《数学专业英语》章节试读

《数学专业英语》的笔记-第197页

(195---197页)1，The subject of ...is concerned with...
2，In subsequent chapters we shall...
3，...and often intuition and experience will play an important role.
4，For the present we observe that ...
5，...in the sense that...
6，for the purpose in hand
7，an improvement in accuracy can be obtained by ...
8，....correct to some specified tolerance
9，due attention must be paid to ...
10，savings achivable 可达节省
11，high degree polynomials 高阶多项式
12，...is a large area of study. The subject includes components in the areas of ...

《数学专业英语》的笔记-第78页

（74----78页）1，The study of differential equations.....
2，....so their study is of practical importance
3，...near the end of this chapter.
4，mathematical theories of much generality
5，A more thorough study of ....is carried out in Volume ..
6，it is customary to write...
7，By.....is meant....
8，we shall begin our study of ...with....
9，the expression ...has various special forms
10，where ...is assumed to be ....
11，the ...equation has infinitely many solutions,...
12，In actual practice,...
13，to fix the position function
14，In some respects,...
15，this terminology originated in mechanics
16，a large variety of ...

《数学专业英语》的笔记-第17页

（12---17页）1，Equations are of two kinds----。。
2，The formulas differ only in the signs preceding their left numbers.
3，词的添加的一个重要理由是需要重复译出。
4，After checking 经检验，
5，An equilateral triangle is one with all its sides equal.
6，signs and symbols of mathematiics
7，The statement of the Gauss Theorem is as follows. (高斯定理叙述如下。)
8，We call a triangle an abtuse triangle when one angle is an obtuse angle.
9，无人称语句“It....”可照样译成无人称句，必要时也可以译成人称句。
10，It should be noted that 应当指出
11，It was found that 人们发现
12，increase (be, go up) by three times 增加3倍
increase three times 增加2倍
13，decrease by 60% 降低了60%
decrease to 40% 降到了原来的40%
14，shortened three times 缩短到(原来的)1/3

《数学专业英语》的笔记-第154页

(151---154页)1，.....be expressed in the form ...
2，...when one seeks ...
3，..a given integer m, say 359
4，testing 359 successively for divisibility by 2,3,5,....
5，On this basis we may assert that ...
6，several theorems regarding primes
7，two positive integers each less than ....
8，By repeated application of this process, we have ...
9，strike out 排除

《数学专业英语》的笔记-第83页

（79----83页）1，It is easy to check that....
2，the solution of...were exhibited as...
3，without any concern regarding the meaning and validity of such formulas.
4，...by present-day standards.
5，....independently and almost simultaneously proposed...
6，....real numbers endowed with certain special properties.
7，From the definition of...., it is possible to deduce ....
8，By use of ....
9，the same strict inequality holds for...
10，but the proof is not easy to give at this stage.
11，the foregoing definition agrees with that given in Chapter 8
12，This is logically equivalent to the statement that...
13，To prove this, it suffices to show that...
14，The results holds trivially ,,,,
15，the subspace spanned by S
16，reduces to....

《数学专业英语》的笔记-第139页

(137----139页)1，equivalence relations
2，....trivially satisfies .....
3，....separates the elements of S into classes,
4，...grouping together objects which agree in some particular respect.
5，.., as claimed
6，a partiton of S
7，every partition on a set S arises in this way form a ...
8，....is partitioned into sets ....
9，....gives rise to...

《数学专业英语》的笔记-第163页

(161----163页)1，consideration of ..... resulted in the development of ....
2，....no point lies on both of them
3，Notice first that....Notice secondly....
4，because of our convention that ...
5，unless stated otherwise
6，the distance between the two lines is everywhere the same
7，We are not committing ourselves either way and shall reserve judgment until we study the matter further.
8，be conditioned to think in ....terms
9，...go on extending them forever
10，intersects both L and m in distinct points
11，The trouble with ... is that ...
12，circular reasoning
13，the questionable nature of...
14，For all vectors .... , we have the following properties

