallowed, the more quickly, anyway, neither boring nor useless.
3. Primary mathematics is the basis of high school mathematics, middle School mathematics is the basis of high school mathematics, high School mathematics is the basis of college mathematics (you can and so on).
So no matter what your goals are, math, math, or the fun of math, fulfilling your dreams from a teenager. Learning to have fun, learning to use, is always to maintain your motivation is not a recession of the two most important factors.

Third, what is advanced mathematics?
Well, let's take a look at the science tree of Standard University math:
Level:
Linear algebra (matrix theory), mathematical analysis, near-generation number (group ring domain), respectively, covering the basic theory of geometry, analysis and algebra. Don't forget that there is a probability theory (a basic discipline based on analysis).
Level Two:
With these foundations, followed by fundamental foundations, abstractions and generalizations: The Theory of measurement (the basis of integral, of course, the basis of probability theory), topology (an extremely important fundamental discipline about set, space, geometry), functional analysis (generalization of linear algebra), complex functions (generalization of analysis), Ordinary differential equations and partial differential equations (generalization of analysis), mathematical statistics and stochastic processes (generalization of probability theory), differential geometry (combination of analysis and geometry).
Then there are some small fresh and applied disciplines: Numerical analysis (algorithms), cryptography, graphics, information theory, time series, graph theory and so on.
Level Three:
Further up is the graduate program, often in algebra, geometry and analysis together: micro-manifold, algebraic geometry, stochastic dynamics and so on.
This science and technology tree three, and primary, middle and high school math is very similar, a layer of learning is not proficient, the next layer to see the heavenly Book.

Iv. How to learn
4.1 Right amount to do the problem
Yes don't be mad about the problem. Play the game of strategic confrontation of the students know that the low-level soldiers to build a few on the line, to save money out of the high-level soldiers in the late win, low-level soldiers not only the damage, there is no fun magic, the meaning of their existence is to let you have the ability to endure to the late. The above list of so many courses, you first spent 5 years to finish Giminovichi six math analysis problem set, you are 30 years old, the back of the two classes have not yet begun to learn. Therefore, do some after-school exercises, to help you review, think, maintain the brain to operate, to continue to study. If you do not understand, return to do exercises to help you clear the clues.
4.2 Understanding Ideas
The essence of mathematics is not the number of questions, but the mastery of ideas. Each branch of mathematics has its own main line of thought and methodology, and different branches have a way of thinking that can be compared and used for reference. Pay attention to it, imitate it, trivial knowledge strung into a necklace, you also mastered a course. Thought is not easy to read a textbook, you have to read several books, to understand some of the applications to realize. Give two examples:
There are a few main lines of calculus: Recognizing that micro and macro are connected, differentiation is used to describe how things change, it enlarges the details to you, and integrals are used to characterize the whole nature of things; differentiation and integration are sometimes different ways of describing a phenomenon, which you may not find easily in mathematical analysis books. But if we learn some physics, we will find that Maxwell's equations have both equivalent differential form and integral form, and integral transforms can establish the relation between different spaces and establish the relation between space and space boundary, which is the Stokes theorem: This formula must be in the differential manifold at the latest to get a glimpse of the whole picture.
Matrix is the abstraction of linear transformation in space, and the whole meaning of linear algebra is to study how to express, simplify and classify the linear transformation operator. SVD decomposition is not only widely used in applied disciplines, but also a powerful tool for understanding matrices. The matrix is a linear operator on the finite dimensional space, and the "space" Understanding not only allows you to re-understand the matrix, more functional analysis of learning to open a good head.
4.3 Progressive roundabout Learning, comparative learning
A lot of times, just read a book, maybe because the author in a place of thinking jumped a bit, and then you will never catch up. One of the tricks of learning math is that you also get several international well-known textbooks, contrast to each other, or read a book and then look at the same theme of the other books, already familiar with the content to jump over, if you do not understand, stop thinking or do exercises, or do not understand the back of a retreat, from the part can be read forward, When you see more, you will find that something appears in many places, the understanding of it deepened. Give two examples:
The outside differential this thing, the domestic some mathematical analysis book may not introduce, I first met is in Peng home expensive "differential geometry", think this is a handy clever tool; later read Cholich's "Mathematical Analysis" and Rudin's "Mathematical Analysis principle", all said this thing, visible in the west outside the differential is a basic knowledge. You have to read it, you may want to understand the matrix first, understand that the determinant is exactly the space volume in the transformation of the matrix under the extension of the multiple, it is a linear form. Finally, when you read the micro-manifolds, you will find that the external differential is the tool to obtain the Stokes theorem on the manifold.
Point set topology this thing, do not use the application. But if you want to learn in depth, this discipline must be mastered, as it provides precise characterization of basic concepts such as open sets, tight sets, continuity, and completeness. Learn functional analysis, micro-manifolds, without these concepts you will be unable to move. First you have to read Mt Chris's masterpiece "Topology", and then read other foreigners write books, more or less will contact some related concepts, your understanding deepened, such as reading Rudin "Functional Analysis", the beginning is to introduce the linear topological space, the knowledge you can use the front.
4.4 Establishing links between different disciplines
See a thing in a lot of places, your understanding of it deepened, slowly can appreciate the subtlety of this thing, and finally you will find that all the basic disciplines are intertwined, and in the follow-up application of mutual help, practical experience they really very basic, very useful. This is a way to experience the pleasures of mathematics.
4.5 Focus on applied disciplines
Nothing inspires your desire for new knowledge or tools more than application. Find some interesting application subject materials, read, broaden your horizons, and accumulate resources for your future. The following combine their own professional (computer vision) and hobbies to say some excellent professional books:

