数学分析中的英文单词和短语

1、数学分析中的英文单词和短语 第一章 实数集与函数Chapter 1 Set of Real Numbers and Functions实数及其性质real numbe and its properties有理数rational number无理数irrational number定义definition命题proposition加plus减minus乘multiplied by/times除over/is to/divided by绝对值与不等式absolute value and inequality三角不等式triangle inequality反三角不等式inverse triangle

2、 inequality伯努利不等式Bernoulli inequality确界原理principles of supremum and infimum开区间closed interval闭区间closed interval半开区间semi-open interval半闭区间semi-closed interval有限区间finite interval无限区间infinite interval邻域neighborhood去心邻域deleted neighborhood和sum差difference积product商quotient数轴number axis/number line封闭性close

3、ness阿基米德性Archimedean property稠密性density上界与下届upper and lower bounds有界集bounded set无界集unbounded set存在域existence domain上确界supremum下确界infimum有序完备集order-complete set实数的完备性completeness of real numbers全序域complete ordered field完备性公理axiom of completeness戴德金分割Dedekind cut戴德金性质Dedekind property常量与变量constant and

4、 variable quantities 函数的定义definition of function定义域domain值域range自变量independent variable因变量dependent variable中间变量intermediate variable 单调性monotonicity初等函数elementary function常量函数constant 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 tri

5、gonometric function反函数inverse function复合函数compound function映射mapping逆映射inverse mapping像image原像primary image分段函数piecewise function符号函数sign function狄利克雷函数Dirichlet function黎曼函数Riemann function有界函数bounded function单调函数monotone function单调增函数monotone increasing function严格单调函数strictly monotone function奇（偶）

6、函数odd(even) function最小正周期minimal positive period绝对值函数absolute value function恒等函数identity function多项式函数polynomial function线性函数linear function二次函数quadratic function有理函数rational function双曲正弦hyperbolic sine双曲余弦hyperbolic cosine三角恒等式trigonometric identity奇偶恒等式odd-even identity余函数恒等式cofunction identity毕达哥

7、拉斯恒等式Pythagorean identity半角恒等式half-angle identity积恒等式product identity和恒等式sum identity加法恒等式addition identity倍角恒等式double-angle identity第二章 数列极限Chapter 2 Limits of Sequences数列极限limit of sequence发散数列divergent sequence无穷小数列infinitesimal nfntesml sequence收敛数列convergent sequence唯一性定理uniqueness theorem有界性定理

8、boundedness theorem保序性inheriting order properties保不等式性inheriting inequality子列subsequence严格递增strictly increasing单调递增数列monotone increasing sequence单调递减数列monotone decreasing sequence严格递增strictly decreasing必要条件necessary condition充分条件sufficient condition夹逼定理squeeze principle柯西收敛准则Cauchy convergence critt

9、erion第三章 函数极限Chapter 3 Limits of Functions函数极限limit of function无穷极限infinite limit单侧极限one-sided limit右（左）极限right(left) limit/right(left) hand limit函数极限的性质property of limit of function局部有界性local boundedness海涅定理Heine theorem无穷大量infinity无穷小量的阶order of infinitesimal高（低）阶无穷小量infinitesimal of higher(lower)

10、 order同阶无穷小量infinitesimal of same order等价无穷小量equivalent infinitesimalk-阶无穷小量k-order infinitesimal垂直渐近线vertical asymptote斜渐近线oblique asymptote水平渐近线horizontal asymptote第四章 函数的连续性Chapter 4 Continuity of Functions自变量的增量increment of independent variable函数的增量increment of function右（左）连续right(left) continuo

11、us间断点及其分类discontinuity point and its classification可去间断点removable discontinuity跳跃间断点jump discontinuity第一类间断点discontinuity of the first kind第二类间断点discontinuity of the second kind连续函数的局部性质local properties of continuous function连续函数的复合性质composition properties of continuous function闭区间上连续函数的性质properties

12、 of continuous function over closed interval极值定理extreme value theorem最大值和最小值定理maximum and minimum value theorem介值性定理intermediate value theorem零点定理zero-point theorem一致连续性定理uniform continuity theorem反函数的连续性continuity of inverse function局部保号性local inheriting order property初等函数的连续性continuity of elementa

13、ry function第五章 导数和微分Chapter 5 Derivatives and Differentials有限增量公式finite increment formula变化率rate of change差商difference quotient左（右）导数left(right) derivativedrvtv导函数derivative function可导函数derivable function导数的集合意义geometric meaning of derivative费马定理Fermat theorem达布定理Darboux theorem导函数的介值定理intermediate