《数学专业英语》的笔记-第95页

（91-95页）1，a propositional function
2，the symbol for negation
3，the universe of discourse consists of...
4，Many mathematical statements assert that....
5，there is an element with a certain property
6，be phrased in term of sth
7，specify values 简称“赋值”

《数学专业英语》的笔记-第53页

（51---53页）1，....find the equation of ....
2，a compact mathematical notation
3，"the graph of the set of ordered pairs"
4，.....will readily infer that...
5，..geometrical figure
6，one may be tempted to believe that...
7，descibe.....in a quantitative fashion
8，If the edges have length x,....
9，.....less than or equal to......
10，....too limited in its scope
11，many stages of genearlziation
12，This may be ....., but it need not be.

《数学专业英语》的笔记-第157页

(155---157页)1，Our purpose is threefold: to understand..., to see..., and to introduce....
2，...be primarily concerned with...
3，Similar definitions hold for .... with the exception that ....
4，...has no degree / has degree zero / ... is of degree two
5，...an algebraic function that arises when .....

《数学专业英语》的笔记-第147页

(146---147页)1，....write Z for the set of ...
2，The set Z admits three operations:...
3，undoes the effect of ...
4，we shall .... once we come to consider....
5，in Ch. 6 we shall...
6，It follows from ....that...
7，independent of 和....无关
8，A similar remark applies to multiplication.
9，by convention
10，...to make our formulae more perspicuous as well as more accurate.
11，removes all doubt
12，with our convention about empty sums

《数学专业英语》的笔记-第130页

(127----130页)1，....was conjectured by ....
2，....has so far resisted all attempts to prove or disprove it.
3，the proofs become more transparent..
4，in everyday affairs
5，the patent absurdity of our conclusion
6，a faulty line of reasoning
7，an abstract line of thoguht
8，by enumerating further possibilities
9，For our purposes we may assume that...
10，.....is not really envisaged,....
11，......is always taken to mean.....
12，With every statement we can associate its opposite or negation by inserting "not" in the appropriate place.
13，a mathematical proof might contain the line....
14，a parallel use in everyday English ....

《数学专业英语》的笔记-第27页

（25---27页）1，The ratio of one quantity to anther like quantity。。
2，。。all the rules governing a fraction apply to ratio
3，the ratio 3 to 4
4，The 3 and 4 are called the terms of the ratio.
5，a ratio is a quotient of like quantities
6，quantities of the same kinds
7，Unless there is an important reason to the contrary,。。
8，the final ratio of width to length is。。
9，parenthesis = round bracket
10，Why study geometry?
11，The Greek word geomerty is derived from geo, meaning "earth", and metron, meaning "measure".
12，by utilizing the truths of geometry
13，Solids and planes
14，Common examples of solids are。。
15，。。are called ....or simply 。。
16，。。is termed a line segment
17，When no confusion may result,。。

《数学专业英语》的笔记-第105页

（100----105页）1，an event is certain to occur.
2，the example of coin tossing
3，the experiment is performed
4，events with probability zero
5，In the preceding sections,
6，...are readily discernible
7，Nature does not follow such a rigid law.
8，from a practical standpoint
9，under collision
10，elucidate these concepts
11，record of some qualitative trait
12，make inference
13，the addition of probabilities
14，the rule for...is..

《数学专业英语》的笔记-第45页

（44---45页）1，Since the natural numbers do not terminate or repeat。。。
2，Let us first turn our attention to fractios,
3，Now for algebraical expressions,。。

《数学专业英语》的笔记-第33页

（28---33页）1，。。joined, straight line segments,。。
2，。。any line segment joining two points on the circle.
3，From the definition it should be apparent that 。。
4，A diameter divides a circle into two arcs termed semicircles,。。
5，vanish 消失，变成零
6，。。a fixed ray l having endpoint O,
7，These functions are of fundamental importance in the study of trigonometry.
8，Let us now take up the solution of right triangles.
9，To solve a triangle is to 。。

