[euler] foundations of differential calculus

《[euler] foundations of differential calculus》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、FoundationsofDifferentialCalculusEulerSpringerPrefaceWhatdifferentialcalculus,and,ingeneral,analysisoftheinfinite,mightbecanhardlybeexplainedtothoseinnocentofanyknowledgeofit.Norcanwehereofferadefinitionatthebeginningofthisdissertationasissometimesdoneinotherdisciplines.It

2、isnotthatthereisnocleardefinitionofthiscalculus;rather,thefactisthatinordertounderstandthedefinitionthereareconceptsthatmustfirstbeunderstood.Besidesthoseideasincommonusage,therearealsoothersfromfiniteanalysisthataremuchlesscommonandareusuallyexplainedinthecourseofthedevel

3、opmentofthedifferentialcalculus.Forthisreason,itisnotpossibletounderstandadefinitionbeforeitsprinciplesaresufficientlyclearlyseen.Inthefirstplace,thiscalculusisconcernedwithvariablequantities.Althougheveryquantitycannaturallybeincreasedordecreasedwithoutlimit,still,sincecal

4、culusisdirectedtoacertainpurpose,wethinkofsomequantitiesasbeingconstantlythesamemagnitude,whileotherschangethroughallthestagesofincreasinganddecreasing.Wenotethisdistinc-tionandcalltheformerconstantquantitiesandthelattervariables.Thischaracteristicdifferenceisnotrequire

5、dbythenatureofthings,butratherbecauseofthespecialquestionaddressedbythecalculus.Inorderthatthisdifferencebetweenconstantquantitiesandvariablesmightbeclearlyillustrated,letusconsiderashotfiredfromacannonwithachargeofgunpowder.Thisexampleseemstobeespeciallyappropriatetocla

6、rifythismatter.Therearemanyquantitiesinvolvedhere:First,thereisthequantityofgunpowder;then,theangleofelevationofthecannonabovethehorizon;third,thedistancetraveledbytheshot;and,fourth,thelengthviPrefaceoftimetheshotisintheair.Unlessthesamecannonisusedthroughouttheexperi

7、ment,wemustalsobringintoourcalculationsthelengthofthebarrelandtheweightoftheshot.Here,wewillnotconsidervariationsinthecannonortheshot,lestwebecomeentailedinverycomplicatedquestions.Hence,ifwealwayskeepthesamequantityofpowder,theelevationofthebarrelwillvarycontinuouslyw

8、iththedistancetraveledandtheshot’sdurationoftimeintheair.Inthiscase,theamountofpowder,ortheforceoftheexplosion,willbe