14、value theorem of derivative function导数的四则运算algebra of derivatives和的导数derivative of sum差的导数derivative of difference积的导数derivative of product商的导数derivative of quotient反函数的导数derivative of inverse function最大值maximum value最小值minimum value复合函数的导数derivative of composite function对数求导法logarithmic derivative圆

15、的参数方程parametric equation of circle椭圆的参数方程parametric equation of ellipse摆线的参数方程parametric equation of cycloid星形线的参数方程parametric equation of asteroid二阶导数second derivative三阶导数third derivativen阶导数n-th derivative莱布尼茨公式Leibniz formula加速度acceleration物理解释physical interpretation微分的概念concept of differential可微

16、函数differentiable function线性主部linear principal part自变量的微分differential of independent variable微分的运算法则operational rules of differential微分形式的不变性invariance of differential form微分的几何意义geometric meaning of differential高阶微分higher-order differential第六章 微分中值定理及其应用Chapter 6 Mean Value Theorems of Differentials

17、 and their Applications罗尔中值定理Rolle mean value theorem拉格朗日中值定理Lagrange mean value theorem柯西中值定理Cauchy mean value theorem泰勒定理Taylor theorem洛必达法则lHospittal rule0/0型不定式极限limit of indeterminate form of type 0/0/型不定式indeterminate form of type /其它类型不定式other indeterminate forms带有皮亚诺型余项的泰勒公式Taylor formula wi

18、th Peano remainder泰勒系数Taylor coefficient泰勒多项式Taylor polynomial泰勒公式的余项remainder of Taylor formula带有拉格朗日型余项的泰勒公式Taylor formula with Lagrange remainder带有拉格朗日型余项的麦克劳林公式Maclaurin formula with Lagrange remainder函数的极值extreme value of function极值判别test of extreme value极值的第一充分条件the first sufficient condition

19、of extreme value函数的凸性和拐点convexity and inflection point of function凸函数convex function凹函数concave function严格凸函数strictly convex function严格凹函数strictly concave function琴生不等式Jenson inequality第七章 实数的完备性Chapter 7 Completeness of Real Numbers实数集完备性的基本定理fundamental theotems of completeness in the set of real n

20、umbers区间套定理nested interval theorem柯西收敛准则Cauchy convergence criterion聚点定理accumulation theorem有限覆盖定理finite covering theorem上极限和下极限upper and lower limits第八章 不定积分Chapter 8 Indefinite Integrals不定积分indefinite integral原函数primitive function被积函数integrand function积分符号integral sign积分表达式expression of integrand积

21、分常数integral constant基本积分表table of basic integrals不定积分的几何意义geometric meaning of indefinite integral积分曲线integral curve初始条件initial condition分部积分法integration by parts换元积分法integration by substitution第一换元公式formula of substitution of the first kind有理函数的不定积分indefinite integral of rational function真分式proper

22、fraction假分式imiproper fraction部分分式分解decomposition into partial fractions待定系数法method of undetermined coefficients第九章 定积分Chapter 9 Definite Integrals定积分definite integral曲边梯形curvilinear trapezoid分割dividing/partition模norm/modulus黎曼和Riemann sum黎曼积分Riemann integral黎曼可积integrability in the sense of Riemann积

23、分区间interval ofintegration上限和下限upper and lower limits定积分的几何意义geometric meaning of definite integral上和upper sum下和lower sum达布上和Darboux upper sum达布下和Darboux lower sum牛顿-莱布尼茨公式Newton-Leibniz formula可积的必要条件necessary condition for integrability可积的充要条件necessary and sufficient conditions for integrability可积函

24、数类integrable function class定积分的线性性质linear property of definite integral积分区间的可加性additive with respect to integral interval/additivity over integralinterval积分中值定理mean value theorem of integral平均值average value变上限积分integral with variant upper limit原函数的存在性existence of primitive function换元积分法integration b

25、y substitution分部积分法integration by parts泰勒公式的积分型余项integral form of remainder of Taylor formula第十章 定积分的应用Chapter 10 Applications of Definite Integrals平面图形的面积area of plane figure由平行截面面积求体积的方法method of finding volume of a solid from the known area of parallel sections平面曲线的弧长与曲率arc length and curvature o

