succulents 多肉植物圖片介紹英文學(xué)習(xí)資料

succulents多肉植物圖片介紹英文新概念英語(yǔ)2冊(cè)動(dòng)詞詞組及固定搭配練習(xí)題(全)新概念英語(yǔ)2冊(cè)動(dòng)詞詞組及固定搭配練習(xí)題(全)

/.新概念英語(yǔ)2冊(cè)動(dòng)詞詞組及固定搭配練習(xí)題(全)動(dòng)詞詞組1Hedidnotknowhowtofight,butheknockedtheboxer_______.(L6)2Thisflowerpotisbroken,Whoknockedit__________?3Iknocked_____earlyyesterdayandwenttoafootballmatch.4Heusuallyknocks_______at6o'clock,buttodayhe'sworkinglate.5Listen!Someoneisknocking________thewindow!6Manyathleteshaveentered________theOlympicGamesthisyear.(L8)7Jansenset________anewworldrecordforthe400metres.(L12)8CaptainAlisonwillset_________ateighto'clock.9TomandIset_____earlyinthemorning.10Willthepersonwhotookmyrulerpleasegiveit_________tome?(L18)11Whenmychildrengrewup,Igavealltheirtoys________.12I'vegiven_______allmyoldfurniture.13Whendowehavetogive______ourcompositions?14Wewerelosingthebattlebutwedidnotgive_____.15Igave______smokinglastyearbutIhavejuststartedagain.16Mrs.Bowerstoldherchildrentoputtheirtoys_______andgotobed.(L27)17Youcanstayheretonight.Wecanputyou______inthespareroom.18I'mnotreadyyet.Ihaven'tputmyshoes_______.19'Openyourexercisebooksandput_____thefollowing,'theteachersaid.20Fatherisputting_____thefirehelitinthegarden.21Whentheyhaveput______thatnewbuilding,itwillspoiltheview.22Ihaveput_____mytriptoJapanuntilnextmonth.23Iamgettingadivorce.Ican'tput______himanylonger.24Icalled_____youfivetimesyesterday.Wereyouout?(L34)25It'stoolatetogotothepictures.Whydon'twecallthewholething______?26Wecalled_______tohimbuthecouldnothearus.271called______thepostofficeonmywaytowork.28Weshallbeholdingapartytomorrow.Iamlooking________itverymuch.(L37)29Look_____!Younearlyknockedthatjugover!30Theysayheisveryfamous.Ishalllookhim____in'Who'sWho'.31MyfriendIngridlivesinStockholm.Whydon'tyoulookher_____whenyou'rethere?32Examinationswillbeheldnextweek.I'mnotlooking_____them.33Havetheyput_____thatforestfire?(L48)34Haven'tyoulearnthowtolook______awordinthedictionary?35Itriedtocallher____butherphonewasoutoforder.36Pick_____allthosetoysandputthem_____.37Ifyoucan'tfindaroomatthehotel,Icanputyou_____.38Look____!Thatbusnearlyhityou!39I'IIpay_____themoneyIborrowedassoonasIcan.40Ican'tput_______thosechildrenanylonger.41He'ssofaraheadofyou,you'llnevercatch_______him.42Weshallhavetodraw______anewplan.(L64)43WhenIrecognizedwhohewasIdrew_______inhorror.44Thecardrew_______outsidethecinema.45Theteacheracceptedtheboy'sapologyandlethim______.(L65)46Hewasgoingtolendmeapoliceman'suniformsothatIcoulddress____fortheparty,butheletme________.47Abigcardrew____outsideourhouse.(L72)48Youcan'trelyonhim.Heissuretoletyou_____.49Don'tletthechildren____ofthegarden.50Weshallhavetodraw_____anewagreement.51Hewasgoingtopunishtheboy,buthelethim______.52Hehasnevergot_____hiswife'sdeath.(L74)53Howdidthethiefmanagetoget____thehouse?54Didyouget_____yourdrivingtest?55Thelidisstuck.Ican'tgetit______.56Hegets______verywell_____allhisnewcolleagues.57Pleasestoptalkingandget_____yourwork,'shesaid.58Ithinkhekept_____________themeetingonpurpose.(L78)59Keep______thefloor.Ihavejustfinishedscrubbingit.60Hehasjustbegunadiary.Iwonderhowlonghewillkeepit______.61Hekept_____makingthesamemistake.62WekeepTim_____oftheroombecauseBettyhasmeasles.63Mostpeoplespendtheirlivestryingtokeep___________theirneighbours.64Threechildrenwerekept______afterschoolforbeingnoisyinclass.65Whowilltake_whenthepresentdirectorleaves?(L79)66Assoonashegotintothelifthetookhishat____.67Youshouldn'tbetaken_____bystorieslikethat.68Lastyearhetook____Russian;nowhe'staking____Chinese.69Noneofmychildrentakes____me.70Thenewrocketwilltake_____fromCapeCanaveral.71'Pleasetake_____thisletterforme,Pamela,'themanagersaid.72Hetakespeople______sowellheoughttogoonthestage.73Whatalotofdirtyplates!Whoisgoingtowash______?(L82)74Don'twearthathat.Peoplewilllaugh_____you.75Thebridgewaswashed_____bytheriver.76Iran_____anoldfriendofmineinarestaurantyesterday.(L88)77Whiledrivingtoworkyesterday,Iran_____petrolandhadtowalktoagarage.78Thiscarwillhavetobeservicedassoonasithasbeenrun______.79She'sbeentakentohospital.Shewasrun_____byacar.80Iran____him,butIcouldnotcatchhim.81Ilostmylecturenotesandhadtomake____anewset.(L91)82lcan'tmake____whatyou'vewritten.83Whenitgotdarkwemade_____home.84Mydaughteroftenmakes_____storiesofherown.85Howcanyoumake____allthoselessonsyoumissed?