《数学专业英语》的笔记-第121页

（117----121页）1，much progress has been made in the field of....
2，Many generalizations have been introduced.
3，To justify that....
4，make no claim to completeness
5，the vastness of the subject
6，To facilitate the reading of ....
7，any reference to a book
8，have given him both moral and material support ....
9，Accompanying this text is ....
10，In what follow,...
11，Our suggestions follow.
12，we acquaint the student with,,,,
13，Towards this end,...
14，change of variables
15，...offers a brief excursion into....
16，skip this chapter
17，in this regard,...
18，there is no need to...
19，they are good practice on the ideas introduced earlier.
20，for compparison purposes
21，Motivation for this material is provided by....
22，sectionally continuous functions 分段连续函数

《数学专业英语》的笔记-第37页

（33---37页）1，Notations for denoting sets
2，elements are designated by 。。
3，When convenient, we shall designate sets by。。
4，Two sets A and B are said to be equal。。
5，According to this definition, 。。
6，the definition requires that we call these sets equal.
7，。。we may form new sets,
8，all even integers less than 10.
9，A is contained in B or that B contains A
10，set inclusion
11，The statemet 。。does not rule out 。。
12，be designated as。。
13，as analogous to 。。
14，conceptually distinct from。。
15，be thought of as 。。
16，the shaded portion
17，。。of this type
18，the underlying set
19，"talk of" 原意是谈论，这里转译为“表述”

《数学专业英语》的笔记-第187页

(184---187页)1，the most important and vibrant fields
2，...has many practical applications in various disciplines
3，to name a few
4，As will become evident after...
5，what is now known as ...
6，a round trip through...
7，traversing each bridge exactly once
8，...distilled the problem into its simplest form
9，This name is bestowed because...
10，performing the ... algorithm on the new problem
11，a minimum problem
12，..be computationally most efficient to ...
13，the two-variable problem

《数学专业英语》的笔记-第142页

(140----142页)1，draw a figure
2，an immediate connection
3，restate it still differently
4，the proposed problem
5，concern yourself with details
6，the principal parts of a "problem to prove"
7，a "problem to find"
8，consider your problem from various sides
9，formerly acquired knowledge
10，the whole of the way or a part of the way
11，seek contacts with your formerly acquired knowledge
12，be led astray
13，....that may be wanting

《数学专业英语》的笔记-第160页

（158---160页）1，By a monoid we understand a set....
2，By a group one understands a monoid...
3，The commutative law is not assumed
4，.....is defined by the following four conditions
5，The notions of .... defined for monoids carry over to groups as a special case.
6，it will cause no confusion to...
7，...is taken to mean ....
8，monoid 幺半群

《数学专业英语》的笔记-第116页

（112---116）1，a grounding in calculus
2，it has been hitherto presented
3，the uninitiated
4，a silod foundation in...
5，tedious calculations
6，Thus the absence of a reference does not imply originality on our part.
7，This book has its roots in...
8，Lively and illuminating discussions ensued.

《数学专业英语》的笔记-第24页

(22---24页)1，the proof of most of geometrical theorems
2，equation 方程（常用）；等式（少用）//
equality 等式（少用）；相等（常用）
3，alike 相同的或一样的
4，。。or simply an equation 或简称方程
5，to move and change the terms 移项和变形
6，The study of arithmetic begings with。。
7，。。instead of special cases, we often consider general statements
8，。。long or complicated statements
9，。。generalizing many specific statements
10，grouping symbols
11，do the addition, do the multiplication

《数学专业英语》的笔记-第150页

(148----150页)1，Then by ... it will be enough to prove...
2，To prove....we shall....
3，for an instance of this see Lemma ....
4，...this does not affect the validity
5，...that they divide, or are divisable by...
6，facilitate our study

《数学专业英语》的笔记-第179页

(176---179页)1，..this simple spring-mass system exhibits behavior of more complex systems.
2，aid in the understanding of ..
3，up-and-down motion of the ocean surface
4，...the now famous formula...
5，..all forces applied to a point mass..
6，(often referred to as just ..)
7，We emphasize the word approximation, for although mathematics is frequently treated as a science of exactness, mathematics is applied to models which only approximate the real world.
8，the forces of action and reaction are equal and opposite
9，...occur widely
10，..that the equation purports to model
11，ease of understanding

