毕业论文：隐函数定理及其应用

齐齐哈尔大学毕业设计（论文）-I-摘要隐函数定理是数学分析和高等数学中的一个重要定理，它不仅是数学分析和高等代数中许多问题的理论基础，并且它也为许多数学分支，如泛函分析、常微分方程、微分几何等的进一步研究提供了坚实的理论依据.隐函数定理有着十分广泛的应用，在经济学、优化理论、条件极值等中均有重要作用.对本课题的研究，可以加深我们对微分学的认识与理解.本文简略地论述了隐函数的概念、隐函数定理的内容及证明方法、以及隐函数定理在各个方面的应用.本文从隐函数定理出发，给出了推论隐函数组定理和反函数组定理以及他们的证明过程.这些推论使隐函数定理的应用更加广泛.并针对隐函数定理在计算导数和偏导数、几何应用、条件极值、以及优化理论这几个方面的应用做了系统的论述.关键词：隐函数定理；应用；优化理论；证明齐齐哈尔大学毕业设计（论文）-II-AbstractImplicitfunctiontheoremofmathematicalanalysisandhighermathematicsisoneoftheimportanttheorem,itisnotonlythemathematicalanalysisandhigheralgebrainthetheoreticalfoundationofthemany,anditalsoformanybranchesofmathematics,suchasfunctionalanalysis,ordinarydifferentialequation,differentialseveralfurtherresearchhowtoprovidethesolidtheoreticalbasis.Implicitfunctiontheoremhasaverywiderangeofapplication,ineconomics,optimizationtheory,suchasextremeconditionswhichisanimportantrole.Thistopicresearch,candeepenourunderstandingofthedifferentialcalculusandunderstanding.Thispaperbrieflydiscussestheconceptofimplicitfunction,thecontentoftheimplicitfunctiontheoremandprovemethod,andimplicitfunctiontheoreminallaspectsoftheapplication.Thispaper,fromtheimplicitfunctiontheoremaregiven,andthecorollaryofimplicitfunctiontheoremandthegroupFanHanShugrouptheoremandproofoftheirprocess.Theseclaimsthattheapplicationofimplicitfunctiontheoremandmoreextensive.Andinthelightofimplicitfunctiontheoreminthecalculationofthederivativeandpartialderivative,geometricapplication,conditionalextreme,andtheseveralaspectsoptimizationtheoryoftheapplicationofthesystemisalsodiscussedinthepaper.Keywords:implicitfunctiontheorem;Application;Optimizationtheory;proof齐齐哈尔大学毕业设计（论文）-III-目录摘要...........................................................................................................................................IAbstract....................................................................................................................................II绪论...........................................................................................................................................1第1章隐函数.......................................................................................................................21.1隐函数.......................................................................................................................21.2隐函数组的概念.......................................................................................................21.3反函数组的概念.......................................................................................................3第2章隐函数定理...............................................................................................................42.1隐函数定理...............................................................................................................42.2隐函数组定理...........................................................................................................62.3反函数组定理...........................................................................................................7第3章隐函数定理的应用...................................................................................................93.1计算导数和偏导数...................................................................................................93.1.1隐函数的导数...................................................................................................93.1.2隐函数组的导数...............................................................................................93.1.3对数求导法.....................................................................................................103.1.4由参数方程所确定的函数的导数.................................................................103.2几何应用.................................................................................................................113.2.1空间曲线的切线与法平面.............................................................................113.2.2空间曲面的切平面与法线.............................................................................143.3条件极值.................................................................................................................153.3.1无条件极值.....................................................................................................153.3.2拉格朗日乘数法.............................................................................................163.4最优化问题.............................................................................................................183.4.1无约束最优化问题.........................................................................................183.4.2约束最优化问题.............................................................................................19结论.........................................................................................................................................21参考文献.................................................................................................................................22致谢...................................................................................................