答案：1-5outoveroff（下班）offat6-10forupoutoffback11-15awayawayininup16-20awayupondownout21-25upoffupwithonoff26-30out,at,forwardto,out,up31-35up,forwardto,out，up，up36-40up…away，up，out，back，upwith41-45upwith,up,back,upoff46-50up…down,up,down,out,up51-55off,over,into,through,off56-60on…with,onwith,awayfrom,off,up61-65on,out,upwith,in,over66-70off,in,up…up,after,off71-75down,off,up,at,away76-80into,outof,in,over,after81-85out,out,for,up,upfor固定搭配1Iwithdrewalotofmoney_______thebankyesterday.(L22)2Irefusetocomment_________hiswork.3Thewaiter'stipisincluded__________thebill.4Hecongratulatedme______havinggotengaged.5Thiswarmcoatwillprotectyou________thecold.6Didanythingemerge______yourdiscussion?7Idreamt________youlastnight.8Youcanneverrely___himtobepunctual.9Nothingwillpreventhim________succeeding.10Areyouinterested_______music?11IsupposeIcancount____youforhelp_______thismatter?12Beware_____thedog.13Hepersisted_____askingquestions.14Iinsist________yourtellingmethetruth.15Ittookmealongtimetogetrid___him.16Doyoumeantosayyouhaveneverheard____Beethoven?17Iseparatedthem______eachotherbecausetheywerefighting.18Theycanonlycurehim_____hisillnessiftheyoperate____him.19Youcandepend______me.20Ihaven'taccusedhim_____anything,butIsuspecthim____havingtakenit.21Whatevermadeyouthink_______suchathing?22Weexpectagreatdeal____you,Smith.23MyhandssmeIl_____soap.24Theydiffer_______eachothersomuch.25Heinvestedalotofmoney______shipping.26Thefilmwasbased_____anovelbyDickens.27Don'tlean___thatshelf!You'llregretit.28Sheoftensuffers_______colds.29Wehaveembarked_______anewscheme.30Ibelieve______takingmytime.31Joneswasdismissed________thefirm.32Theybeganbyexperimenting______rats.33Pleaseconcentrate______whatyouaredoing.34Shepridesherself_______hercleanhouse.35Theclimberfailed_______hisattempttoreachthesummit.36Manypeopleescaped______prisonsduringthelastfiveyears.37Wemusteconomize_____fuel.38He'sneverdoneanywork.Helives______hismother.39Hewasemployed______afactorybeforehejoinedthearmy.40Andwhatdoesthishorribledrinkconsist_______.41Ishallcertainlyact________youradvice.42Don'twrite_____thedesk!43Youshouldnotboast_______oursuccess.44Youmustencouragehim_____hisefforts.45Heshouldbeinstructed______hisduties.46Twoorthreepeoplewereinvolved______theaccident.47Childrenshouldbeprohibited_____smoking.48lcanassureyou_____mysupport.49Doyouapprove____hunting?50Idespair_____everteachinghimanything!51Hasthisplayeverbeenperformed______thestage?52Iwarnedhim_____thedanger,buthewouldn'tlistentome.53Howmuchhaveyouborrowed_______mealready?54Hedelights_______annoyingme.55Idon'tagree______you.(L46)56Shepreferred___wait____him.57Wehavebeencorresponding_____eachotherforyears.58Howdoyouaccount______this?59Doyouobject______mysmoking?60I'msurprised_______you!61Youmustreply_____hisletter.62Hehassomeimportantbusinesstoattend_______.63Doyoumeantosayyouexchangedthatlovelycar_____this?64Hasitoccurred____youthatshemusthavearrived_____LondonAirportbynow?65Iwasshocked______hisindifference!66Youmustcomply_____therulesofthegame.67PoorMary!Shehassomuchtocope_______.68Pleasedon'tmentionit_____myhusband,butIpaid￡50_____thishat.69Shewasquiteunprepared_______thenews.70Don'tblameme______theaccident!71I'mdisgusted_____yourbehavior!72YouforgottothankAuntJane_____herpresent.73ltisrudetostare____people.74I'mnotsatisfied_____yourwork.75Hisdebtnowamounts______$100!76Mixthecontentsofthispacket_____alittlewater.77Iknocked_____thedoor.78Whomdoesthisbookbelong_______?79Ireasoned_____him,buthewouldnotlisten_______me.80She'saccustomed_______livingincomfort.She'llneverpart____herpreciouspossessions.81Atwhattimewillyoucall_____me?82ThespySurrenderedhimself_____theenemyandwascondemned_____death.83I'velooked_____iteverywhere,butlcan'tfindit.84I'llsee______thecookingtonight.85Imustapologize_____keepingyouwaiting.86Theclassfailedtorespond_______theteacher'snewmethods.87Heturned_____meforhelp,evenafterIhadquarreled._____him..88LikeMicawber,Ihope______somethingbetter.89Pleaseapply____thesecretaryforinformation.90Therewasanoteattached______theparcel.91Justguess_____thepriceofthiscarpet.92Howlonghaveyoubeenworking______thisexercise?93Theconcertbegan_____apiecebyanunknowncomposer.94Howmuchdidtheychargeyou________that?95Georgeisjealous______hisyoungersister.