高等数学(微积分学)专业术语名词、概念、定理等英汉对照.doc

高等数学(微积分学)专业术语名词、概念、定理等英汉对照 目目录第一部分英汉微积分词汇Part1English-Chinese CalculusVocabulary第一章函数与极限Chapter1function andLimit1第二章导数与微分Chapter2Derivative andDifferential2第三章微分中值定理Chapter3Mean Valuetheorem ofdifferentials and the Application of Derivatives3第四章不定积分Chapter4Indefinite Integrals3第五章定积分Chapter5Definite Integral3第六章定积分的应用Chapter6Application of the Definite Integrals4第七章空间解析几何与向量代数Chapter7Space AnalyticGeometry andVector Algebra4第八章多元函数微分法及其应用Chapter8Differentiation offunctions Severalvariables and Its Application5第九章重积分Multiple Integrals6第十章曲线积分与曲面积分Chapter10Line(Curve)Integrals and Surface Integrals6第十一章无穷级数Chapter11Infinite Series6第十二章微分方程Chapter12Differential Equation7第二部分定理定义公式的英文表达Part2English Expressionfor Theorem,Definition andFormula第一章函数与极限Chapter1Function andLimit t191.1映射与函数(Mapping andFunction)191.2数列的极限(Limit of the Sequenceof Number)201.3函数的极限(Limit of Function)211.4无穷小与无穷大(Infinitesimal andInfinity)231.5极限运算法则(Operation Ruleof Limit)241.6极限存在准则两个重要的极限(Rule forthe Existenceof LimitsTwo ImportantLimits)251.7无穷小的比较(The Comparisonof infinitesimal)261.8函数的连续性与间断点(Continuity of Function AndDiscontinuity Points)281.9连续函数的运酸与初等函数的连续性(Operation OfContinuous Functions and Continuityof ElementaryFunctions)281.10闭区间上连续函数的性质(Properties ofContinuous Functionson aClosed Interval)30第二章导数与数分Chapter2Derivative andDifferential312.1导数的概念(The Concept of Derivative)312.2函数的求导法则(Rules forFinding Derivatives)332.3高阶导数(Higher-order Derivatives)342.4隐函数及由参数方程所确定的函数的导数相关变化率(Derivatives ofImplicit Functionsand FunctionsDetermined byParametric Equation and CorrelativeChange Rate)342.5函数的微分(Differential of a Function)35第三章微分中值定理与导数的应用Chapter3Mean Value Theorem of Differentials and the Applicationof Derivatives363.1微分中值定理(The MeanValueTheorem)363.2洛必达法则(LHospitals Rule)383.3泰勒公式(Taylors Formula)413.4函数的单调性和曲线的凹凸性(Monotonicity of FunctionsandConcavity ofCurves)433.5函数的极值与最大最小值(Extrema,Maxima andMinima of Functions)463.6函数图形的描绘(Graphing Functions)493.7曲率(Curvature)503.8方程的近似解(Solving Equation Numerically)53第四章不定积分Chapter4Indefinite Integrals544.1不定积分的概念与性质(The Conceptand Properties of Indefinite Integrals)544.2换元积分法(Substitution Rulefor IndefiniteIntegrals)564.3分部积分法(Integration by Parts)574.4有理函数的积分(Integration ofRational Functions)58第五章定积分Chapter5Definite Integrals615.1定积分的概念和性质(ConceptofDefinite Integraland itsProperties)615.2微积分基本定理(Fundamental Theoremof Calculus)675.3定积分的换元法和分部积分法(Integration bySubstitution andDefinite IntegralsbyParts)695.4反常积分(Improper Integrals)70第六章定积分的应用Chapter6Applications of the Definite Integrals756.1定积分的元素法(The ElementMethod ofDefiniteIntegra756.2定积分在几何学上的应用(Applications of the DefiniteIntegrals toGeometry)766.3定积分在物理学上的应用(Applications of the DefiniteIntegrals toPhysics)79第七章空间解析几何与向量代数Chapter7Space AnalyticGeometry andVector Algebar807.1向量及其线性运算(Vector and Its LinearOperation)807.2数量积向量积(Dot Productand CrossProduct)867.3曲面及其方程(Surface and Its Equation)897.4空间曲线及其方程(The Curvein Three-space and Its Equation917.5平面及其方程（Plane inSpace andIts Equation)937.6空间直线及其方程（Lines inand TheirEquations)95第八章多元函数微分法及其应用Chapter8Differentiation ofFunctions of Several VariablesandItsApplication998.1多元函数的基本概念（The BasicConcepts ofFunctions ofSeveral Variables)998.2偏导数(Partial Derivative)1028.3全微分(Total Differential)1038.4链式法则（The ChainRule)1048.5隐函数的求导公式(Derivative Formulafor