《数学专业英语》的笔记-第99页

（96----99页）1，The following are...
2，the compound proposition ... and its converse...
3，all choices of truth values of the variables.....
4，the final columns of...
5，where the truth values of P and Q agree
6，to reason with these senteces (reasoning with logical expressions)
7，.....pertains to all people.
8，this is more compact
9，rule of inference
10，in equal measure

《数学专业英语》的笔记-第73页

（70----73页）1，we conclude that...
2，Thus the line segment ....has the direction....
3，In the notation of logarithms,...
4，It is important to realize....
5，it follows that....
6，computation by means of logarithms
7，...is used mainly for theoretical purposes.
8，in the counterclockwise sense
9，in the form of an equation
10，These equations are called differential equations, and their study forms one of the most challenging branches of mathematics.
11，....are classified under two main headings: ......

《数学专业英语》的笔记-第133页

(131----133页)1，too restrictive to account for....
2，lead to confusion
3，With each newborn baby associate its weight in grams to the nearest gram.
4，n-fold product
5，On closer examination we see that .....
6，.....is just an expression of the fact that ...
7，...., then we can compose them to get....
8，A graphic way of expressing this equation is ....
9，in the accompanying diagram
10，...we express this by saying that ...
11，As a rule....
12，Further we note that.....
13，As an example, ....
14，let ..... be given by....
15，the equation (3) holds

《数学专业英语》的笔记-第175页

(171----175页)1，...of degree greater than two
2，Using such arguments, it can be shown that ...
3，The next two sections contain further details on the graphing of higher plane curves.
4，But letting z=a gives ...
5，Continuing we get...
6，on the threshold of proving ...
7，So the question arises:...
8，...expended a great deal of effort

《数学专业英语》的笔记-第194页

(192--194页)1，a best or optimal solution to..
2，This point will be amplified in ...
3，outside of those values
4，the net force acting on a body
5，Be careful about...
6，we can not extrapolate broad generalizations from the particular evidence we gather about our model
7，We summarize the steps for ... in Figure..
8，the problem under study
9，amplify 详细陈述

《数学专业英语》的笔记-第12页

（8---12页）
A takes its value 3.
matched them against their fingers
To be more accurate
长句翻译：（总-分-合）
1，先抓主句，了解中心大意
2，次抓从句，理清主从关系
3，逐步细化，分析破解疑难
4，全局综合处理，完善理解与表达
5，校对和改善

《数学专业英语》的笔记-第50页

（46---50页）1，As mentioned earlier,...
2，.....we do not talk about area by itself, instead, we talk about the area of something.
3，....the subject of analytic geometry
4，Figure 2-5 illustrates some examples.
5，The point with coordinates (3,2) lies........
6，...we agree that....
7，Points (a,b) with both a and b positive.....
8，with appropriate regard for signs,..
9，We shall discuss ....in more detail later on; for the present we confine our attention to.....
10，For example, consider a circle of radius r with its center at the origin......
11，and suppose P has coordinates (x,y)
12，a right triangle whose legs have lengths .....
13，and hence,......
14，......be intemately intertwined.
15，as needed.
16，elementary concepts of plane analytic geometry
17，the rudiments of calculus
18，drawing graph of function

《数学专业英语》的笔记-第43页

（38---43页）1，。。listing each of elements would be tedious.
2，。。would be impossible because they go on forever.
3，。。reduce many mathematical concepts to their logical foundations
4，outside the scope of this book
5，a single entity
6，belong to or to be contained in the set
7，contain or be composed of its elements
8，..and from this generality the theory derive its power.
9，reduce to 化简为，归结为
10，Strictly speaking,..
11，in a careful treatment of。。
12，we shall not enter into the details of such proofs
13，。。all the field axioms and the order axioms are satisfied by Q
14，to the left / right of
15，as illustrated in Figrue ..
16，it is customary to 。。
17，the ordering relation
18，the point x lies to the left of ..
19，without any reference to geometry
20，geometric arguments
21，..are presented in analytic form
22，这里 expressions 可译成 表述 说法

