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1、微积分词汇第一章函数与极限Chapter1 Function and Limit集合set元素element子集subset空集empty set并集union交集intersection差集difference of set基本集basic set补集complement 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
2、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mappingX 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composite mapping自变量independent variab
3、le因变量dependent variable定义域domain函数值value of function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 valued function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stepcurve当且仅当if and only if(iff)分段函数piecewise
4、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com
5、posite function中间变量intermediate variable函数的运算operation of function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hyperb
6、olic sine双曲余弦hyperbolic cosine双曲正切hyperbolic tangent反双曲正弦inverse hyperbolic sine反双曲余弦inverse hyperbolic cosine反双曲正切inverse hyperbolic tangent名师资料总结-精品资料欢迎下载-名师精心整理-第 1 页，共 8 页 -极限limit数列sequenceof number收敛convergence收敛于a converge to a发散divergent极限的唯一性uniquenessof limits收 敛 数 列 的 有 界 性boundedness of
7、aconvergent sequence子列subsequence函数的极限limits of functions函 数当 x 趋 于 x0 时 的 极 限limitoffunctions asx approachesx0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squeezerule单调数列monotonic sequence高阶无穷小infinitesimal of higher o
8、rder低阶无穷小infinitesimal of lower order同阶无穷小infinitesimal of the sameorder作者：新少年特工2007-10-8 18:37回复此发言-2 高等数学-翻译等阶无穷小equivalent infinitesimal函数的连续性continuity of a function增量increment函数在 x0 连续the function is continuous atx0左连续left continuous右连续right continuous区间上的连续函数continuous function函 数在 该 区 间 上 连 续
9、functioniscontinuous on an interval不连续点discontinuity point第一类间断点discontinuity point of the firstkind第 二 类 间 断 点discontinuitypoint of thesecondkind初 等 函 数 的 连 续 性continuityof theelementary functions定义区间defined interval最 大 值global maximumvalue(absolutemaximum)最 小 值global minimumvalue(absoluteminimum)零
10、点定理the zero point theorem介值定理intermediate value theorem第二章导数与微分Chapter2 Derivativeand Differential速度velocity匀速运动uniform motion平均速度averagevelocity瞬时速度instantaneousvelocity圆的切线tangent line of a circle切线tangent line切线的斜率slope of the tangent line位置函数position function导数derivative可导derivable函数的 变化率问题probl
11、em of the changerate of a 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在闭区间【a,b】上可导is derivable on theclosed interval a,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 functi
12、on二阶函数secondderivative三阶导数third derivative名师资料总结-精品资料欢迎下载-名师精心整理-第 2 页，共 8 页 -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 of function自 变 量 的 微 分differentialof independentvariable微商differe
13、ntial 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Chapter3MeanValueTheoremofDifferentialsandtheApplicationofDerivatives罗马定理Rolles theorem费马引理Fermats lemma拉格朗日中值定理Lagranges meanvaluetheorem驻点stationary point稳定点stable point临界点critical point辅助函数auxiliary fu
14、nction拉格朗日中值公式Lagranges meanvalueformula柯 西 中 值 定 理Cauchy s mean valuetheorem洛必达法则LHospital s Rule0/0 型不定式indeterminate form of type 0/0不定式indeterminate form泰勒中值定理Taylorsmean value theorem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Maclaurin sformula佩亚诺公式Peanoremainder term凹
15、凸性concavity凹向上的concave upward,cancaveup凹向下的，向上凸的concave downward concave down拐点inflection point函数的极值extremum of function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averagecurvature曲率园circ
16、le of curvature曲率中心center of curvature曲率半径radius of curvature渐屈线evolute渐伸线involute根的隔离isolation of root隔离区间isolation interval切线法tangent line method第四章不定积分Chapter4 IndefiniteIntegrals原函数primitive function(antiderivative)积分号sign of integration被积函数integrand积分变量integral variable积分曲线integral curve积分表tabl
17、e of integrals换元积分法integration by substitution分部积分法integration by parts分部积分公式formula of integration by parts有理函数rational function真分式proper fraction假分式improper fraction第五章定积分Chapter5 Definite Integrals曲边梯形trapezoid with曲边curve edge窄矩形narrow rectangle曲 边 梯 形 的 面 积area of trapezoid withcurved edge积分下限l