(L70)96Sheiswell-qualified______languagesandveryefficient______herwork.97Wearefortunate______havingsufficientsuppliesoffuel______thewinter.98Contrary______myexpectations,there’rewasnoneedtobeuneasy________theresultsofthematch.99Iamnotfamiliar________hisnovelsandnotverykeen________readingthem.100Iknowheisslow______understanding,butyouhavetobepatient_____him.101Itisobvious_______everyonethatheisnotresponsible______thismistake.102Ithinkhe'squitehonest________hisintentions.103Heisnotonlyindifferent_____otherpeople;heisoftenextremelyrude____themaswell.104I'mnotveryfond________dancing.105Childrenremaindependent_______theirparentsforalongtime.106Ishouldbegrateful____anyadviceyoucangive.107Heisnotdifferent_____anyoneelse.108Hemaybequick_____understanding,buthe'snotcapable_____rememberinganything.109Heisintent______passingtheexamination,butI'mdoubtful_______hischances.110Hesayshe'ssorry_____whathedid,sodon'tgetangry______him.111I'veneverbeengood_______arithmetic.112Thatboyisfarahead______everyoneelseintheclass.113Eventhoughhewasoftencruel_____hisdog,itremainedfaithful______him.114Yourconclusionsarenotconsistent_____thefacts.115Thisdiarywillproveuseful______you.116I'mnotsure____theexactdate,butIthinkhearrivedherein1993.117Hewasfoundguilty_____greatmanycrimes.118Youshouldnotbesosensitive_____criticism.119Thispassportisvalid______allcountries.120Manypeopledonotapprove____bloodsports.(L94)121Hewasfoundguilty_____murderandcondemned_____death.122Hasiteveroccurred____youthatthosetwinsarequitedifferent______eachotherinmanyways?123Iconsultedmylawyer_____thematterandIshallact____hisadvice.124Itisimpossibletopreventthem____quarrelling____eachother.125Heisresponding____treatmentandwillsoonbecured___hisillness.126Eventhoughheisthirty-five,helives____hismotherandiscompletelydependent___her.127Itriedtoreason___him,buthewasveryrude____me.128Iamgrateful____youforbeingsopatient______him.129Hemightbegood____hisjob,butyoucan'trely_____him.130Iamthinking_____looking_______anewjob.131lfyouinterfere_____otherpeople'saffairs,youwillregretit.132Doyoubelieve____allthatnonsense?133Itshouldbeobvious___youthatifyoupersist____botheringhim,hewillgetangry____you.134Youdemandtoomuch______him;heisnotreallyequal_____thetask.135Don'tbesosure____yourself!136Hehasprovided______everyemergency.137lwasafraid______mentioningit_____him.138Don'tblamehim____this;Iamresponsible_____whathashappened.139Heissokeen_____learning,youshouldencouragehim____hisefforts.140Beware_____peoplewhoappeartobeenthusiastic____yoursuccess.141Iappealed______himforhelp.142Iamsorry_____havingaskedhim.Iwasshocked____hisrefusal.143Areyouinterested-_____opera?144Areyouaware____thedifficultiesthatlieahead____you?145He'sentitled_____apension,buthewon'tdream____retiringyet.146Whoisgoingtopay____thedamage?147Thiscarisinferior_____theoneIboughtlastyear.148I'mafraidIcan'tcomment_____yourworkjustyet.149Shemayprideherself_____herabilities,butshe'snotcapable_____bringingupchildren.150Weareaccustomed_______badweather.151Howcanyouagree_____suchanideawhenyouareignorant______thebasicfacts?152Heconfessed_____methathehadjustbeenconverted______somestrangereligion.153Shewantedtoborrowtherecord_______mebutshewasshy____asking.154Ifyoufail____thisattempt,don'tcount______meforhelp.155Hislieswouldnottake__anybodybutafool.(L96)156Theexplorersreturnedbecausetheyhadrun____food.157Heisverygoodatmaking_____storiesforchildren.158Ittookhimalongtimetoget______hisillness.159Mrs.Jonestoldherdaughtertokeep____thestove.160Duringthestorm,theshipmade-_____thenearestharbor.161Hasthiscarbeenrun_____yet?162Thebirdaccidentallyflewintotheroomandcouldn'tget_____.163Hespokesoquickly,Ididn'thavetimetotake____whathesaid.164Thoughweallran_______thethief,wecouldnotcatchhim.165Ihadn'tseenhimforyearsandIaccidentallyran______himinthestreetthismorning.166Ronaldwilltake_______thefamilybusinessnowthathisfatherhasdied.167Canyoumake______theaddressonthisenvelope?168I'vetaken________paintinginmysparetime.答案：1-5fromoninonfrom6-10fromofonfromin11-15on…inofinonof16-20offromof…ononof…of21-25ofof/fromoffromin26-30ononfromonin31-35fromonononin36-40fromononinof41-45ononofinin46-50infromofofof51-54on,of,from,in,with55-60to…for,with,for,to,at61-65to,to,for,to…at,at66-70with,with,to…for,for,for71-75with,for,at,with,to76-80with,at,to,with…to,to…with,81-85for,to…to,for,to,for86-90to,to…with,for,to,to91-95at,at,with,for,of96-100in…at,in…for,to…about,with…on,at…with101-105to…for,in,to…to,of,on106-110for,from,at…of,on…about/of,for…with111-115at,of,to…to,with,to,116-120of,of,to,for,of121-125of…to,to…from,on…on,from…with,to…of126-130with…on,with…to,to…with,at…on,of…for131-135in/with,in,to…in…with,of…to,of,136-140for,of…to,for…for,on…in,of…about141-145to,for/about…at/by,in,of…of,to…of146-150for,to,on,on…of,to151-155to/with…of,to…to,from…of,in…on,in156-160outof,up,over,awayfrom,for161-165in,out,down,after,into166-168over,out,up