If you learn calculus, you can read the first volume of Feynman physics, and understand the mysteries of force, heat, light and time and space. By learning the partial differential equation, you can read the second volume of the Feynman Physics handout, and learn the mysteries of electricity; Learn the matrix theory, you can buy a "computer vision in the multi-View geometry", Understanding the mysteries of imaging, programming the three-dimensional reconstruction of image sequences; students who have learned the probability theory should have heard of the Bayesian school and the frequency school, the two schools of people brought the battlefield to the field of machine learning, the achievement of two classics "Pattern Recognition and machines Learning and the Elements of statistical learning, reading them, I was deeply impressed by the fruitful results and insights provided by basic mathematics for the field of machine learning; I read "Ray tracing from the Ground up", I wrote a ray-tracer to render the real scene, which is based on a little calculus and matrices ...
The application of advanced mathematics is too much, if you like programming, automation, robotics, computer vision, pattern recognition, data mining, graphic images, information theory and cryptography ... There are plenty of models for you to play around, and you just need a little bit of advanced math. In these areas, you may find it more interesting and easier to find a job goal than a math book.
4.6 Find interesting books to see
Mathematicians write books are sometimes more rigid, but there are always some textbooks, their authors have a strong desire to show you "this thing is actually very interesting", "this thing is not what you think of it" and so on, they succeeded, and some authors, they like to put a thing in different fields of application, And the application of different things in a certain field is concentrated to show you. This book offers you plenty of fun to read. Typical representative is a set of "Turing Mathematical Statistics series" published in China, this set of books is really great, such as "linear algebra should learn" "complex Analysis: Visualization method", "differential equations, dynamical systems and Chaos Introduction", the individual think are learning mathematics must read the classic textbook, very very interesting.

Five, read more books.
If only one sentence summarizes how to develop mathematical ability, then this is the sentence: Read more, reading good books. So I would like to take this step alone to say a few more words.
Presumably everyone is very proficient and proficient in the application of primary mathematics. If you want to read algebra geometry, or step back and want to read the basics of information theory, you need to pick a few good basic textbooks, preferably written by foreigners, and master it as you master math in elementary School. Don't just look at one, find three different authors of the book, compared to see, line by word to see. Some places must not understand, write down, perhaps in another book somewhere on this thing from another point of view.
If you're going to be back later, every basic theorem you see now will be used in the future.
Every basic book, you give up today, and tomorrow will be good to start again.
Like reading Scripture, cross-reading contrasts the similarities and differences of different textbook contents.

5.1. Recommended textbook (In fact, I have read the feeling of good books):
First Level:
"Linear algebra should be learned"
Cholich Mathematics Analysis (two volumes) (read English version of it, not difficult.) Some friends say this is not too simple, you can first read a domestic textbook, and then back to see this.
The probability theory of Fudan University

Second level:
Mount Chris "Topology"
Some fascicles in Turing's series
Kostricking on the introduction of algebra
Vapnik "The essence of statistical learning theory"
Rudin "Principles of Mathematical analysis"
Rudin "Functional Analysis"
Gamelin "Complex Analysis"
Peng Jia gui "differential geometry"
Cover "Information theory basis"
Third level:
"Differential prevalence and Riemannian geometry"
"Modern geometry, methods and applications" three volumes

5.2. Read some popular science textbooks
"What is mathematics"
"Elementary mathematics under high opinion"
"Bach, Escher, and Del."
The Story of E

5.3. Read the most interesting, lively, and most enjoyable materials and books that make you more knowledgeable, application-focused, and most readable in all fields.
"Feynman Physics Handout" three copies
Chaos and Fractals: The New Frontiers of Science
Introduction of differential equations, dynamical systems and chaos
"Complex analysis: a visual approach"

Finally, mathematics is a bottomless pit that consumes your precious youth. You may be inspirational to understand modern mathematics, but you will be deterred, and the rest of the time is not proficient in another science. And even if you are proficient in pure mathematics, it is not easy to find a job without a few good articles.
My advice is to explore in the process of reading math, pure mathematics and applied mathematics have a look, to find interest, the application of a wide range of work (to the money) in the direction of a fierce plunge down into your career. For example, the math is solid, the ability of programming is also very promising people.
Edited on 2014-07-01
Author reserves the right

Math Learning (turn to know)