18、ower limit of integral积分上限upper limit of integral积分区间integral interval名师资料总结-精品资料欢迎下载-名师精心整理-第 3 页，共 8 页 -分割partition积分和integral sum可积integrable矩形法rectangle method积 分 中 值 定 理mean value theorem ofintegrals函数在区间上的平均值average value of afunction on an integvals牛 顿 莱 布 尼 茨 公 式Newton-Leibnizformula微 积 分 基
19、本 公 式fundamental formula ofcalculus换 元 公 式formulaforintegrationbysubstitution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improper integral on aninfinite interval无界函数的反常积分improper integral ofunbounded func
20、tions绝对收敛absolutely convergent第六章定积分的应用Chapter6ApplicationsoftheDefiniteIntegrals元素法the element method面积元素element of area平面图形的面积areaof a luane figure直 角 坐 标 又 称“笛 卡 儿 坐 标(Cartesiancoordinates)”极坐标polar coordinates抛物线parabola椭圆ellipse旋转体的面积volume of a solid of rotation旋转椭球体ellipsoid of revolution,ell
21、ipsoidof rotation曲线的弧长arc length of 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 AnalyticGeometryandVector 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 poeration of vector三角法则triangle rule平行四
22、边形法则parallelogram rule交换律commutative 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 of vector向量 a 与 b 的夹角angle between vector aand b方向余弦direction cosine方向角direction angle向 量 在 轴 上 的 投 影projection o
23、f a vectoronto an axis数 量 积，外 积，叉 积scalar product,dotproduct,inner product曲面方程equation for a surface球面sphere旋转曲面surface of revolution母线generating line轴 axis圆锥面cone顶点vertex旋转单叶双曲面revolution hyperboloids ofone sheet旋转双叶双曲面revolution hyperboloids of名师资料总结-精品资料欢迎下载-名师精心整理-第 4 页，共 8 页 -two sheets柱面cylind
24、rical surface,cylinder圆柱面cylindricalsurface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 of one sheet双叶双曲面hyperboloid of two sheets旋转椭球面ellipsoid of revolution椭圆抛物面elliptic paraboloid旋转抛物面paraboloid of revolution双曲抛物面hyperbolic paraboloid马鞍面saddle
25、surface椭圆柱面elliptic cylinder双曲柱面hyperbolic cylinder抛物柱面parabolic cylinder空间曲线spacecurve空间曲线的一般方程general form equationsof a spacecurve空间曲线的参数方程parametric equationsof a spacecurve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 form eqyationof a plane法向量normal vector平面的一般方程general
26、form equation of aplane两平面的夹角angle between two planes点到平面的距离distance from a point to aplane空间直线的一般方程general equation of aline in space方向向量direction vector直 线 的 点 向 式 方 程pointdirectionformequations of aline方向数direction number直线的参数方程parametric equations of aline两直线的夹角angle between two lines垂直perpendic
27、ular直线与平面的夹角angle between a line anda planes平面束pencil of planes平面束的方程equation of a pencil of planes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Chapter8Differentiationof FunctionsofSeveral Variables and Its Application一元函数function of one variable多元函数function of several variables内点interior
28、 point外点exterior point边界点frontier point,boundary point聚点point of accumulation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 of function ofseveal连续函数continuous function不连续点discontinuity point一
29、致连续uniformly continuous偏导数partial derivative对自变量x 的偏导数partial derivative withrespectto independent variable x高阶偏导数partial derivative of higher order二阶偏导数secondorder partial derivative混合偏导数hybrid partial derivative全微分total differential偏增量oartial increment偏微分partial differential全增量total increment可微分di
30、fferentiable必要条件necessarycondition名师资料总结-精品资料欢迎下载-名师精心整理-第 5 页，共 8 页 -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 of the existence ofimplicit function曲线的切向量tangentvector of a curve法平面normal plane向量方程vector equation向量值函数vector-valued
31、function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 plane to asurface at a point曲线在一点的法线normal line to a surfaceat apoint无条件极值unconditional ex
32、treme values条件极值conditional extreme values拉格朗日乘数法Lagrange multiplier method拉格朗日乘子Lagrange multiplier经验公式empirical formula最小二乘法method of least squares均方误差mean squareerror第九章重积分Chapter9 MultipleIntegrals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
