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专业英语2.4~2,10 翻译(标准版)

专业英语2.4~2,10 翻译(标准版)
专业英语2.4~2,10 翻译(标准版)

4-A Integers and rational numbers There exist certain subsets of R which are`````and the rational numbers

翻译:有一些R的子集很著名,因为他们具有实数所不具备的特殊性质。在本节我们将讨论这样的子集,整数集和有理数集。

To introduce the positive integers····and they are called the positive integers.

翻译:我们从数字1开始介绍正整数,公理4保证了1的存在性。1+1用2表示,2+1用3表示,以此类推,由1重复累加的方式得到的数字1,2,3,…都是正的,它们被叫做正整数。Strictly speaking, this description····or “repeated addition of 1”.

翻译:严格地说,这种关于正整数的描述是不完整的,因为我们没有详细解释“等等”或者“1的重复累加”的含义。

Although the intuitive meaning····ntroduce first the notion of an inductive set.

翻译:虽然这些说法的直观意思似乎是清楚的,但是在认真处理实数系统时必须给出一个更准确的关于正整数的定义。有很多种方式来给出这个定义,一个简便的方法是先引进归纳集的概念。

DEFINITION OF AN INDUCTIVE SET···numbers which belong to every inductive set

现在我们来定义正整数,就是属于每一个归纳集的实数。

Let P denote the set of all positive integers···to P as the smallest inductive set.

用P表示所有正整数的集合。那么P本身是一个归纳集,因为其中含1,满足(a);只要包含x就包含x+1, 满足(b)。由于P中的元素属于每一个归纳集,因此P是最小的归纳集。

This property of P forms···Part 4 of this introduction

P的这种性质形成了一种推理的逻辑基础,数学家称之为归纳证明,在介绍的第四部分将给出这种方法的详细论述。

The negatives of the positive integers are called the negative integers. The positive integers, together with the negative integers and 0 (zero), form a set Z which we call simply the set of integers.

正整数的相反数被叫做负整数。正整数,负整数和零构成了一个集合Z,简称为整数集。

In a thorough treatment of the real-number system, it would be necessary at this stage to prove certain theorems about integers. For example, the sum, difference, or product of two integers is an integer, but the quotient of two integers need not to ne an integer. However, we shall not enter into the details of such proofs.

在实数系统中,为了周密性,此时有必要证明一些整数的定理。例如,两个整数的和、差和积仍是整数,但是商不一定是整数。然而还不能给出证明的细节。

Quotients of integers a/b (where b≠0) are called rational numbers. The set of rational numbers, denoted by Q, contains Z as a subset. The reader should realize that all the field axioms and the order axioms are satisfied by Q. For this reason, we say that the set of rational numbers is an ordered field. Real numbers that are not in Q are called irrational.

整数a与b的商被叫做有理数,有理数集用Q表示,Z是Q的子集。读者应该认识到Q满足所有的域公理和序公理。因此说有理数集是一个有序的域。不是有理数的实数被称为无理数。

4-B Geometric interpretation of real

numbers as points on a line

The reader is undoubtedly familiar with the geometric interpretation of real numbers by means of points on a straight line. A point is selected to represent 0 and another, to the right of 0, to represent 1, as illustrated in Figure 2-4-1. This choice determines the scale.

毫无疑问,读者都熟悉通过在直线上描点的方式表示实数的几何意义。如图2-4-1所示,选择一个点表示0,在0右边的另一个点表示1。这种做法决定了刻度。

If one adopts an appropriate set of axioms for Euclidean geometry, then each real number corresponds to exactly one point on this line and, conversely, each point on the line corresponds to one and only one real number.

如果采用欧式几何公理中一个恰当的集合,那么每一个实数刚好对应直线上的一个点,反之,直线上的每一个点也对应且只对应一个实数。

For this reason the line is often called the real line or the real axis, and it is customary to use the words real number and point interchangeably. Thus we often speak of the point x rather than the point corresponding to the real number.

为此直线通常被叫做实直线或者实轴,习惯上使用“实数”这个单词,而不是“点”。因此我们经常说点x不是指与实数对应的那个点。

This device for representing real numbers geometrically is a very worthwhile aid that helps us to discover and understand better certain properties of real numbers. However, the reader should realize that all properties of real numbers that are to be accepted as theorems must be deducible from the axioms without any references to geometry.