單純的課本內(nèi)容，并不能滿(mǎn)足學(xué)生的需要，通過(guò)補(bǔ)充，達(dá)到內(nèi)容的完善教育之通病是教用腦的人不用手，不教用手的人用腦，所以一無(wú)所能。教育革命的對(duì)策是手腦聯(lián)盟，結(jié)果是手與腦的力量都可以大到不可思議。

柱坐標(biāo)和球坐標(biāo)系下拉普拉斯算符表達(dá)式的簡(jiǎn)單推導(dǎo)柱坐標(biāo)和球坐標(biāo)系下拉普拉斯算符表達(dá)式的簡(jiǎn)單推導(dǎo)柱坐標(biāo)和球坐標(biāo)系下拉普拉斯算符表達(dá)式的簡(jiǎn)單推導(dǎo)柱坐標(biāo)和球坐標(biāo)系下拉普拉斯算符表達(dá)式的簡(jiǎn)單推導(dǎo)[摘要]：本文采用多元微積分，利用球坐標(biāo)與柱坐標(biāo)、柱坐標(biāo)與直角坐標(biāo)變量轉(zhuǎn)換的相同關(guān)系，以拉普拉斯算符為例，簡(jiǎn)化了在柱坐標(biāo)和球坐標(biāo)系下拉普拉斯算符表達(dá)式的推導(dǎo)。本文提出了此法在柱坐標(biāo)和球坐標(biāo)系下梯度、旋度、散度算符表達(dá)式的推導(dǎo)中的適用性，適合廣大非數(shù)學(xué)專(zhuān)業(yè)本科生學(xué)習(xí)與掌握。[關(guān)鍵詞]：拉普拉斯算符；球坐標(biāo)；柱坐標(biāo)；多元微積分[中圖分類(lèi)號(hào)]：O13[文獻(xiàn)標(biāo)識(shí)碼]：A[文章編號(hào)]:1672-1452(2015)**-****-041引言在材料科學(xué)基礎(chǔ)、近代物理、量子力學(xué)等課程的內(nèi)容中，菲克第二定律和薛定諤方程中的拉普拉斯算符在柱坐標(biāo)系和球坐標(biāo)系中的表達(dá)式十分重要。在近代物理的課本[1]和材料科學(xué)基礎(chǔ)的課本[2]上，提到了拉普拉斯算符在柱坐標(biāo)和球坐標(biāo)系下的表達(dá)式，但沒(méi)有給出具體的推導(dǎo)過(guò)程。在電動(dòng)力學(xué)課本[3]中,這方面的內(nèi)容是通過(guò)引入“正交曲線(xiàn)坐標(biāo)系”得出關(guān)于拉普拉斯算符的一般結(jié)論，再推導(dǎo)出球坐標(biāo)和柱坐標(biāo)下的表達(dá)式。但是利用正交曲線(xiàn)坐標(biāo)系的一般結(jié)論進(jìn)行推導(dǎo)比較抽象，對(duì)于非數(shù)學(xué)專(zhuān)業(yè)的同學(xué)來(lái)說(shuō)，理解一般性的結(jié)論需要較高的數(shù)學(xué)水平。現(xiàn)有的文獻(xiàn)[4][5]中，有采用多元復(fù)合函數(shù)微商法則完成推導(dǎo)的，雖然此法在對(duì)學(xué)生的微積分要求較低，但是所給出的證明計(jì)算繁瑣，無(wú)助于學(xué)生直接理解公式的正確性和自主完成推導(dǎo)。本文給出了用多元微積分導(dǎo)出拉普拉斯算符在柱面坐標(biāo)系和球面坐標(biāo)系中表達(dá)式的簡(jiǎn)單方法。此法僅要求學(xué)生掌握基本的多元微積分知識(shí)，計(jì)算過(guò)程簡(jiǎn)潔美觀，便于廣大的非數(shù)學(xué)系專(zhuān)業(yè)的學(xué)生掌握和理解。建議在近代物理、量子力學(xué)、材料科學(xué)基礎(chǔ)等課程教材和教學(xué)中應(yīng)用。2柱坐標(biāo)和球坐標(biāo)下拉普拉斯算符的推導(dǎo)2.1柱坐標(biāo)系下的拉普拉斯算符表達(dá)式的推導(dǎo)首先，直角坐標(biāo)系的分量與柱坐標(biāo)系的分量有如下的轉(zhuǎn)換關(guān)系： （1） （2） （3） （4）（1）式兩端分別對(duì)x和y求偏導(dǎo)，得 （5） （6）（2）兩端對(duì)x求偏導(dǎo)，并將（5）式代入，得