ImplicitFunctions).1048.6多元函数微分学的几何应用(Geometric Applications of Differentiation of Ffunctionsof Severalvariables)1068.7方向导数与梯度(Directional Derivativesand Gradients)1078.8多元函数的极值(Extreme ValueofFunctions ofSeveral Variables)108第九章重积分Chapter9Multiple Integrals1119.1二重积分的概念与性质(The Conceptof DoubleIntegrals andIts Properities)1119.2二重积分的计算法(Evaluation ofdouble Integrals)1149.3三重积分(Triple Integrals)1159.4重积分的应用(Applications ofMultiple Itegrals)120第十章曲线积分与曲面积分Chapte10Line Integrals andSurface Integrals12110.1对弧长的曲线积分(line Intergralswith Respectto ArcLength)12110.2对坐标的曲线积分(Line Integralswith respectto CoordinateVariables)12310.3格林公式及其应用(Greens FormulaandItsApplications)12410.4对面积的曲面积分(Surface Integralswith Respectto Aarea)12610.5对坐标的曲面积分(Surface Integralswith Respectto CoordinateVariables)12810.6高斯公式通量与散度(Gausss FormulaFlux andDivirgence)13010.7斯托克斯公式环流量与旋度(Stokess FormulaCirculation andRotation)131第十一章无穷级数Chapter11Infinite Series13311.1常数项级数的概念与性质(The conceptand Properties of TheConstant series)13311.2常数项级数的审敛法(Test forConvergence of the ConstantSeries)13711.3幂级数(power Series).14311.4函数展开成幂级数(Represent theFunction asPower Series)14811.5函数的幂级数展开式的应用(the Appliacationof thePower Seriesrepresentation of a Function)14811.6函数项级数的一致收敛性及一致收敛级数的基本性质(The UnanimousConvergence of the SeriesofFunctionsandItsproperties)14911.7傅立叶级数（Fourier Series）15211.8一般周期函数的傅立叶级数(Fourier Seriesof PeriodicFunctions)153第十二章微分方程Chapter12Differential Equation15512.1微分方程的基本概念（The Conceptof DifferentialEquation)15512.2可分离变量的微分方程(Separable Differential Equation)15612.3齐次方程(Homogeneous Equation)15612.4一次线性微分方程(Linear Differential Equation of the FirstOrder)15712.5全微分方程(Total Differential Equation)15812.6可降阶的高阶微分方程(Higher-order Differential Equation Turnedto Lower-order DifferentialEquation)15912.7高阶线性微分方程(Linear DifferentialEquation ofHigher Order)15912.8常系数齐次线性微分方程(Homogeneous LinearDifferentialEquationwith ConstantCoefficient)16312.9常系数非齐次线性微分方程(Non HomogeneousDifferentialEquationwith ConstantCoefficient)16412.10欧拉方程(Euler Equation)16412.11微分方程的幂级数解法(Power SeriesSolution toDifferentialEquation)164第三部分常用数学符号的英文表达Part3English ExpressionoftheMathematical Symbolin CommonUse第一部分英汉微积分词汇Part1English-Chinese CalculusVocabulary第一章函数与极限Chapter1Function andLimit集合set元素element子集subset空集empty set并集union交集intersection差集difference ofset基本集basic set补集plement set直积direct product笛卡儿积Cartesian product开区间open interval闭区间closed interval半开区间half openinterval有限区间finite interval区间的长度length of an interval无限区间infinite interval领域neighborhood领域的中心centre of a neighborhood领域的半径radius of a neighborhood左领域left neighborhood右领域right neighborhood映射mapping X到Y的映射mapping of X ontoY满射surjection单射injection一一映射one-to-one mapping双射bijection算子operator变化transformation函数function逆映射inverse mapping复合映射posite mapping自变量independent variable因变量dependent variable定义域domain函数值value offunction函数关系function relation值域range自然定义域natural domain单值函数single valued function多值函数multiple valuedfunction单值分支one-valued branch函数图形graph of a function绝对值函数absolute value符号函数sigh function整数部分integral part阶梯曲线step curve当且仅当if andonly if(iff)分段函数piecewise function上界upper bound下界lower bound有界boundedness无界unbounded函数的单调性monotonicity of a function单调增加的increasing单调减少的decreasing单调函数monotone function函数的奇偶性parity(odevity)of a function对称symmetry偶函数even function奇函数odd function函数的周期性periodicity of a function周期period反函数inverse function直接函数direct function复合函数posite