这种几何化的表示实数的方法是非常值得推崇的,它有助于帮助我们发现和理解实数的某些性质。然而,读者应该认识到,拟被采用作为定理的所有关于实数的性质都必须不借助于几何就能从公理推出。

This does not mean that one should not make use of geometry in studying properties of real numbers. On the contrary, the geometry often suggests the method of proof of a particular theorem, and sometimes a geometric argument is more illuminating than a purely analytic proof (one depending entirely on the axioms for the real numbers).

这并不意味着研究实数的性质时不会应用到几何。相反,几何经常会为证明一些定理提供思路,有时几何讨论比纯分析式的证明更清楚。

In this book, geometric arguments are used to a large extent to help motivate or clarity a particular discuss. Nevertheless, the proofs of all the important theorems are presented in analytic form.

在本书中,几何在很大程度上被用于激发或者阐明一些特殊的讨论。不过,所有重要定理的证明必须以分析的形式给出。

4—C Fractions and decimal fractions

分数和小数

首先,让我们把注意力转移到分数。你肯定见过表达式"一半"和"季度",分母等于 2 或 4 时使用它们。1/3 是读"三分之一"。其他分数是以相同的方式读取的。因此,我们读1/5,1/6,1/7,1/10,1/25,1/100 为五分之一,六分之一,六分之一、七分之一,十分之一,1:25 上午和百分之一。这些表达式被视为名词,因此可能产生的复数形式。因此,我们作为三分之二;读了2/3同样 5 6,9 105/100 是读取六分之五、九十分之五百分之一。但是,如果分母的最后一位是1 或2,然后我们不要读分数以上述方式。例如,我们宣布5/21 作为"五个超过21"。此方法还在其它情况下使用。如果分数不是一项共同(e.g.,1/1089年或501/1205年),然后我们说"一对千和89 个"或"五一百以上十二百和五"。

下一步,让我们检查小数。他们是很简单的发音。只读取数的整数部分普通的方式,然后说"点"(代表"小数点"),然后后另读取一个小数位。因此12.65 读取十二-点-六-五;Π正确,6 位小数,等于three-point-one-four-one-five-nine-two,正确的五个重要的数字,等于three-point-one-four-one-six。小于一个小数部分时,一般不写了,例如,0.56,但仅限于英格兰。56)。56 读"点-5-6",.0007 读"

点-团团转-团团转-团团转-七"或通常更-三-0 的七。

现在代数表达式,分数再次读取"指针经过"。(2a-1)/(ax+b) 对ax + b 读取2a-1 和括号表示"到"一词如(a+b)(a-b) 读"加进减号b b"。权力指数或指数由表示。"平方"和3"桂鱼",或"到第三个"指数的读取索引2。其它指数",第四,第五,减号第二部分,第n,以"读取。身份^3++ b ^3 = (+ b)(a^2-ab+b^2) 读取"多次元的罗宋汤的脓b 到平方的ab 加b 平方等于a + b"。或与方程x^(-2/3) + √^2 = 0 读取"x,减号的三分之二,加上第五根的平方等于零"。

6-A Informal description of functions

V arious fields of human have to do with relationships that exist between one collection of objects and another.

各行各业的人们都必须处理一类事物与另一类事物之间存在的关系。

Graphs, charts, curves, tables, formulas, and Gallup polls are familiar to everyone who reads the newspapers.

几乎每个人都熟悉图形,图表,曲线,公式和盖洛普民意测验

These are merely devices for describing special relations in a quantitative fashion. Mathematicians refer to certain types of these relations as functions.

这些只是以定量的方式描述特定关系的方法。数学家将这些关系中的某些类型视作函数。In this section, we give an informal description of the function concept. A formal definition is

given in Section 3.

在本节中,我们给出一个非正式的描述函数的概念。在第3节给出一个正式的定义。EXAMPLE 1. The force F necessary to stretch a steel spring a distance x beyond its natural length is proportional to x. That is, F=cx, where c is a number independent of x called the spring constant.