（7）同理可知， （8）假設(shè)所研究的函數(shù)為由于z關(guān)于x，y是獨(dú)立的變量，故 （9）同理 （10）利用公式（5）（7）（9），對(duì)f求x的二次偏導(dǎo) （11）類(lèi)似地，計(jì)算f關(guān)于y的二階偏導(dǎo)數(shù)。計(jì)算過(guò)程與上面相同，將x換為y，換為，換為即可。于是（12）結(jié)合（11）（12）式，合并同類(lèi)項(xiàng)后可以得到（13）式 （13）最后，拉普拉斯算子的柱坐標(biāo)可表示如下 （14-a）或者 （14-b）2.2球坐標(biāo)系下的拉普拉斯算子的推導(dǎo)上述柱坐標(biāo)系下拉普拉斯算符表達(dá)式的推導(dǎo)方法較為常用，運(yùn)算量不大，也是大多數(shù)理工類(lèi)本科生可以獨(dú)立完成的。采用多元復(fù)合函數(shù)微商法則進(jìn)行證明的現(xiàn)有文獻(xiàn)中，推導(dǎo)路徑是從直角坐標(biāo)公式分別推導(dǎo)出柱坐標(biāo)和球坐標(biāo)系下的算符表達(dá)式。但是從直角坐標(biāo)系出發(fā)推導(dǎo)球坐標(biāo)系下的拉普拉斯算符表達(dá)式較為繁瑣，運(yùn)算量過(guò)大，不利于學(xué)生理解整個(gè)證明過(guò)程，許多同學(xué)難以完成。本文采取從（14）式出發(fā)，利用與的關(guān)系式和與的關(guān)系式（1）（2）（3）（4）間的相似性，給出利用柱坐標(biāo)系下的拉普拉斯算符的表達(dá)式導(dǎo)出球坐標(biāo)系下拉普拉斯算符表達(dá)式的方法。此法亦適用于旋度、梯度、散度算符在球坐標(biāo)系下表達(dá)式的推導(dǎo)。球坐標(biāo)系下的變量和柱坐標(biāo)系下的變量間的關(guān)系如下： （15） （16） （17） （18）對(duì)比（14）（15）（16）（17）式與（1）（2）（3）（4）式，可以發(fā)現(xiàn)球坐標(biāo)系和柱坐標(biāo)系的變量關(guān)系與柱坐標(biāo)系和直角坐標(biāo)系的變量關(guān)系相同。于是在利用（13）式推導(dǎo)球坐標(biāo)系的拉普拉斯算符表達(dá)式時(shí)，可以將上一節(jié)的推導(dǎo)過(guò)程中得到的公式套用過(guò)來(lái)。將（10）（13）式中的分別換為可得， （19） （20）將（19）（20）式代入（14-a）式，并注意到，有最后整理得， （21）（21）式即為球坐標(biāo)系下的拉普拉斯算子。3結(jié)論本文利用多元微積分，從直角坐標(biāo)系出發(fā)導(dǎo)出柱坐標(biāo)系下的拉普拉斯算符表達(dá)式，利用球坐標(biāo)系變量與柱坐標(biāo)系變量的關(guān)系式與柱坐標(biāo)系變量與直角坐標(biāo)系變量的關(guān)系式相同的特點(diǎn)，從柱坐標(biāo)系的算符表達(dá)式出發(fā)導(dǎo)出球坐標(biāo)系下的算符表達(dá)式，證明簡(jiǎn)潔易懂。此外，此法在證明梯度、旋度、散度算符的柱坐標(biāo)和球坐標(biāo)的表達(dá)式時(shí)同樣能簡(jiǎn)化計(jì)算，適合理工類(lèi)本科生學(xué)習(xí)和掌握。[參考文獻(xiàn)][1]Beiser,A.Conceptsofmodernphysics[M].NewYork:McGraw-HillEducation,2003.[2]潘金生，田民波，仝健民.材料科學(xué)基礎(chǔ)[M].北京：清華大學(xué)出版社，2011.[3]郭碩鴻，電動(dòng)力學(xué)[M].北京：高等教育出版社，2008.[4]江俊勤.柱面坐標(biāo)系和球面坐標(biāo)系中的拉普拉斯算符[J].廣東教育學(xué)院學(xué)報(bào),2003，2：32-34.[5]姚久民，石鳳良.球坐標(biāo)系中拉普拉斯算符表達(dá)式的推導(dǎo)[J].唐山師范學(xué)院學(xué)報(bào),2005，5：67-71.AConciseDerivationoftheExpressionsofLaplacianOperatorinCylindricalandSphericalCoordinateSystemsZhangJinyuSchoolofMaterialScienceandEngineering,TsinghuaUniversity，Beijing100084,China;Abstract:Inthispaper,theexpressionsofLaplacianoperatorincylindricalandsphericalcoordinatesystemarederivedbyusingtheknowledgeofmultivariablecalculus.TherelationshipbetweenvariablesinsphericalcoordinatesystemandthoseinsphericalcoordinatesystemissimilartothatbetweenvariablesincylindricalcoordinatesystemandthoseinCartesiancoordinate,whichsimplifiesthederivationgreatly.Itisproposedthatthismethodcanalsopredigestthederivationoftheexpressionsofthecurl,gradient,divergenceoperatorinthetwocurvilinearcoordinatesystemsmentionedabove.Insum,thismethodissuitableforundergraduatesofengineeringtograsp.Itissuggestedthatthismethodcouldbeinvolvedintherelativetextbooksandteachingprocesses.Keywords:Laplacianoperator;sphericalcoordinatesystem;cylindricalcoordinatesystem;multivariablecalculus

