数学专业英语－Phrases or clauses frequently used in mathematiscs

yunjiang · 发表于 2004-5-6 10:03:18

<

> 
<

>I.常见的涉及运算的短语或句子(句中(A),(B)表示某些表达式,如不等式,等式,方程等)
<

><v:shapetype><v:stroke joinstyle="miter"></v:stroke><v:path connecttype="rect" gradientshapeok="t"></v:path></v:shapetype><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1026"></v:textbox></v:shape>
<TABLE cellSpacing=0 cellPadding=0 width="100%">

<TR>
<TD >
<DIV class=shape v:shape="_x0000_s1026">
Differentiate
Integrate</DIV></TD></TR></TABLE> 1. both sides of the equation and we get … 
 
 
<v:shape><v:textbox style="mso-next-textbox: #_x0000_s1027"></v:textbox></v:shape>
<TABLE cellSpacing=0 cellPadding=0 width="100%">

<TR>
<TD >
<DIV class=shape v:shape="_x0000_s1027">
Differentiating 
Integrating </DIV></TD></TR></TABLE> 2.
 both sides of the equation, we get …
 
 
 3.Add (A) to (B) and we have …
 4.Substract (B) from (A) and we have …
 5.Multiplying each term of equation by …,we obtain …
 6.Dividing the equation through by …,we have…
<v:shape><v:textbox style="mso-next-textbox: #_x0000_s1028"></v:textbox></v:shape>
<TABLE cellSpacing=0 cellPadding=0 width="100%">

<TR>
<TD >
<DIV class=shape v:shape="_x0000_s1028">
Give
yield
imply</DIV></TD></TR></TABLE> 7.(A) and (B) together …
 
 
 
 
 8.Comparing (A) with (B) ,it is easy to see that …
 9.Substituting (A) into (B) ,we obtain …
 10.Eliminating (the parameter) from (A) and (B) ,we have …
<v:shape><v:textbox style="mso-next-textbox: #_x0000_s1029"></v:textbox></v:shape>
<TABLE cellSpacing=0 cellPadding=0 width="100%">

<TR>
<TD >
<DIV class=shape v:shape="_x0000_s1029">
as follows 
in the following form</DIV></TD></TR></TABLE> 11.By introducing a new variable …,we can then rewrite (A) 
 
 
 
 12.By a simple calculation,we obtain from (A)…
II.定理证明过程中常见的短语和句子.
1. 下面的句型可用来表达”根据什么即可得到什么”的意思.
<v:shape><v:textbox style="mso-next-textbox: #_x0000_s1030"></v:textbox></v:shape>
<TABLE cellSpacing=0 cellPadding=0 width="100%">

<TR>
<TD >
<DIV class=shape v:shape="_x0000_s1030">
definiton
hypothesis
assumptions
theorem (N)
lemma (A)
corollary (B)
the remark
the fact that
……………..</DIV></TD></TR></TABLE>According to ,it follows …
 
 
 
 
 
 
 
 
 
Since …, it follows …
 
2. 如果一个论断可以通过一些简单运算或简单推理而获得,由于这些运算或推理比较简单,读者可以自行推算,因而只需直接写出论断来,这时可用下面名句型:
<v:shape><v:textbox style="mso-next-textbox: #_x0000_s1031"></v:textbox></v:shape>
<TABLE cellSpacing=0 cellPadding=0 width="100%">

<TR>
<TD >
<DIV class=shape v:shape="_x0000_s1031">
see
show 
prove 
verify 
check </DIV></TD></TR></TABLE>It is easy to that …
 
 
 
 
 
 
<v:shape><v:textbox style="mso-next-textbox: #_x0000_s1032"></v:textbox></v:shape>
<TABLE cellSpacing=0 cellPadding=0 width="100%">

<TR>
<TD >
<DIV class=shape v:shape="_x0000_s1032">
seen
shown
proved 
verified 
checked</DIV></TD></TR></TABLE>It can easily be that …
 