把一个钢制的弹簧拉伸到超过其自然长度的距离为x时所需要的力F与x成正比。即,F=cx,这里c是不依赖与x的数,叫做弹性系数。

This formula, discovered by Robert Hooke in the mid-17th century, is called Hooke’s law, and it is said to express the force as a function of the displacement.

这个公式是在17世纪中叶被胡克发现的,叫做胡克定律,它用来表示力关于位移的函数。EXAMPLE 2. The volume of a cube is a function of its edge-length. If the edges have length x, the volume V is given by the formula V=x3.

立方体的体积是它棱长的函数。如果棱长为x,那么体积的公式为:V=x3。

EXAMPLE 3. A prime is any integer n>1 that cannot be expressed in the form n=ab, where a and b are positive integers, both less than n. The first few primes are 2,3,5,7,11,13,17,19.

素数是大于1且不能表示成n=ab形式的整数,这里a和b都是小于n的正整数。开始的几个素数是2,3,5,7,11,13,17,19.

For a given real number x>0, it is possible to count the number of primes less than or equal to x. This number is said to be a function of x even though no simple algebraic formula is known for computing it (without counting) when x is known.

对于一个给定的实数x>0,数出小于或者等于x的素数的个数是有可能的。这个数称为x的函数,尽管还没有一个简单代数式可以由已知的x计算(不通过计数求)出它的值。

The word “function” was introduced into mathematics by Leibniz, who used the term primarily to refer to certain kinds of mathematical formulas.

“函数”这个词是由莱布尼茨引入到数学中的,他主要使用这个术语来指代某种数学公式。It was later realized that Leibniz’s idea of function was much too limited in its scope, and the meaning of the word has since undergone many stages of generalization.

后来人们才认识到,莱布尼茨的函数思想适用的范围太过局限了,这个术语的含义从那时起已经过了多次推广。

Today, the meaning of function is essentially this: Given two sets, say X and Y, a function is a correspondence which associates with each element of X one and one only element of Y.

如今,从本质上讲,函数的定义如下:给定两个集合X 和Y,函数是X中元素与Y中元素的一一对应。

The set X is called the domain of the function. Those elements of Y associated with the elements in X form a set called the range of the function. (This may be all of Y, but it need not be)

集合X叫做函数的定义域,与X中的元素相对应的Y中的元素的集合叫做函数的值域。(值域可能是整个集合Y,也可能不是。)

Letters of the English and Greek alphabets are often used to denote functions. The particular letters f,g,h,F,G,H, and are frequently used for this purpose.

英语字母和希腊字母表通常用于表示函数。为此,一些特定的字母如:f,g,h,…频繁使用。If f is a given function and if x is an object of its domain, the notation f(x) is used to designate that object in the range which is associated to x by the function f; and it is called the value of f at x or the image of x under f. The symbol f(x) is read as “f of x.”

如果f是一个给定的函数,x是它定义域中的一个点,符号f(x)表示值域中按照f对应于x

的点,它叫做f在x点的值或者x在f下的像。符号f(x)读作“f of x.”

6-C The concept of function

Seldom has a single concept played so important a role in mathematics as has the concept of function. It is desirable to know how the concept has developed.

在数学中,很少有个概念象函数的概念那样,起那么重要的作用的。因此,需要知道这个概念是如何发展起来的。

This concept, like many others, originates in physics.

这个概念像许多其他概念一样,起源于物理学。

The physical quantities were the forerunners of mathematical variables, and relation among them was called a function relation in the later 16th century.

物理量是数学变量的先驱,他们之间的关系在16世纪后期称为函数关系。

For example, the formula s=16t2 for the number of feet s a body falls in any number of seconds t is a function relation between s and t, it describes the way s varies with t.

例如, 代表一物体在若干秒t中下落若干英尺s的公式s=16t2 就是s和t之间的函数关系, 它描述了s随t 变化的公式。

The study of such relations led people in the 18th century to think of a function relation as nothing but a formula.

对这种关系的研究导致了18世纪的人们认为函数关系只不过是一个公式罢了。

Not specified by this definition is the manner of setting up the correspondence.

至于如何建立这种对应关系,这个定义并未详细规定。

It may be done by a formula as the 18th century mathematics presumed but it can equally well be done by a tabulation such as a statistical chart, or by some other form of description.

