Syllabus-of-Advanced-Mathematics(II)

1SyllabusofAdvancedMathematics（II）Code:AdvancedMathematics:130701X2Credits:5Credithours:80Prerequisites:ElementaryMathematicsSuitableFor：EngineeringStudents一、课程简介本门课程是我校工科类各专业学生必修的一门重要的基础课，也是培养学生理性思维和创造能力的重要载体。课程系统地介绍了多元函数微积分与无穷级数的基本概念、必要的基础理论和常用的运算方法。其内容主要包括：多元函数微分学、曲线积分、曲面积分、无穷级数等。CourseDescriptionAsanimportantcompulsorybasiccourseofvariouskindsofengineeringstudentsinouruniversity,CalculusIistodevelopstudents’abilityofrationalthoughtandinnovation.Itprovidesthebasicconceptsofmulti-variablecalculusandpowerseriesandnecessarytheoreticalfoundationandcommoncomputationalmethods.Thiscourseconsistsofdifferentialofmulti-variablefunctions,multipleintegrals,curveintegrals,surfaceintegralsandpowerseries.二、课程目标通过本门课程的学习，要使学生掌握微积分的基本理论和基本方法，为学习后续课程和进一步获取数学知识奠定基础。同时，通过各个教学环节逐步培养学生的抽象概括问题的能力、逻辑推理能力、空间想象能力、创造性思维能力和自学能力，还要特别注意培养学生具有比较熟练的运算能力和综合运用所学数学知识分析问题和解决问题的能力。CourseGoal2Theaimofthiscourseistoenablestudentstomasterthebasictheoriesandmethodologiesofcalculus,andsettlefoundationforfurtherstudyoffollow-upcoursesandmathematicalknowledge.Meanwhile,throughvariousteachinglink,inordertodevelopthestudents'abilitytoabstractproblems,logicalreasoningability,spaceimaginationability,creativethinkingabilityandself-educatedabilitiesgradually.Especially,aimtocultivatestudentsformoreskilledoperationabilityandtheintegratedapplicationofmathematicalknowledgetoanalyzeandsolveproblems.三、课程内容和要求CourseContentsandRequirements（一）多元函数微分学(1)二元函数、多元函数的概念.(2)二元函数的极限、连续性、有界闭区域上连续函数的性质.(3)二元函数偏导数与全微分的概念，全微分存在的必要条件与充分条件.(4)一元向量值函数及其导数的概念，计算方法.(5)复合函数一阶偏导数、二阶偏导数的求法.(6)隐函数(方程组确定的隐函数)的一阶偏导数、二阶偏导数方法.(7)二元函数极值、条件极值的概念、求二元函数的极值方法，求条件极值的拉格朗日乘数法，求最大值、最小值的应用问题.MultivariateFunctionDifferential(1)Conceptsofbinaryfunction,multivariatefunction.(2)Thelimitandcontinuityofbinaryfunction,propertiesofcontinuousfunctionsonboundedinterval.(3)Conceptsofbinaryfunctionpartialderivativeandtotaldifferential,thenecessaryandsufficientconditionsfortheexistenceoftotaldifferential.(4)Theconceptandcalculationmethodoftheunaryvectorvaluefunctionanditsderivatives.(5)Methodofsolvingthefirstorderpartialderivativeandsecondorderpartialderivativeofcompositefunction.(6)Implicitfunction(theimplicitfunctiondefinedbytheequationset),thesecond-orderpartialderivativemethod.(7)Theconceptofbinaryfunctionextremevalue,conditionalextremevalue,methodforsolvingbinaryfunctionextremevalue,Lagrangianmultipliermethodforsolvingtheconditionalextremevalue,theapplicationofthemaximumvalueandtheminimumvalue.3（二）多元数量值函数积分学(1)二重积分，三重积分，重积分的性质.(2)二重积分、三重积分的计算（直角坐标，极坐标.(3)第一类曲线积分，第一类曲面积分的概念、性质、计算方法.(4)科学问题中建立重积分与曲线、曲面积分的元素法（微元法.MultipleValueFunctionIntegral(1)Thenaturalofdoubleintegral,tripleintegral,multipleintegral.(2)Thecalculationofdoubleintegral,tripleintegral(Cartesiancoordinate,polarcoordinate).(3)Thefirsttypeofcurveintegral,theconcept,nature,calculationmethodofthefirsttypeofsurfaceintegral.(4)Elementmethodofestablishingmultipleintegral,curveintegralandsurfaceintegralinscientificissues(micro-elementmethod).（三）多元向量值函数积分学（1）第二类曲线积分的概念、性质、计算方法.（2）格林（Green）公式、平面曲线积分及第二类曲线积分与路径无关的条件.（3）全微分方程的概念、解法.（4）第二类曲面积分的概念、性质、计算方法.（5）高斯（Gauss）公式,斯诺克（Stokes）公式.（6）简单物理量积分表达式建立的思想.MultivariateVectorValueFunctionIntegral(1)Theconcept,propertiesandcalculationmethodofthesecondtypeofcurveintegral.(2)Greenformula,planecurveintegral,thesecondtypeofcurveintegralandpath-independentconditions.(3)Theconceptandsolutionsoftotaldifferentialequations.(4)Theconcept,propertiesandcalculationmethodofthesecondtypeofsurfaceintegral.(5)Gaussformula,Strokesformula.(6)Ideaestablishedbysimplephysicalvolumeintegralexpression.(四)无穷级数(1)无穷级数的性质，以及收敛、发散、和的概念.(2)正项级数的比较审敛法以及几何级数、p−级数的敛散性.(3)交错级数的莱布尼茨定理，估计交错级数的截断误差的方法绝对收敛、条件收敛的关系.4(4)函数项级数的收敛域，幂级数收敛区间的求法，幂级数在收敛区间内的性质.(5)ex、sinx、cosx、ln(1x)与麦克劳林（Maclaurin）展开式将函数展开成幂级数.(6)傅里叶（Fourier）级数，正弦余弦函数.InfiniteSeries(1)Thepropertyofinfiniteseries,aswellastheconceptofconvergence,divergence,andsum.(2)Thecomparisonconvergencetestofthepositiveseriesandtheconvergenceofgeometricprogression,p-series.(3)Leibniztheoremofthealternateseries,themethodofestimatingthetruncationerrorofthealternateseries,therelationshipbetweentheabsolutelyconvergenceandconditionalconvergence.(4)Theconvergencerangeoffunctionseries,solutionofpowerseriesinconvergenceinterval.(5)Expandfunctionintopowerseriesbyex,sinx,cosx,ln(1+x)andMaclaurinexpansion.(6)Fourierseries,sinecosinefunction.