 
 
 
 
3.如果所要提及的结论比较显浅,或是众所周知,无需作过一步的证明,这时可用下面名句型:
<v:shape><v:textbox style="mso-next-textbox: #_x0000_s1033"></v:textbox></v:shape>
<TABLE cellSpacing=0 cellPadding=0 width="100%">

<TR>
<TD >
<DIV class=shape v:shape="_x0000_s1033">
clear
obvious
evident
well-known
 </DIV></TD></TR></TABLE> It is that …
 
 
 
 
<v:shape><v:textbox style="mso-next-textbox: #_x0000_s1034"></v:textbox></v:shape>
<TABLE cellSpacing=0 cellPadding=0 width="100%">

<TR>
<TD >
<DIV class=shape v:shape="_x0000_s1034">
Clerly
Obviously
Evidently</DIV></TD></TR></TABLE> , …
 
 
 
4.证明一个定理有时需要引进辅助函数,这时可用下面句型:
Let us first define the fuction …
Let us introduce a new function …
Let us consider the function …
Let us first investigate the function …
 Or simply:
Let …
Set …
Define …
Put …
Consider …
5.在一个定理中,有几个结论需要证明,其中有些结论比较明显,可不用证明,仅需证明余下结论便可,这时可用如下句型:
<v:shape><v:textbox style="mso-next-textbox: #_x0000_s1035"></v:textbox></v:shape>
<TABLE cellSpacing=0 cellPadding=0 width="100%">

<TR>
<TD >
<DIV class=shape v:shape="_x0000_s1035">
Obvious
trivial</DIV></TD></TR></TABLE> Since (A) and (B) are ,we need only prove (C)
 
 
 
 Since (A) and (B) are trivial ,it suffices to prove (C)
 
6.为了证明一个定理,有时我们并不是直接去证明,而是证明一个新的论断,一旦新的论断得到证明,已给定理不难由此而得证,这时可用下面句型:
 
<v:shape><v:textbox style="mso-next-textbox: #_x0000_s1036"></v:textbox></v:shape>
<TABLE cellSpacing=0 cellPadding=0 width="100%">

<TR>
<TD >
<DIV class=shape v:shape="_x0000_s1036">
Theorem
result</DIV></TD></TR></TABLE>The will be proved if we can show …
 
 
The theorem will be proved by showing that …
If we can prove …then the theorem follows immediately 
以下各句用于新的论断被证明之后:
The theorem is now a direct consequence of what we have proved.
The theorem follows immediately from what we have proved.
The theorem is now evident from what we have proved.
It is evident to see that the theorem holds .

yunjiang · 发表于 2004-5-6 10:03:52

<

0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt">7.在证明过程中,有时要用到一些早已学过的知识或技巧,这里可用下面句子,以提醒读者:<