一些數(shù)學(xué)的體積和表面積計(jì)算公式一些數(shù)學(xué)的體積和表面積計(jì)算公式

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一些數(shù)學(xué)的體積和表面積計(jì)算公式一些數(shù)學(xué)的體積和表面積計(jì)算公式長(zhǎng)方形的周長(zhǎng)=（長(zhǎng)+寬）×2

正方形的周長(zhǎng)=邊長(zhǎng)×4

長(zhǎng)方形的面積=長(zhǎng)×寬

正方形的面積=邊長(zhǎng)×邊長(zhǎng)

三角形的面積=底×高÷2

平行四邊形的面積=底×高

梯形的面積=（上底+下底）×高÷2

直徑=半徑×2半徑=直徑÷2

圓的周長(zhǎng)=圓周率×直徑=圓周率×半徑×2

圓的面積=圓周率×半徑×半徑

長(zhǎng)方體的表面積=（長(zhǎng)×寬+長(zhǎng)×高＋寬×高）×2

長(zhǎng)方體的體積=長(zhǎng)×寬×高

正方體的表面積=棱長(zhǎng)×棱長(zhǎng)×6

正方體的體積=棱長(zhǎng)×棱長(zhǎng)×棱長(zhǎng)

圓柱的側(cè)面積=底面圓的周長(zhǎng)×高

圓柱的表面積=上下底面面積+側(cè)面積

圓柱的體積=底面積×高

圓錐的體積=底面積×高÷3

長(zhǎng)方體（正方體、圓柱體）的體積=底面積×高

平面圖形

名稱(chēng)符號(hào)周長(zhǎng)C和面積S正方形a—邊長(zhǎng)C＝4aS＝a2

長(zhǎng)方形a和b－邊長(zhǎng)C＝2(a+b)S＝ab

三角形a,b,c－三邊長(zhǎng)h－a邊上的高s－周長(zhǎng)的一半

A,B,C－內(nèi)角其中s＝(a+b+c)/2S=ah/2=ab/2·sinC=[s(s-a)(s-b)(s-c)]1/2=a2sinBsinC/(2sinA)

四邊形d,D－對(duì)角線(xiàn)長(zhǎng)