《数学专业英语》的笔记-第126页

(124---126页)1，It may happen that...
2，the set with no members
3，In this situation....
4，We also note....
5，a certain analogy with....
6，not to impose this restriction
7，taking A, B, C to be the set of ....
8，n-tuple n元组，n重的

《数学专业英语》的笔记-第5页

半专业术语
functional在数学中常常不表示“函数的”，而作“泛函”（名词）解

《数学专业英语》的笔记-第170页

(167----170页)1，....equation in two variables has the form...
2，a limiting case of ...
3，the distance ...increases without bound
4，by passing the plane ... through P
5，the equation ...takes on the form...
6，Thus the real graph (when such exists)
7，a very readable treatment of ...
8，Graph the following conic sections
>

《数学专业英语》的笔记-第8页

(6---8页)
1，The function 。。approaches infinity as x tends to zero.(tends to, approaches趋于)
2，Suppose D is an open set with its closure in G.（with）
3，分词“satisfying”可代替“such that”
4，(suppose, let, set, assume)的命令式表示假定、设（大前提）
5，since, as, for表原因，since从句最通用；because频率很低，只在非常强调因果关系时才用。
6，by 短语，有时也用according to表示推理的根据
7，用现在分词表示“经过。。。而得到。。”（推论）
Integrating the above inequality twice, we see that 。。
（将上一不等式两次积分得到。。）
8，if and only if 可用 iff 代替，但应事先作出说明。
9，such that 。。* whenever。。** (使得只要。。** 就有。。*)
10，For any number。。可改用 Given。。
11，for all x with 。。** （对所有满足。。** 的x）
12，逻辑符号的使用

《数学专业英语》的笔记-第3页

we have // one has
名次化结构The rotation of the line on x-axis forms a conicoid.
Now we investigate the integrability of the functions. (现在，我们来研究....)

《数学专业英语》的笔记-第166页

(164----166页)1，...are easy consequences of the definition
2，To carry out this proof
3，....has important applications to .....
4，....is a consequence of the Cauchy-Schwarz inequality
5，The converse statement is also true.

《数学专业英语》的笔记-第69页

（64---69页）1，A quantity that can take on ..... values
2，....these values need not have any definite relations one to another.
3，As stated before, ....
4，....formula ..is valid
5，all we need to prove is that...
6，as illustrated in Fig. 2-7-1
7，If..., the proof is similar,...
8，in the foregoing section,..
9，when x changes from x to x+h
10，....may be stated as follows
11，the rate of change of position with respect to time
12，In the example worked out in section 7.2
13，a way of obtaining.....
14，We make the convention that ...
15，This agrees with the fact that...
16，the difference quotient ....has the value m....
17，A line for which ...has slope 1.

《数学专业英语》的笔记-第136页

(134----136页)1，this shows g to be....
2，find a ...to satisfy.....
3，...which will be taken up in greater detail in Vol. 2
4，Later we shall meet many applications of this lemma.
5，...is 1 dollar better off
6，come out even
7，In a complete account...
8，what can be said about ....
9，Show that ....证明.....
10，composing mapping 复合映射

《数学专业英语》的笔记-第145页

(143----145页)1，....to fit it into your formerly acquired knowledge
2，teacher's purpose in the classroom
3，vary the words
4，obtain the same effect
5，to collect and to group questions and suggestions....