function中间变量intermediate variable函数的运算operation offunction基本初等函数basic elementary function初等函数elementary function幂函数power function指数函数exponential function对数函数logarithmic function三角函数trigonometric function反三角函数inverse trigonometric function常数函数constant function双曲函数hyperbolic function双曲正弦hyperbolic sine双曲余弦hyperbolic cosine双曲正切hyperbolic tangent反双曲正弦inverse hyperbolicsine反双曲余弦inverse hyperboliosine反双曲正切inverse hyperbolictangent极限limit数列sequence of number收敛convergence收敛于a convergeto a发散divergent极限的唯一性uniqueness oflimits收敛数列的有界性boundedness of a convergentsequence子列subsequence函数的极限limits offunctions函数()f x当x趋于x0时的极限limit offunctions()f xas xapproaches x0左极限left limit右极限right limit单侧极限one-sided limits水平渐近线horizontal asymptote无穷小infinitesimal无穷大infinity铅直渐近线vertical asymptote夹逼准则squeeze rule单调数列monotonic sequence高阶无穷小infinitesimal of higher order低阶无穷小infinitesimal oflower order同阶无穷小infinitesimal ofthe sameorder等阶无穷小equivalent infinitesimal函数的连续性continuity ofa function增量increment函数()f x在x0连续the function()f xis continuousat x0左连续left continuous右连续right continuous区间上的连续函数continuous function函数()f x在该区间上连续function()f xis continuouson aninterval不连续点discontinuity point第一类间断点discontinuity point ofthefirst kind第二类间断点discontinuity point ofthesecond kind初等函数的连续性continuity ofthe elementary functions定义区间defined interval最大值global maximumvalue(absolute maximum)最小值global minimumvalue(absolute minimum)零点定理the zeropoint theorem介值定理intermediate value theorem第二章导数与微分Chapter2Derivative andDifferential速度velocity匀速运动uniform motion平均速度average velocity瞬时速度instantaneous velocity圆的切线tangent lineofacircle切线tangent line切线的斜率slope ofthe tangent line位置函数position function导数derivative可导derivable函数的变化率问题problem ofthe changerate ofa function导函数derived function左导数left-hand derivative右导数right-hand derivative单侧导数one-sided derivatives()f x在闭区间【a,b】上可导()f xis derivableon theclosed intervala,b切线方程tangent equation角速度angular velocity成本函数cost function边际成本marginal cost链式法则chain rule隐函数implicit function显函数explicit function二阶函数second derivative三阶导数third derivative高阶导数nth derivative莱布尼茨公式Leibniz formula对数求导法log-derivative参数方程parametric equation相关变化率correlative changerata微分differential可微的differentiable函数的微分differential offunction自变量的微分differential ofindependent variable微商differential quotient间接测量误差indirect measurementerror绝对误差absolute error相对误差relative error第三章微分中值定理与导数的应用Chapter3MeanValue TheoremofDifferentialsandtheApplicationofDerivatives罗马定理Rolles theorem费马引理Fermats lemma拉格朗日中值定理Lagranges mean value theorem驻点stationary point稳定点stable point临界点critical point辅助函数auxiliary function拉格朗日中值公式Lagranges meanvalue formula柯西中值定理Cauchys meanvalue theorem洛必达法则LHospitals Rule0/0型不定式indeterminate formof type0/0不定式indeterminate form泰勒中值定理Taylors meanvaluetheorem泰勒公式Taylor formula余项remainder term拉格朗日余项Lagrange remainder term麦克劳林公式Maclaurins formula佩亚诺公式Peano remainderterm凹凸性concavity凹向上的concave upward,cancave up凹向下的，向上凸的concave downwardconcave down拐点inflection point函数的极值extremum offunction极大值local(relative)maximum最大值global(absolute)mximum极小值local(relative)minimum最小值global(absolute)minimum目标函数objective function曲率curvature弧微分arc differential平均曲率average curvature曲率园circle ofcurvature曲率中心center ofcurvature曲率半径radius ofcurvature渐屈线evolute渐伸线involute根的隔离isolation ofroot隔离区间isolation interval切线法tangentlinemethod第四章不定积分Chapter4IndefiniteIntegrals原函数primitive function(antiderivative)积分号sign