α－對(duì)角線(xiàn)夾角S＝dD/2·sinα

平行四邊形a,b－邊長(zhǎng)h－a邊的高

α－兩邊夾角S＝ah＝absinα

菱形a－邊長(zhǎng)α－夾角D－長(zhǎng)對(duì)角線(xiàn)長(zhǎng)

d－短對(duì)角線(xiàn)長(zhǎng)S＝Dd/2＝a2sinα

梯形a和b－上、下底長(zhǎng)

h－高m－中位線(xiàn)長(zhǎng)S＝(a+b)h/2＝mh

圓r－半徑d－直徑C＝πd＝2πr

S＝πr2＝πd2/4

扇形r—扇形半徑a—圓心角度數(shù)

C＝2r＋2πr×(a/360)

S＝πr2×(a/360)

弓形l－弧長(zhǎng)b－弦長(zhǎng)h－矢高r－半徑

α－圓心角的度數(shù)S＝r2/2·(πα/180-sinα)

＝r2arccos[(r-h)/r]-(r-h)(2rh-h2)1/2

＝παr2/360-b/2·[r2-(b/2)2]1/2＝r(l-b)/2+bh/2

≈2bh/3

圓環(huán)R－外圓半徑r－內(nèi)圓半徑D－外圓直徑d－內(nèi)圓直徑S＝π(R2-r2)＝π(D2-d2)/4

橢圓D－長(zhǎng)軸d－短軸S＝πDd/4

立方圖形

名稱(chēng)符號(hào)面積S和體積V

正方體a－邊長(zhǎng)S＝6a2V＝a3

長(zhǎng)方體a－長(zhǎng)b－寬c－高S＝2(ab+ac+bc)V＝abc

棱柱S－底面積h－高V＝Sh

棱錐S－底面積h－高V＝Sh/3

棱臺(tái)S1和S2－上、下底面積

h－高V＝h[S1+S2+(S1S2)1/2]/3

擬柱體S1－上底面積S2－下底面積S0－中截面積h－高V＝h(S1+S2+4S0)/6

圓柱r－底半徑h－高C—底面周長(zhǎng)S底—底面積S側(cè)—側(cè)面積S表—表面積C＝2πr

S底＝πr2S側(cè)＝ChS表＝Ch+2S底

V＝S底h＝πr2h

空心圓柱R－外圓半徑r－內(nèi)圓半徑h－高V＝πh(R2-r2)

直圓錐r－底半徑h－高V＝πr2h/3

圓臺(tái)r－上底半徑R－下底半徑h－高V＝πh(R2＋Rr＋r2)/3

球r－半徑d－直徑V＝4/3πr3＝πd2/6

球缺h－球缺高r－球半徑a－球缺底半徑V＝πh(3a2+h2)/6＝πh2(3r-h)/3

a2＝h(2r-h)

球臺(tái)r1和r2－球臺(tái)上、下底半徑h－高V＝πh[3(r12＋r22)+h2]/6

圓環(huán)體R－環(huán)體半徑D－環(huán)體直徑r－環(huán)體截面半徑d－環(huán)體截面直徑V＝2π2Rr2＝π2Dd2/4

桶狀體D－桶腹直徑d－桶底直徑h－桶高V＝πh(2D2＋d2)/12(母線(xiàn)是圓弧形,圓心是桶的中心)

V＝πh(2D2＋Dd＋3d2/4)/15(母線(xiàn)是拋物線(xiàn)形)

