数学及应用数学专业外文翻译--多元函数的极值

《数学及应用数学专业外文翻译--多元函数的极值》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。

1、外文原文EXTREMEVALUESOFFUNCTIONSOFSEVERALREALVARIABLES1.StationaryPointsDefinition1.1Letand.Thepointaissaidtobe:(1)alocalmaximumifforallpointssufficientlycloseto;(2)alocalminimumifforallpointssufficientlycloseto;(3)aglobal(orabsolute)maximumifforallpoints;(4)ag

2、lobal(orabsolute)minimumifforallpoints;;(5)alocalorglobalextremumifitisalocalorglobalmaximumorminimum.Definition1.2Letand.Thepointaissaidtobecriticalorstationarypointifandasingularpointifdoesnotexistat.Fact1.3Letand.Ifhasalocalorglobalextremumatthepoint,the

3、nmustbeeither:(1)acriticalpointof,or(2)asingularpointof,or(3)aboundarypointof.Fact1.4Ifisacontinuousfunctiononaclosedboundedsetthenisboundedandattainsitsbounds.Definition1.5Acriticalpointwhichisneitheralocalmaximumnorminimumiscalledasaddlepoint.Fact1.6Acrit

4、icalpointisasaddlepointifandonlyiftherearearbitrarilysmallvaluesofforwhichtakesbothpositiveandnegativevalues.Definition1.7Ifisafunctionoftwovariablessuchthatallsecondorderpartialderivativesexistatthepoint,thentheHessianmatrixofatisthematrixwherethederivativ

5、esareevaluatedat.Ifisafunctionofthreevariablessuchthatallsecondorderpartialderivativesexistatthepoint,thentheHessianoffatisthematrixwherethederivativesareevaluatedat.Definition1.8Letbeanmatrixand,foreach,letbethematrixformedfromthefirstrowsandcolumnsof.Thed

6、eterminantsdet(),,arecalledtheleadingminorsofTheorem1.9(TheLeadingMinorTest).SupposethatisasufficientlysmoothfunctionoftwovariableswithacriticalpointatandHtheHessianofat.If,thenis:(1)alocalmaximumif0>det(H1)=fxxand0

7、d0det(H1),0det(H3);(2)alocalminimum

8、if0det(H3);(3)asaddlepointifneitheroftheabovehold.wherethepartialderivativesareevaluatedat.KeyPoints.·Acontinuousfunctiononaclosedboundedsetisboundedandachievesitsbounds.·Tofindt