重学微积分

• Chapter One: A Preview
• 1-1 Introduction
• 1-2 Summation - 求和
• 1-3 Functions - 函数
• 1-4 Composite Functions - 复合函数
• 1-5 Variables and Loci - 变量和位点
• 1-6 Integrals - 积分
• 1-7 Derivatives - 微分
• 1-8 The Fundamental Theorem - 基础定理（牛顿-莱布尼茨公式）
• 1-9 Calculus of Variables - 变量的微积分

Chapter One: A Preview

因为本人英语不是很好，理解上难免出现错误，欢迎各位在评论处留言。

1-1 Introduction

The eighteenth century is frequently thought of as the golden age of formal development in calculus, while the nineteenth century is regarded as the most important era of basic development. It should be noted, however, that the twentieth century has seen a significant basic development in calculus.

1-2 Summation - 求和

$sum_{i=j}^{n}a_i$

符号	含义
$sum$	求和符号 It is called a summation sign.
$i$	求和索引（我们都叫下标） It is called the summation index.
$j$ and $n$	上下限 The lower and upper limits of summation, respectively.
-	求和符号的意思是把从下限到上限的连续整数赋给$i$。 The summation sign means: Assign to the summation index successive integer values from the lower to the upper limit of summation, inclusive.

1-3 Functions - 函数

函数三要素：定义域，值域，函数法则（陈老师强调过）。
$f(x) = y$

术语	含义
function	函数被定义为一个有序对的集合，这些有序对的第一项各不相同，也就是$x$是唯一的，不存在$x_1=x_2$，一个有序对可表示为$(x, y)$。 A function is defined as a set of ordered pairs of numbers no two of which have the same first entry. The notation $(x, y)$ is commonly used for the ordered pair whose first entry is $x$ and whose second entry is $y$.
domain	函数的定义域为$x$的集合。 The domain of a function is the set of all first entries in its ordered pairs.
range	函数的取值范围，简称值域，是$y$的集合。 The range of a function is the set of all its second entries.
entry	查到的中文解释对应不上，我个人理解的是“项”：第一项，第二项。
identity function	恒等函数
polynomial_function	多项式函数

Frequently, a single letter is used to denote a particular function. The ones in most common use are $f, g, F, G, phi, Phi$. If $f$ is a function and $a$ is a number in its domain, then the symbol $f(a)$ is used to denote the entry in the range corresponding to a. The symbol $f(a)$ is read, $f$ of $a$, and is called the value of $f$ at $a$.
And, given $a$, the operation of getting $f(a)$ is called application of $f$ to $a$.

例：f（a,b）, g(a,c)两个函数的计算。
$egin{array}{cccc} f & g & f+g & fg \ hline ext{-5 -1} & ext{-5 -1} & ext{-5 -2}& ext{-5 -2}\ ext{-1 4} & ext{-1 3} & ext{-1 -2}& ext{-1 -2}\ ext{ 0 5} & ext{ 0 -2} & ext{ 0 -2} & ext{ 0 -2}\ ext{ 2 -1}\ ext{ 3 -2} & ext{ 3 0} & ext{ 3 -2} & ext{ 3 0}\ & ext{ 4 4} end{array}$
注意：两个函数相除时，如果除数的值域中包含0，则不能进行运算。

Unless otherwise specified, the domain of a function will be the set of all real numbers for which the formula for function values yields real numbers.

1-4 Composite Functions - 复合函数

$f circ g,$ if $f(a)=b, and, g(b)=c,$ then, $a o b o c.$

术语	含义
composite_function	If $f$ and $g$ are functions such that the domain of $f$ overlaps the range of $g$, then the composite function ($f circ g$) read $f$ circle $g$, is defined as the set of all ordered pairs $(a, c)$ such that for some $b$, $(a,b)$ is in $g$ and $(b,c)$ is in $f$.

1-5 Variables and Loci - 变量和位点

（我的英文差，也不知道翻译得对不对，嫌弃的话，直接看英文吧?）

映射就是一个有序对集合，且集合中不存在$x_1=x_2$。定义域和值域与之前函数的定义相同。
如果两个映射有重合的定义域，且值域为数集，那么他们的四则运算规则和函数相同。

A mapping is defined as a set of ordered pairs no two of which have the same first entry. The words “domain” and “range” apply to mappings in general as to the special case of functions.
If two mappings have overlapping domains and if their ranges are sets of numbers, then their sum, difference, product and quotient are defined as in the case of functions.

术语	含义
sum	和
difference	差
product	乘积
quotient	商
origin	原点 the point of intersection of the coordinate axes.
abscissa	横坐标 the horizontal measurement.
ordinate	纵坐标 the vertical measurement.
coordinates	坐标，包括横坐标和纵坐标。
abscissa variable	横坐标变量
ordinate variable	纵坐标变量

1-6 Integrals - 积分

$A=lim_{n o infty}sum_{i=1}^n f(c_i)(ci-c_{i-1})$ 上图中，当矩形宽度无穷小时，所有矩形面积之和$A$会无限接近于包围矩形，且上下限为$(a,b)$的曲线与横轴围成的面积，即:$int_a^bf=lim_{n o infty}sum_{i=1}^n f(c_i)(ci-c_{i-1})，$it called the integral of $a$ from a to $b$, and the symbol $int$ is called an integral sign. The numbers $a$ and $b$ appended tp the integral sign are called the ==limits of integration.==This terminology is unfortunate, but standard.

1-7 Derivatives - 微分

$f'(a)=lim_{h o0}frac{f(a+h)-f(a)}{h}$
The number $f'(a)$is called the derivative of $f$ at $a$. It is also referred to as the slope of the curve at $a$.

The set of all number pairs $[a,f'(a)]$ is a function $f'$called merely the derivative of f.
The operation of finding $f'$ from f is called differentiation.

In general, the $n$-th derivative of $f$, designated by $f^{(n)}$, is the function obtained by starting with $f$ and applying the operation of differentiation $n$ times. Parentheses are used in the notation $f^{(n)}$ to distinguish it from $f^n$, which means $n$-th power of $f$.

1-8 The Fundamental Theorem - 基础定理（牛顿-莱布尼茨公式）

method of exhaustions - 穷举法(这个是经典希腊的算法不是基础定理，就记一下穷举法怎么说。)
下面的牛顿-莱布尼茨公式（Newton and Leibnitz）才是基础定理。
当曲线穿过矩形的$(ar x$, $f(ar x)$)时，存在：$frac{A(x_0+h)-A(x_0)}{h}=frac{hf(ar x)}{h}=f(ar x)$因为，$A(x_0+h)-A(x_0)$表示图a的阴影面积，而观察图b可以发现如果曲线的位置适当，图a的阴影区域面积近似于图b的矩形面积。进而有：$A'(x_0)=lim_{h o 0}frac{A(x_0+h)-A(x_0)}{h}=lim_{ar x o x_0}f(ar x)=f(x_0)$常见的变换：
$int x^n=frac{x^{n-1}}{n}$$int_a^bcf=cint_a^bf quad quad; quad int_a^b(f+g)=int_a^bf+int_a^bg$$int_a^bf+int_b^cf=int_a^cfquad quad; quad int_a^bf=F|_a^b=F(b)-F(a)$

1-9 Calculus of Variables - 变量的微积分

The derivative of $v$ with respect to $u$ is denoted by $D_uv$ and defined by
$D_uv=f'(u)$上式为变量$v$对变量$u$的微分。
The $n$-th derivative of $v$ with respect to $u$ is then $D_u^nv$
and $int_a^bf(x)dx$is called the integral from $a$ to $b$ of $f(x)$ with respect to $x$.