可以如18世纪的数学所假定的那样,用公式来建立,但同样也可以用统计表那样的表格或用某种其他的描述方式来建立。

A typical example is the room temperature, which obviously is a function of time. But this function admits of no formula representation, although it can be recorded in a tabular form or traced out graphically by an automatic device.

典型的例子是室温,这显然是一个时间的函数。但是这个函数不能用公式来表示,不过可以用表格的形式来记录或者用一种自动装置以图标形式来追踪。

The modern definition of a function y of x is simply a mapping from a space X to another space Y.

A mapping is defined when every point x of X has a definite image y, a point of Y.

现代给x的一个函数y所下的定义只是从一个空间X到另一个空间Y的映射。当X空间的每一个点x有一个确定的像点y,即Y空间的一点,那么映射就确定了。

The mapping concept is close to intuition, and therefore desirable to serve as a basis of the function concept. Moreover, as the space concept is incorporated in this modern definition, its generality contributes much to the generality of the function concept.

这个映射概念接近于直观,因此,值得作为函数概念的一个基础。此外,由于这个现代的定义中体现了空间的概念,所以,它的普及性对函数概念的普及性有很大的贡献。

8-A函数的导数

前一节中描述的例子给出了引进导数概念的方法。我们从至少定义在x-轴上的某个开区间(a,b)内的函数f(x)开始,然后我们在这个区间内选择一点x,引进差商

这里,数h (可以是正的或者负的但不能是0)要使得x+h还在(a, b)内。这个商的分子测量了当x从x变到x+h时函数的变化。称这个商为f在连接x与x+h的区间内的平均变化率。现在让h→0,看看这个商会发生什么。如果商趋于某个确定的值作为极限(这就推得无论h 是从正的方向还是负的方向趋于0,这个极限是一样的),成这个极限为f在x点的导数,记为f / (x)(读作“f一撇x”)。因此,f / (x)的正规定义可以陈述如下:

导数定义。如果

存在极限,导数f / (x)由等式(8.2)定义。数f / (x)也称为f在x点的变化率。

对比(8.2)与前一节的(7.3),我们看到瞬时速度仅仅是导数概念的一个例子。速度v(t)等于f / (t),这里f是位移函数,这就是常常被描述为速度是位移关于时间的变化率。在7.2节算出的例子中,位移函数由等式f (t)=144t-32t2表示,而它的导数f / 是由 f / (t) =144-32t给出的新的函数(速度)。

一般地,从f(x)产生f / (x)的极限过程给我们从一个给定函数f获得一个新函数f / 的方法。这个过程称为微分法,f / 称为f的一阶导数。依次地,如果f / 定义在开区间上,我们可以设法求出它的一阶导数,记为 f // 并称其为f的二阶导数。类似地,由f (n-1)定义的一阶导数是f的n阶导数记为f (n),我们规定f (0)= f,即零阶导数是函数本身。

对于直线运动,速度的一阶导数(位移的二阶导数)称为加速度。例如,要计算7.2节中的例子的加速度,我们可以用等式(7.2)形成差商

因为这个差商对每一个h≠0都是常数值-32,因此当h→0时它的极限也是-32.于是在这个问题中,加速度是常数且等于-32. 这个结论告诉我们速度是以每秒32尺/秒的速率递减的。9秒内,速度总共减少了9·32=288尺/秒。这与运动9秒期间,速度从v(0)=144变到v(9)=-144是一致的。

8-B 导数作为斜率的几何意义

通常定义导数的过程给出了一个几何意义,就是以自然的方式导出关于曲线的切线的思想。图2-8-1是一个函数的部分图像。两个坐标(x,f(x)) 和(x+h,f(x+h))分别表示P, Q两个点坐标,考虑斜边为PQ的直角三角形,它的高度:f(x+h)- f(x),表示P, Q两个点纵坐标的差,因此差商

表示PQ与水平线的夹角α的正切,实数tanα 称为通过P, Q两点直线的斜率,而它提供了一种测量这条直线“陡度”的方法。例如,如果f是线性函数,记为f=mx+b,则(8.4)的差商是m, 所以m是这条直线的斜率。图2-8-2表示的是一些各种斜率的直线的例子。对于水平线而言,α=0,因而tanα也是0. 如果α位于0与π/2之间, 直线是从左到右上升的,斜率是正的。如果α位于π/2与π之间,直线是从左到右下降的,斜率是负的。对于α=π/4的直线,斜率是1. 当α从0增加到π /2时,tanα递增且无界,斜率为tanα相应的直线趋于垂直的位置,因为tanπ/2没有定义,所以我们说垂直的直线没有斜率。

