学堂在线：微积分-1 @陈酌
微积分基本内容: 极限论, 函数微分、积分理论, 级数, 场论, 常微分方程 ...
e.g.和i.e.是英语中两个容易混淆的拉丁语缩写:
- 前者是exempli gratia的缩写，意思是“for example” (举例)
- 后者是id est的缩写，意思是“that is” (即)
Unit 1 Review of Real Numbers (回顾实数)
§1 Real Numbers (实数体系)
术语：
- R: real numbers (实数);
- Q: rational numbers (有理数),
numbers that can be expressed as a/b,
(a over b, called the quotient (商) of a and b),
where a and b are integers and b ≠ 0, - e.g. 2/3 (two-thirds), -5/4 (minus five over four, minus five fourths);
- Z: integers (整数),
- e.g. -3, -2, -1, 0, 1, 2, ...;
- Z+, +右上标: nonnegative integers (非负整数),
- e.g. 0, 1, 2, ...;
- N: natural numbers (自然数), also called positive integers (正整数),
- e.g. 1, 2, 3, 4, ...;
- R\Q=Qc, c右上标: Irrational numbers (无理数),
numbers that cannot be expressed as a/b,
where a and b are integers and b ≠ 0, - e.g. 根号2 (square root of two), pi.
[注] Q 以外的实数，用 R backslash Q 表示 Q 的余集
§2 The R2 Plane (R2平面)
R2, 2右上标.
术语：
- x-axis (x轴): horizontal axis;
- y-axis (y 轴): vertical axis;
- (x, y): the coordinate (坐标) of a point;
- Origin (原点): the point of intersection of two axes, resented by (0, 0).
§3 Operations With Real Numbers (实数的基本运算)
If a, b, c belong to R, we have:
(a plus parentheses b plus c equals parentheses a plus b plus c)
这个叫做 单位律. ↑
§4 Inequalities (不等式)
描述两个数大小的时候, 建立不等式.
§5 Absolute Values of Real Numbers (实数的绝对值)
- |a|: the absolute value (绝对值) of a.
- Two commonly used inequalities:
(the absolute value of a plus b is always less than or equal to the absolute value of a plus the absolute value of b)
(the absolute value of the difference of the absolute value of a and the absolute value of b is always less than or equal to the absolute value of the difference of a and b)
附录:
| 运算结果 | 运算符 | 运算法 |
|---|---|---|
| Sum 和 | plus +, add 加上 | addition 加法 |
| Difference 差 | minus -, subtract 减去 | subtraction 减法 |
| Product 积 | times *, multiplied by 乘以 | multiplication 乘法 |
| Quotient 商 | over /, divided by 除以 | division 除法 |
Unit 1 Summary: 实数, 运算, 两个重要的不等式, 英文术语
Unit 2 Exponentiations, Logarithms and Sets(指数、对数与集合)