《数学专业英语》的笔记-第111页

（106----111页）1，to attack this problem
2，it can readily be solved by..
3，the more general problem...
4，an n-element set
5，choosing the elements one at a time
6，......, as asserted
7，straightforward approach 直接的方法
8，ungrouped data 未分类的数据
9，....are in error
10，a is approximately equal to b
11，a plus b into a minus b （into 代表“乘”）

《数学专业英语》的笔记-第183页

(180---183页)1，a wide range of applications
2，horizontal translation of a solution
3，it is not reasonable to think that...
4，per unit population
5，the effective growth rate
6，an initial population of ...
7，The two constant solutions,... have significance
8，the carrying capacity of the environment
9，a population (such as humanity)
10，separate variables to get...
11，Then integrate to get...
12，the flaw in this reasoning is that...
13，...this simple analysis would dictate
14，multiply through by b

《数学专业英语》的笔记-第63页

（59----63页）1，In everyday usage of the language,....
2，....these words have ...meanings.
3，the common use of the term...
4，The important thing here is that.....
5，....how to carry on after a given start.
6，the essential thing is that....
7，For brevity, the notation.... is used to denote...
8，Unless otherwise specified, ......
9，The main question we are concerned with here is to....
10，To treat this problem,....
11，The analogy carries over to.... as well,.
12，and we leave it for the reader to define the symbols.....
13，The basic rules for dealing with........also hold for.....
14，formulate these theorems

《数学专业英语》的笔记-第21页

（18---21页）1，To deal with some more complex practical problems，。。
2，。。analytic geometry and calculus, which belong to the higher mathematics.
3，Mathematicians study conceptions and propositions.
4，Axioms, postulates, definitions and theorems are all propositions.
5，symbols Vs signs (例句看区别)
6，the centric place of mathematical methods。。
7，An equation is a statement of the equality between。。
8， on the peprformance of the indicated operation
9，An identity is true。。
10，An equation which is true。。
11，To obtain the root or roots of an equation。。
12，To solve an equation。。
13，。。almost every problem gives us one or more statements that 。。（结合7体会statement用法）

《数学专业英语》的笔记-第59页

（54----59页）1，be illustrated schematically
2，....are called ....., or, more briefly,...
3，schematic representations of...
4，.....expressed in function notation.
5，plot 画（草图）
6，.....has the property that...
7，the inverse function to f
8，Observe that....
9，in the general case,...
10，the equation... may have no solutions at all
11，.....to admit an inverse function
12，It is desirable to know that ...
13，the rise of modern analysis
14，In the extended sense,...
15，.....serves many a practical purpose
16，Not speciified by....
17，some other form of description
18，admit of no...
19，desireable to serve as .....
20，.....is incorporated in.....

《数学专业英语》的笔记-第191页

(188---191页)1，...is depicted in Figure ..
2，This is in contrast to the situation for..., in which...
3，Recall also that ...
4，at each succeeding step,..
5，The main difficulty with ... is the need to ...
6，Once that is done,..
7，convert such a problem into the current form
8，for more information, the reader may see ...
9，To be specific,..
10，In the problem of ..., we produce:...
11，The reason for doing this is given in ...
12，Also,...
13，Now let's focus attention on..
14，an outline of a procedure
15，real-world examples
16，real-life situations
17，the verbal statements
18，mathematical symbology
19，...fall into two main activities
20，...seeks to explain are...
21，You may choose to ...
22，the effect of ...may be relatively small

《数学专业英语》的笔记-第90页

（84----90页）1，the proof ...is valid for...
2，The same argument with S and T interchanged...
3，We denote ....as an n-tuple...
4，by contraction from...
5，hold trivially 显然成立
6，the underlying linear sapce 基础线性空间
7，by induction on...
8，there is associated...
9，in many diverse fields...
10，make wide use of...
11，use ... to good advantage
12，read with understanding
13，be reduced to...
14，this belief is now known to be untenable...
15，Specifically for this case,...
16，equations in n unknwowns
17，in the sense that...
18，polynomial in x [关于]x 的多项式

《数学专业英语》的笔记-第123页

（122---123页）1，the definition is sufficiently clear...
2，a grouping together
3，keep to this convention
4，in the contrary case,..
5，no considerations of order or multiplicity enter
6，In a more precise form this can be stated by saying that ....
7，in the real world
8，too free a use of set-theoretic notions can easily lead to contraditions.
9，There are several ways of doing this,...