假设f在x点有导数,这就意味着,当h→0时,P点保持不动,Q沿曲线向P移动,通过P, Q两点直线不断改变方向,结果其斜率趋于极限f / (x)。基于这个原因,将曲线在点P的斜率定义为数f / (x)似乎是自然的。通过P点具有这个斜率的直线称为过点P的切线。

2.10—C篇

In recent years the applications of matrices in mathematics and in many diverse fields have increased with remarkable speed. Matrix theory plays a central role in modern physics in the study of quantum mechanics.

近年来,在数学和许多各种不同的领域中,矩阵的应用一直以惊人的速度不断增加。在研究量子力学时,矩阵理论在现代物理学上起着主要的作用。

Matrix methods are used to solve problems in applied differential equations, specifically, in the area of aerodynamics, stress and structure analysis. One of the most powerful mathematical methods for psychological studies is factor analysis, a subject that makes wide use of matrix methods.

解决应用微分方程,特别是在空气动力学,应力和结构分析中的问题,要用矩阵方法。心理学研究上一种最强有力的数学方法是因子分析,这也广泛的使用矩阵(方)法.

Recent developments in mathematical economics and in problems of business administration have led to extensive use of matrix methods. The biological sciences, and in particular genetics, use matrix techniques to good advantage.

近年来,在数量经济学和企业管理问题方面的发展已经导致广泛的使用矩阵法。生物科学,特别在遗传学方面,用矩阵的技术很有成效。

No matter what the students’ field of major interest is , knowledge of the rudiments of matrices is likely to broaden the range of literature that he can read with understanding .

不管学生主要兴趣是什么,矩阵基本原理的知识都可能扩大他能读懂的文献的范围。

The solution of n simultaneous linear equations in n unknowns is one of the important problems of applied mathematics.

解一有n个未知数的n个联立(线性)方程组是应用数学的一个重要问题。

Descartes, the inventor of analytic geometry and one of the founders of modern algebraic notation, believed that all problems could ultimately be reduced to the solution of a set of simultaneous linear equations.

解析几何的发明者和现代代数计数法的创始人之一笛卡儿相信,所有的问题最后都能简化为解一组联立方程。

Although this belief is now known to be untenable , we know that a large group of significant applied problems from many different disciplines are reducible to such equations.

虽然这种信念现在认为是站不住脚的,但是,我们知道,从许多不同的学科里的一大群重要的应用问题都可以约简为这类的方程。

Many of the applications, require the solution of a large number of simultaneous linear equations, sometimes in the hundreds . The advent of computers has made the matrix methods effective in the solution of these formidable problems.

许多应用要求解大量的,往往数以百计的联立方程,计算机的发明已经使得矩阵方法在解这些难以解决的问题方面非常活跃。

From the above discussion, we see that the problem of solving n simultaneous linear equation in n unknowns is reduced to the problem of finding the inverse of the matrix of coefficients. (P89 下数第9行)

从上面的讨论,我们看到解有n个未知数的n个联立方程问题简化成求系数矩阵的逆矩阵的问题。

It is therefore not surprising that in books on the theory of matrices the techniques of finding inverse matrices occupy considerable space.

因此,在矩阵论的书中,用大量的篇幅来讲求逆矩阵的技巧就不奇怪了。

Of course , we will not in our limited treatment discuss such techniques.

当然,我们在这有限的叙述中不会讨论这类的技巧。

※Not only are matrix methods useful in solving simultaneous equations , but they are also useful in discovering whether or not the set of equations are consistent, in the sense that they lead to solutions, and in discovering whether or not the set of equations are determinate, in the sense that they lead to unique solution.

矩阵方法不仅在解联立方程中有用,而且在发现方程组是否相容,即方程组是否有解的问题,以及方程组是否是确定的,即是否有惟一解等方面,都是有用的。

电气工程及其自动化专业英语第二章课文翻译

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专业英语翻译

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