冠詞練習(xí)題冠詞練習(xí)題

/冠詞練習(xí)題冠詞專(zhuān)項(xiàng)復(fù)習(xí)?題組訓(xùn)練一單項(xiàng)選擇?(??)1.?Last?night?I?sawUFO?flying?in?the?sky?and?took?pictures?of?it.???A.?a???????????B.?an???????????C.?the?????????D.?不填?(??)2.?Tim?had?tried?twice,?and?his?father?asks?him?to?have??third?try.?A.?the????????????B.??a??????C.?another???????D.?/?(??)3.?Many?students?in?our?school?can?play?________?guitar.?A.?the??????B.?a?????C.?不填?(??)4.?Liu?Changchun?is?_________?first?Chinese?to?join?the?Olympics.?A.?a?????????B.?an???????C.?the????????D.?不填?(??)5.?I?think?it's?_________?useful?advice,?so?I?will?always?remember?it.?A.?a???????????B.?an?????????C.?the?????????D.?/?(??)6.?—Will?you?get?there?by?_______?train??—No,?I'll?take?_______?taxi.??A.?/;?a???????B.?a;?the???????C.?/;?/??????D.?the;?a?(??)7.?I?read?_____?story.?It?is?_____?interesting?story.?A.?a;?an???B.?a;?a???C.?the;?the???D.?/;?an?(??)8.?We?always?have?_____?rice?for?______?lunch.?A.?/;?/???B.?the;?/???C.?/;?a????D.?/;?the?(??)9.?______?train?is?faster?than?_______?bus.?A.?A;?a???B.?The;?/????C.?A;?the????D.?/;?the?(??)10.?January?is?______?first?month?of?the?year.?A.?a?????B.?/???C.?the????D.?/?(??)11.?Einstein?won?______?Nobel?Prize?in?Physics?in?1921.?A.?a???B.?an???C.?the???D.?/?(??)12.?I?like?music,?but?I?don't?like?______?music?of?that?TV?play.?A.?a???B.?an??C.?the???D.?/?(??)13.?______?girl?over?there?is?______?English?teacher.?A.?The;?an????B.?A;?/???C.?The;?/????D.?A;?a?(??)14.?We?often?have?sports?after?class,?and?I?like?to?play?_____?basketball.?A.?a???B.?an????C.?the???D.?/?(??)15.?More?college?graduates?would?like?to?work?in?____?west?part?of?our?country?___?next?year.?A.?the;?the???B.?/;?/???C.?/;?the???D.?the;?/?(??)16.?We?work?five?days?________?week.?A.?a???B.?an???C.?the?????D.?/?(??)17.?There's?_______?“u”?and?_______?“s”?in?______?word?“use”.?A.?an;a;?the????B.?a;?an;?/???C.?a;?a;?a,??????D.?a;?an;?the?(??)18.?That's?______?useful?book.?I?have?read?it?already.A.?the??B.?an??C.?a????D.?/?(??)19.?______?Whites?have?moved?to?a?new?house.?A.?A?????B.?An?????C.?The????D.?/?(??)20.?Avatar?is?such?_______?wonderful?science?fiction?movie?that?I?want?to?see?it?_____?second?time.?A.?a;?a????B.?a;?the???C.?/;?the????D.?/;?a題組訓(xùn)練二單項(xiàng)選擇?(??)1.?I?got?the?news?on?______?radio.?A.?/?????????B.?a??????????C.?an????????D.?the?(??)2.?—Who?is?_____?man?in?a?black?hat??It's?Jim?Brown.?We?often?play?_____?basketball?together?in?our?free?time.?A.?an;?a??B.?a;?/??C.?the;?a??D.?the;?/?(??)3.?______?exciting?news!?We're?never?had?_______?long?vacation?before.???A.?What;?such?a???B.?What?an;?such?a?????C.?How;?such?a??????D.?What;?so?a?(??)4.?What?_______?interesting?game!?Let?me?have?______?try,?OK??A.?an;?a??????????B.?a;?an??????C.?/;?a??????????D.?an;?/?(??)5.?—What?do?you?think?of?the?cartoon?????—Oh,?it?is?really?____?fantastic?one.????A.?a???B.?an????C.?the?????D.?/?(??)6.?________?sad?news?it?is!?We?must?try?our?best?to?help?them?out?of?trouble.????A.?What?an????????B.?What????????C.?How?an????????D.?How?(??)7.?—What?can?I?do?for?you????—I?want?______?MP?4?player?for?my?daughter.???A.?a????????????B.?an?????????????C.?the??????????D.不填?(??)8.—I?hear?there’ll?be?_______?talk?on?teenager?problems?next?Monday.????—Do?you?mean?________?talk?our?teacher?asked?us?to?listen?to?????A.?a;?the?????B.?a;?a??????C.?the;?the??????????D.?the;?a?(??)9.?—What?was?Marc?doing?when?his?mother?got?home?yesterday????—He?was?playing?______?guitar.???A.?the????B.?a?????C.?an??????D.?/?(??)10.?Look,?Simon,?______?Walkman?that?I?bought?last?year?isn't?working?properly.???A.?a????????????B.?an???????????C.?the??????????D.?不填?(??)11.?—Did?you?see?________?dog?just?now?????—Is?it?______?black?dog??????A.?the;?a???B.?a;?a????????C.?the;?不填?(??)12.?He?is?______?kind?man.?He?often?gives?money?to?charity.????A.?a????B.?an??C.?the???D.?/?(??)13.?—Do?you?know?________?woman?in?pink?over?there????—Yes.?She?is?_______?office?worker?of?Lijing?Hotel.???A.?a;?an???B.?the;?the??????C.?a;?the???D.?the;?an?(??)14.?Don't?talk?to?Tom?like?that.?He?is?only?_______?eleven-year-old?boy.???A.?a?????B.?an???C.?the????D.?不填?(??)15.?—Why?did?you?laugh?just?now??Ted?wanted?to?tell?us?________?very?funny?story,?but?he?forgot?_________?end?himself.?A.?/;?the?????????B.?a;?an????????C.?a;?the?????????D.?the;?the題組訓(xùn)練三（與中考鏈接）?一、單項(xiàng)選擇?【中考預(yù)測(cè)】?(??)1.?There?is?________?“s”?in?the?word?“six”?and?______?“s”?is?the?first?letter?of?the?word.?A.?a;?the??????B.?a;?an????????C.?an;?the?????D.?a;?a?(??)2.?—Is?it?_______?English?dictionary??—Yes,?and?it?is?______?us