ofintegration被积函数integrand积分变量integral variable积分曲线integral curve积分表table of integrals换元积分法integration bysubstitution分部积分法integration byparts分部积分公式formula ofintegration byparts有理函数rational function真分式proper fraction假分式improper fraction第五章定积分Chapter5DefiniteIntegrals曲边梯形trapezoid with曲边curve edge窄矩形narrow rectangle曲边梯形的面积area oftrapezoid withcurved edge积分下限lower limitof integral积分上限upper limitofintegral积分区间integral interval分割partition积分和integral sum可积integrable矩形法rectangle method积分中值定理meanvaluetheorem ofintegrals函数在区间上的平均值average valueofafunction onan integvals牛顿莱布尼茨公式Newton-Leibniz formula微积分基本公式fundamental formulaof calculus换元公式formula forintegration bysubstitution递推公式recurrence formula反常积分improper integral反常积分发散the improper integral isdivergent反常积分收敛the improper integral isconvergent无穷限的反常积分improperintegralonaninfinite interval无界函数的反常积分improperintegralof unboundedfunctions绝对收敛absolutely convergent第六章定积分的应用Chapter6Applicationsofthe DefiniteIntegrals元素法the elementmethod面积元素element ofarea平面图形的面积area ofa luanefigure直角坐标又称“笛卡儿坐标(Cartesian coordinates)”极坐标polar coordinates抛物线parabola椭圆ellipse旋转体的面积volume ofa solidof rotation旋转椭球体ellipsoid ofrevolution,ellipsoid ofrotation曲线的弧长arc lengthof acurve可求长的rectifiable光滑smooth功work水压力water pressure引力gravitation变力variable force第七章空间解析几何与向量代数Chapter7Space AnalyticGeometry andVector Algebra向量vector自由向量free vector单位向量unit vector零向量zero vector相等equal平行parallel向量的线性运算linear poerationof vector三角法则triangle rule平行四边形法则parallelogram rule交换律mutative law结合律associative law负向量negative vector差difference分配律distributive law空间直角坐标系space rectangularcoordinates坐标面coordinate plane卦限octant向量的模modulus ofvector向量a与b的夹角angle betweenvector aand b方向余弦direction cosine方向角direction angle向量在轴上的投影projection ofa vectoronto anaxis数量积，外积，叉积scalar product,dot product,inner product曲面方程equation fora surface球面sphere旋转曲面surface ofrevolution母线generating line轴axis圆锥面cone顶点vertex旋转单叶双曲面revolution hyperboloidsof one sheet旋转双叶双曲面revolution hyperboloidsof two sheets柱面cylindrical surface,cylinder圆柱面cylindrical surface准线directrix抛物柱面parabolic cylinder二次曲面quadric surface椭圆锥面dlliptic cone椭球面ellipsoid单叶双曲面hyperboloid ofonesheet双叶双曲面hyperboloid oftwosheets旋转椭球面ellipsoid ofrevolution椭圆抛物面elliptic paraboloid旋转抛物面paraboloid ofrevolution双曲抛物面hyperbolic paraboloid马鞍面saddle surface椭圆柱面elliptic cylinder双曲柱面hyperbolic cylinder抛物柱面parabolic cylinder空间曲线space curve空间曲线的一般方程general formequations ofa space curve空间曲线的参数方程parametric equations ofaspacecurve螺转线spiral螺矩pitch投影柱面projecting cylinder投影projection平面的点法式方程pointnorm formeqyation ofa plane法向量normal vector平面的一般方程general formequation ofa plane两平面的夹角angle betweentwo planes点到平面的距离distance froma pointto aplane空间直线的一般方程general equation ofa line inspace方向向量direction vector直线的点向式方程pointdirection formequationsofa line方向数direction number直线的参数方程parametric equationsofaline两直线的夹角angle betweentwo lines垂直perpendicular直线与平面的夹角angle betweenalineand aplanes平面束pencil of planes平面束的方程equation ofa pencilofplanes行列式determinant系数行列式coefficient determinant第八章多元函数微分法及其应用Chapter8DifferentiationofFunctionsofSeveralVariablesandItsApplication一元函数function ofone variable多元函数function ofseveral variables内点interior point外点exterior point边界点frontier point,boundary point聚点point ofaumulation开集openset闭集closed set连通集connected set开区域open region闭区域closed region有界集bounded set无界集unbounded setn维空间n-dimentional space二重极限double limit多元函数的连续性continuity