1、微积分第一章函数与极限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 open interval有限区间finite interval区间的长度length of an interval无限区间infinite interval领域
2、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 variabl
3、e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step curve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co
5、mposite 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hyper
6、bolic 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极限limit数列sequence of number收敛convergence收敛于a converge to a发散divergent极限的唯一性uniqueness of limits收敛数列的有界性 boundedness of a convergent sequence 子列 subsequence函数的
7、极限limits of functions函数当 x 趋于 x0 时的极限limit of functions as x approaches 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 of lower orde
8、r同阶无穷小infinitesimal of the same order-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 at x0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 function is continuous on an interval 不连续点 discontinuity point第一类间
9、断点discontinuity point of the first kind第二类间断点discontinuity point of the second kind初等函数的连续性continuity of the elementary functions定义区间defined interval最大值global maximum value (absolute maximum)最小值global minimum value (absolute minimum)零点定理the zero point theorem介值定理intermediate value theorem第二章导数与微分Cha
10、pter2 Derivative and Differential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 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 problem of the change rate of a function 导函数 derived function左导数left
11、-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 the closed 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 function二阶函数second derivative三阶导数third derivative高阶导数nth derivat
12、ive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 change rata微分differential可微的differentiable函数的微分differential of function自变量的微分differential of independent variable微商differential quotient间接测量误差indirect measurement error绝对误差absolute error相对误差relative error第三章微分中值定理与导数
13、的应用Chapter3 MeanValue Theorem of Differentials and the Application of Derivatives 罗马定理 Rolle s theorem费马引理Fermat s lemma拉格朗日中值定理Lagrange s 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Lagrange s mean value formula柯西中值定理Cauchy s mean value theorem洛
14、必达法则L Hospital s Rule0/0 型不定式indeterminate form of type 0/0不定式indeterminate form泰勒中值定理Taylor s mean 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 s formula佩亚诺公式Peano remainder term凹凸性concavity凹向上的concave upward, cancave up凹向下的，向上凸的 concave downward conca
15、ve 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average curvature曲率园circle of curvature曲率中心center of curvature曲率半径radius of curvature渐屈
16、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 Indefinite Integrals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table of integrals换元积分法integration by substitution分部积分法integration
17、 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 with curved edge积分下限lower limit of integral积分上限upper limit of integral积分区间integral inter
18、val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 of integrals函数在区间上的平均值 average value of a function on an integvals 牛顿莱布尼茨公式 Newton-Leibniz formula微积分基本公式fundamental formula of calculus换元公式formula for integration by substitution递推公式recurrence formula反常积分improper i
19、ntegral反常积分发散the improper integral is divergent反常积分收敛the improper integral is convergent无穷限的反常积分improper integral on an infinite interval无界函数的反常积分 improper integral of unbounded functions 绝对收敛 absolutely convergent第六章定积分的应用Chapter6 Applications of the Definite Integrals元素法the element method面积元素eleme
20、nt of area平面图形的面积area of a luane figure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 of a solid of rotation旋转椭球体ellipsoid of revolution, ellipsoid of 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 forc
21、e第七章空间解析几何与向量代数Chapter7 Space Analytic Geometry and Vector 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
22、标系space rectangular coordinates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 a and b方向余弦direction cosine方向角direction angle向量在轴上的投影projection of a vector onto an axis数量积，外积，叉积 scalar product,dot product,inner product 曲面方程 equation for a surface 球面 sphere旋转曲面surface
23、of revolution母线generating line轴 axis圆锥面cone顶点vertex旋转单叶双曲面revolution hyperboloids of one sheet旋转双叶双曲面revolution hyperboloids of 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 of one shee
24、t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 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 form equations of a space curve空间曲线的参数方程paramet
25、ric equations of a space curve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 eqyation of a plane法向量normal vector平面的一般方程general form equation of a plane两平面的夹角angle between two planes点到平面的距离distance from a point to a plane空间直线的一般方程general equation of a line in space方向向量direc
26、tion vector直线的点向式方程pointdirection form equations of a line方向数direction number直线的参数方程parametric equations of a line两直线的夹角angle between two lines垂直perpendicular直线与平面的夹角angle between a line and a 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
27、元函数微分法及其应用Chapter8 Differentiation of Functions of Several Variables and Its Application 一元函数 function of one variable多元函数function of several variables内点interior 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 r
28、egion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 of seveal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 derivative with respect to independent variable x 高阶偏导数 partial derivative
29、 of higher order二阶偏导数second order 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
30、隐函数存在定理 theorem of the existence of implicit function 曲线的切向量 tangent vector of a curve 法平面 normal plane向量方程vector equation向量值函数vector-valued 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
31、引力势gravitational potential曲面在一点的切平面tangent plane to a surface at a point曲线在一点的法线normal line to a surface at a point无条件极值unconditional extreme 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 square
32、 error第九章重积分Chapter9 Multiple 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 element in rectangular coordinate system 柱面坐标 cylindrical coordinates柱面坐标系中的体积元素 volume element in cylindrical coordinate system 球面坐标 spherical coordina
33、tes球面坐标系中的体积元素 volume element in spherical coordinate system 反常二重积分 improper double integral 曲面的面积 area of a surface质心centre of mass静矩static moment密度density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Chapter10 Line(Curve)Integrals and Surface Integrals对弧长的曲线积分line integrals with
34、 respect to arc hength第一类曲线积分line integrals of the first type对坐标的曲线积分line integrals with respect to x,y,and z第二类曲线积分line integrals of the second type有向曲线弧单连通区域复连通区域directed arcsimple connected regioncomplex connected region格林公式Green formula第一类曲面积分surface integrals of the first type对面的曲面积分surface int
35、egrals with respect to area有向曲面directed surface对坐标的曲面积分 surface integrals with respect to coordinate elements 第二类曲面积分 surface integrals of the second type 有向曲面元 element of directed surface高斯公式gauss formula拉普拉斯算子Laplace operator格林第一公式Green s first formula通量flux散度divergence斯托克斯公式Stokes formula环流量circu
36、lation旋度rotation,curl第十一章无穷级数Chapter11 Infinite Series一般项general term部分和partial sum余项remainder term等比级数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D Alembert test柯西判别法Cauchy
37、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 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inte
38、rval of convergence泰勒级数Taylor series麦克劳林级数二项展开式Maclaurin seriesbinomial expansion近似计算approximate calculation舍入误差round-off error,rounding error欧拉公式Euler s 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direc
39、t component基波fundamental wave二次谐波second harmonic三角函数系trigonometric function system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complex form of Fourier series第十二章微分方程Chapter12 Differential Equation
40、解微分方程solve a dirrerential equation常微分方程ordinary differential equation偏微分方程partial differential equation,PDE微分方程的阶order of a differential equation微分方程的解solution of a differential equation微分方程的通解general solution of a differential equation初始条件initial condition微分方程的特解 particular solution of a differential equation 初值问题 initial value problem微分方程的积分曲线 integral curve of a differential equation 可分离变量的微分方程 variable separable differential equation 隐式解 implicit solution隐式通解inplicit general solution衰变系数decay coefficient衰变decay齐次方程homogeneous equation一阶线性方程linear differential equation of