0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"> Recall that …<

0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 118.5pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"> Notice that … Note that … Observe that … In order to prove the theorem ,we need the knowledge of … (e.g ..the knowledge of Variations) In order to obtain the following equation ,we need … (e.g .. Leibniz’s rule for differentiating under integralsign)8.如果需要证明的定理的假设条件是一般条件,但是,只要定理在特殊条件下成立,就不难推出定理在一般条件下也成立,这时仅需在特殊情况下去证明定理就够了,为此可用下面句型:<v:shape><v:textbox style="mso-next-textbox: #_x0000_s1037"></v:textbox></v:shape><TABLE cellSpacing=0 cellPadding=0 width="100%"><TR><TD #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent"><DIV class=shape 7.2pt; PADDING-LEFT: 7.2pt; PADDING-BOTTOM: 3.6pt; PADDING-TOP: 3.6pt" v:shape="_x0000_s1037">Considerassume</DIV></TD></TR></TABLE> Without loss of generality ,we may … Without loss generality ,we may prove the theorem in the case … It suffices to prove the theorem in the theorem in the case … We need only consider the case … For simplicity consider the case … (e.g .. we may take the domain to be the disk ) 9.如果特征的论断可用以前用过的相似的方法或步骤进行证明,则可用下面句型:<v:shape><v:textbox style="mso-next-textbox: #_x0000_s1039"></v:textbox></v:shape><TABLE cellSpacing=0 cellPadding=0 width="100%"><TR><TD #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent"><DIV class=shape 7.2pt; PADDING-LEFT: 7.2pt; PADDING-BOTTOM: 3.6pt; PADDING-TOP: 3.6pt" v:shape="_x0000_s1039">the same waya similar way </DIV></TD></TR></TABLE><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1038"></v:textbox></v:shape><TABLE cellSpacing=0 cellPadding=0 width="100%"><TR><TD #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent"><DIV class=shape 7.2pt; PADDING-LEFT: 7.2pt; PADDING-BOTTOM: 3.6pt; PADDING-TOP: 3.6pt" v:shape="_x0000_s1038">theoremstatementlemma…</DIV></TD></TR></TABLE> The can be proved in as shown before. This theorem can be proved by the same method as employed in the last section. This theorem can be completed by the method analogous to that used above. Using the same argument as in the proof of theorem N, we can easily carry out the proof of this theorem. We now proceed as in the proof of theorem N. We shall adopt the same procedure as in the proof of theorem N.

yunjiang · 发表于 2004-5-6 10:04:04

<

0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 237.0pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt">10.如果我们用的是反证法,则其开头及结尾可用下面句型:<

0cm 0cm 0pt 10.5pt; TEXT-INDENT: -10.5pt; tab-stops: 135.0pt 276.75pt; mso-char-indent-count: -1.0; mso-char-indent-size: 10.5pt"><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1041"></v:textbox></v:shape><TABLE cellSpacing=0 cellPadding=0 width="100%"><TR><TD #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent"><DIV class=shape 7.2pt; PADDING-LEFT: 7.2pt; PADDING-BOTTOM: 3.6pt; PADDING-TOP: 3.6pt" v:shape="_x0000_s1041"><

0cm 0cm 0pt">falsenot truenot right</DIV></TD></TR></TABLE><v:shape><v:textbox style="mso-next-textbox: #_x0000_s1040"></v:textbox></v:shape><TABLE cellSpacing=0 cellPadding=0 width="100%"><TR><TD #ece9d8; BORDER-TOP: #ece9d8; BORDER-LEFT: #ece9d8; BORDER-BOTTOM: #ece9d8; BACKGROUND-COLOR: transparent"><DIV class=shape 7.2pt; PADDING-LEFT: 7.2pt; PADDING-BOTTOM: 3.6pt; PADDING-TOP: 3.6pt" v:shape="_x0000_s1040">statementassertionconclusion</DIV></TD></TR></TABLE>If the were then … If the assertion would not hold ,then …This is contrary to …This contradicts the fact that ..This leads to a contradiction.11.表示定理已证毕或者把前面所证的总结为一结论. We have thus proved the theorem . This completes the proof . The proof of the theorem is now completed. It is now obvious that the theorem holds . Thus we have derived that … Consequently ,we infer that … Thus we conclude that … Thus we are led to the conclusion that … Thus we arrive at the conclusion that … Thus we can summarize what we have proved as the following theorem.12.其它 There exist (s) … such that … We claim … in fact … We are now in a position to … If otherwise … Provided that …

zwindling · 发表于 2004-5-19 18:39:27

感谢分享！[em07]

iwfine · 发表于 2004-12-19 06:28:50

<

>很好！<

>谢谢！

beibeihon · 发表于 2006-2-9 23:28:06

真好！谢谢！

amao · 发表于 2008-4-23 09:05:05

这个不错，学习了。

0107 · 发表于 2009-12-16 15:48:39

不错，学习啊~~

		自动登录	找回密码
密码			注-册-帐-号