offunction ofseveal连续函数continuous function不连续点discontinuity point一致连续uniformly continuous偏导数partial derivative对自变量x的偏导数partial derivativewith respectto independent variable x高阶偏导数partial derivativeof higher order二阶偏导数second orderpartial derivative混合偏导数hybrid partialderivative全微分total differential偏增量oartial increment偏微分partial differential全增量total increment可微分differentiable必要条件necessary condition充分条件sufficient condition叠加原理superpostition principle全导数total derivative中间变量intermediate variable隐函数存在定理theorem ofthe existenceof implicitfunction曲线的切向量tangent vectorofacurve法平面normal plane向量方程vector equation向量值函数vector-valuedfunction切平面tangent plane法线normal line方向导数directional derivative梯度gradient数量场scalar field梯度场gradient field向量场vector field势场potential field引力场gravitational field引力势gravitational potential曲面在一点的切平面tangent plao a surface ata point曲线在一点的法线normal lio asurface ata point无条件极值unconditional extremevalues条件极值conditional extremevalues拉格朗日乘数法Lagrange multipliermethod拉格朗日乘子Lagrange multiplier经验公式empirical formula最小二乘法method ofleast squares均方误差mean squareerror第九章重积分Chapter9Multiple Integrals二重积分double integral可加性additivity累次积分iterated integral体积元素volume element三重积分triple integral直角坐标系中的体积元素volume elementin rectangularcoordinate system柱面坐标cylindrical coordinates柱面坐标系中的体积元素volume elementin cylindricalcoordinate system球面坐标spherical coordinates球面坐标系中的体积元素volume elementin sphericalcoordinate system反常二重积分improper doubleintegral曲面的面积area ofasurface质心centre ofmass静矩static moment密度density形心centroid转动惯量moment ofinertia参变量parametric variable第十章曲线积分与曲面积分Chapter10Line(Curve)IntegralsandSurfaceIntegrals对弧长的曲线积分line integralswith respectto archength第一类曲线积分line integralsofthefirst type对坐标的曲线积分line integralswith respectto x,y,and z第二类曲线积分line integralsofthesecond type有向曲线弧directed arc单连通区域simple connectedregion复连通区域plex connectedregion格林公式Green formula第一类曲面积分surface integralsofthefirst type对面的曲面积分surface integralswith respectto area有向曲面directed surface对坐标的曲面积分surface integralswith respectto coordinateelements第二类曲面积分surface integralsofthesecond type有向曲面元element ofdirected surface高斯公式gauss formula拉普拉斯算子Laplace operator格林第一公式Greens firstformula通量flux散度divergence斯托克斯公式Stokes formula环流量circulation旋度rotation,curl第十一章无穷级数Chapter11Infinite Series一般项general term部分和partial sum余项remainderterm等比级数geometric series几何级数geometric series公比mon ratio调和级数harmonic series柯西收敛准则Cauchy convergencecriteria,Cauchy criteriafor convergence正项级数series ofpositive terms达朗贝尔判别法DAlembert test柯西判别法Cauchy test交错级数alternating series绝对收敛absolutely convergent条件收敛conditionally convergent柯西乘积Cauchy product函数项级数series offunctions发散点pointofdivergence收敛点pointofconvergence收敛域convergence domain和函数sum function幂级数power series幂级数的系数coeffcients ofpower series阿贝尔定理Abel Theorem收敛半径radius ofconvergence收敛区间interval ofconvergence泰勒级数Taylor series麦克劳林级数Maclaurin series二项展开式binomial expansion近似计算approximate calculation舍入误差round-off error,rounding error欧拉公式Eulers formula魏尔斯特拉丝判别法Weierstrass test三角级数trigonometric series振幅amplitude角频率angular frequency初相initial phase矩形波square wave谐波分析harmonic analysis直流分量direct ponent基波fundamental wave二次谐波second harmonic三角函数系trigonometricfunctionsystem傅立叶系数Fourier coefficient傅立叶级数Forrier series周期延拓periodic prolongation正弦级数sine series余弦级数cosine series奇延拓odd prolongation偶延拓even prolongation傅立叶级数的复数形式plex formof Fourierseries第十二章微分方程Chapter12DifferentialEquatio