# 数学之美

快速浏览完一遍李航的《统计学习方法》，总体感觉比周志华的《机器学习》易读，容易理解，不啰嗦。周的书读来有点晦涩，有空下次再读一遍《机器学习》。

数学基础很重要。本科教学方面，重大（电气）学生使用学校自编的《高等数学》《线性代数》，课堂上以传授计算技巧为主，而非引导学生思考数学问题；西交采用《一元函数微积分与无穷级数》《多元函数微积分与常微分方程》《线性代数与解析几何》，从教材题目上看，增加了“解析几何”的内容，从几何直观的角度引导学生思考代数的问题。优秀的教材是打好数学基础的必要条件。

吴军的《数学之美》不是真正的数学之美，建议去看张景中院士的《数学家的眼光》和G. Polya的《怎样解题》，体会数学之美。

Summary taken from G. Polya, “How to Solve It”, 2nd ed., Princeton University Press, 1957, ISBN 0-691-08097-6.

- UNDERSTANDING THE PROBLEM
**First**. You have to understand the problem.- What is the unknown? What are the data? What is the condition?
- Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
- Draw a figure. Introduce suitable notation.
- Separate the various parts of the condition. Can you write them down?

- DEVISING A PLAN
**Second**. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.- Have you seen it before? Or have you seen the same problem in a slightly different form?
- Do you know a related problem? Do you know a theorem that could be useful?
- Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.
- Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?
- Could you restate the problem? Could you restate it still differently? Go back to definitions.
- If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?
- Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

- CARRYING OUT THE PLAN
**Third**. Carry out your plan.- Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

- Looking Back
**Fourth**. Examine the solution obtained.- Can you check the result? Can you check the argument?
- Can you derive the solution differently? Can you see it at a glance?
- Can you use the result, or the method, for some other problem?