33、元素volume elementin rectangular coordinate 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 spherical coordinate system反常二重积分improper double integral曲面的面积areaof a surface质心centre of mass静矩static moment密度densi
34、ty形心centroid转动惯量moment of inertia参变量parametric variable第十章曲线积分与曲面积分Chapter10Line(Curve)IntegralsandSurface Integrals对 弧 长 的 曲 线 积 分line integrals withrespectto arc hength第 一 类 曲 线 积 分line integrals of the firsttype对 坐 标 的 曲 线 积 分line integrals withrespectto x,y,and z第二类曲线积分line integrals of the seco
35、ndtype有向曲线弧directed arc单连通区域simple connected region复连通区域complex connected region格林公式Green formula第一类曲面积分surface integrals of the firsttype对 面 的 曲 面 积 分surface integrals withrespectto area有向曲面directed surface对 坐 标 的 曲 面 积 分surface integrals withrespectto coordinate elements第 二 类 曲 面 积 分surface integr
36、als of thesecondtype有向曲面元element of directed surface高斯公式gaussformula拉普拉斯算子Laplace operator格林第一公式Green sfirst formula通量flux散度divergence斯托克斯公式Stokes formula环流量circulation名师资料总结-精品资料欢迎下载-名师精心整理-第 6 页，共 8 页 -旋度rotation,curl第十一章无穷级数Chapter11 InfiniteSeries一般项general term部分和partial sum余项remainder term等比级数
37、geometric series几何级数geometric series公比common ratio调和级数harmonic series柯西收敛准则Cauchy convergence criteria,Cauchy criteria for convergence正项级数series of positive 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ser
38、ies of functions发散点point of divergence收敛点point of convergence收敛域convergence domain和函数sum function幂级数power series幂级数的系数coeffcients of power series阿贝尔定理Abel Theorem收敛半径radius of convergence收敛区间interval of convergence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 calculat
39、ion舍入误差round-off error,rounding error欧拉公式Eulers formula魏尔斯特拉丝判别法Weierstrasstest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 component基波fundamental wave二次谐波secondharmonic三角函数系trigonometric function system傅立叶系数Fourier coefficient傅立
40、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complex form ofFourier series第十二章微分方程Chapter12 DifferentialEquation解微分方程solve a dirrerential equation常微分方程ordinary differential equation偏微分方程partial differential equati
41、on,PDE微分方程的阶order of adifferential equation微 分 方 程 的 解solution of a differentialequation微 分 方 程 的 通 解general solutionof adifferential equation初始条件initial condition微 分 方 程 的 特 解particular solution of adifferential equation初值问题initial value problem微 分 方 程 的 积 分 曲 线integral curve of adifferential equat
42、ion可分离变量的微分方程variable separabledifferential equation隐式解implicit solution隐式通解inplicit generalsolution衰变系数decaycoefficient衰变decay齐次方程homogeneousequation一阶线性方程linear differential equation offirst order非齐次non-homogeneous齐次线性方程homogeneouslinear equation非 齐 次 线 性 方 程non-homogeneous linearequation常数变易法meth
43、od of variation of constant暂态电流transient statacurrent名师资料总结-精品资料欢迎下载-名师精心整理-第 7 页，共 8 页 -稳态电流steady statecurrent伯努利方程Bernoulli equation全微分方程total differential equation积分因子integrating factor高阶微分方程differential equation of higherorder悬链线catenary高 阶 线 性 微 分 方 程linera differentialequation of higher order
44、自由振动的微分方程differential equation offree vibration强迫振动的微分方程differential equation offorced oscillation串联电路的振荡方程oscillation equation ofseries circuit二 阶 线 性 微 分 方 程second order lineradifferential equation线性相关linearly dependence线性无关linearly independce二阶常系数齐次线性微分方程secondorderhomogeneour linear differential
45、 equation withconstant coefficient二阶变系数齐次线性微分方程secondorderhomogeneous linear differential equation withvariable coefficient特征方程characteristic equation无 阻 尼 自 由 振 动 的 微 分 方 程differentialequation of free vibration with zero damping固有频率natural frequency简谐振动simple harmonic oscillation,simpleharmonic vib
46、ration微分算子differential operator待 定 系 数 法method ofundeterminedcoefficient共振现象resonancephenomenon欧拉方程Euler equation幂级数解法power series solution数值解法numerial solution勒让德方程Legendre equation微分方程组system of differential equations常 系 数 线 性 微 分 方 程 组system of lineradifferential equationswith constant coefficient名师资料总结-精品资料欢迎下载-名师精心整理-第 8 页，共 8 页 -