问题 问答题

设函数f(x)在(-∞,+∞)内具有一阶连续导数,L是上半平面(y>0)内的有向分段光滑曲线,其中起点为(a,b)),终点为(a,d).记:


(1) 证明曲线积分I与路径L无关:
(2) 当ab=cd时,求I的值.

答案

参考答案:(1) 将I表示为I=

Pdx+Qdy.因上半平面y>0是单连通区域,又


(y>0).所以积分I在y>0与路径无关.
(2) 由于I与路径无关,取特殊的一条积分路径,即由(a,b)到(c,b)再到(c,d)的折线段,得


又ab=cd,故I=

解析:[考点提示] 曲线积分.

阅读理解

Ever since humans have lived on the earth, they have made use of various forms of communication.Generally, this expression of thoughts and feelings has been in the form of oral speech.When there is a language barrier(障碍), communication is completed through sign language in which motions stand for letters, words, and ideas.Tourists and the people unable to hear or speak have had to turn to this form of expression.Many of these symbols of whole words are very lively and exact and can be used internationally; spelling, however, cannot.

Body language sends ideas or thoughts by certain actions, either meaningfully or not.A wink(眨眼) can be a way of showing that the party is only joking.A nod means agreement, while shaking the head indicates disagreement.

Other forms of nonlinguistic(非语言的) language can be found in Braille (a system of raised dots read with the fingertips), signal flags, Morse code, and smoke signals.Road maps and picture signs also guide and warn people.While language is the most common form of communication, other systems and techniques also express human thoughts and feelings.

小题1:Which of the following statements is NOT true?

A.There are many forms of communication in use today.

B.Language is the most common form of communication.

C.Tourists are unable to use an oral form of communication.

D.Ideas and thoughts can be expressed by body language.小题2:Which form other than oral speech could be most commonly used among deaf people?

A.Picture signs.

B.Body language

C.Braille

D.Signal flags.小题3:Sign language is said to be very lively and exact and can be used meaningfully except for ____.

A.spelling

B.idea

C.whole words

D.expressions小题4:How many different forms of communication are mentioned here?

A.Five

B.Eight

C.Nine

D.Three

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