26、f plane curve光滑曲线smooth curve弧微分differential of arc曲率圆circle of curvature曲率半径radius of curvature曲率中心center of curvature旋转曲面的面积area of revolution surface定积分在物理中的某些应用some applications of definite integral in physics引力gravitation功work定积分的近似计算approximate computation of definite integral梯形法trapezoidal me

27、thod抛物线法parabola method第十一章 反常积分Chapter 11 Improper Integrals反常积分的概念notion of improper integral无穷区间上的反常积分improper integral on infinite interval无界函数的反常积分improper integral of unbounded function无穷积分的性质和收敛判别property of infinite integral and test of convergence绝对收敛absolutely convergent比较判别法comparison tes

28、t柯西判别法Cauchy test狄利克雷判别法Dirichlet test阿贝尔判别法Abel test第十二章 数项级数Chapter 12 Series of Number Terms数项级数series with number terms无穷级数infinite series级数的收敛性convergence of series部分和序列sequence of partial sum几何级数geometric series调和级数harmonic series级数收敛的柯西准则Cauchy onvergence criterion for series正项级数series with p

29、ositive terms收敛的必要条件necessary condition for convergence根式判别法root test达朗贝尔判别法（比式判别法）Dalembert(ratio) test比式判别法的极限形式limit form of ratio test高斯判别法Gauss test积分判别法integral test拉阿伯判别法Raabe testp级数p-series一般项级数series with arbitraty terms交错级数alternating series莱布尼茨判别法Leibnitz test级数的重排rearrangement of series

30、阿贝尔判别法Abel test狄利克雷判别法Dirichlet test第十三章 函数列与函数项级数Chapter 13 Sequences of Functions and Series of Functions在点x0收敛convergent at x0收敛域convergence domain/region of convergence和函数sum function极限函数limit function函数项级数series of functions一致收敛性uniform convergence一致收敛判别法test of uniform convergence维尔斯特拉斯判别法Weie

31、rstrasss test/Weierstrass uniform convergence criterion/Weierstrass m-test for uniform convergence一致有界uniform boundedness狄利克雷判别法Dirichlet test第十四章 幂级数Chapter 14 Power Series幂级数power series收敛区间interval of convergence收敛半径radius of convergence阿贝尔定理Abel theorem幂级数的运算operations of power series泰勒级数taylor

32、series初等函数的幂级数展开式expansion of power series of elementary function复变量的指数函数exponential function of complex variable欧拉公式Euler formula第十五章 傅里叶级数Chapter 15 Fourier Series傅里叶级数Fourier series三角级数trigonometric series正交函数系system of orthogonal functions简谐振动simple harmonic vibration以2为周期的函数的傅里叶级数Fourier series

33、 for functions of period 2角频率angular frequency分段光滑piecewise smooth傅里叶系数Fourier coefficient收敛定理convergence theorem奇函数与偶函数的傅里叶级数Fourier series of even and odd functions振幅amplitude正弦级数sine series余弦级数cosine series周期延拓periodic extension第十六章 多元函数的极限与连续Chapter 16 Limits and Continuity of Functions of Sever

34、al Variavles多元函数functions of several variables平面点集plane point set坐标平面coordinate plane内点interior point外点outer point界点boundary point边界boundary孤立点isolated point开集open set闭集closed set连通性connectedness连通开集connected open set开域open domain(region)闭域closed domain(region)有界点集bounded point set无界点集unbounded poin

35、t set闭域套定理nested closed domain theorem二元函数function of two variablesn-维向量空间n-dimensional vector space非正常极限improper limit二重极限double limit有界闭域上连续函数的性质properties of continuous functions on bounded closed region累次极限repeated limits全增量total increment偏增量partial increment第十七章 多元函数微分学Chapter 17 Differential C

36、alculus of Functions of Several Variables全微分total differential偏导数partial derivative连续可微continuously differentiable曲面的切平面tangent plane of surface曲线的法平面normal plane of curve复合函数微分法differentiation of composite function多元函数的链式法则chain rule for fucntions of several variables一阶微分形式不变性invariance of differen

37、tial form of first order方向导数与梯度directional derivative and gradient可微性differentialbility泰勒公式Taylor formula极值问题problem of extreme value高阶偏导数partial derivative of higher order混合偏导数mixed partial derivative第十八章 隐函数定理及其应用Chapter 18 Implicit Funciton Theorems and their Applications隐函数存在唯一性定理existence and u

38、niqueness theorem of implicit fucntions隐函数组system of implicit functions函数行列式（雅可比行列式）functional determinant (Jacobian determinant)反函数组system of inverse functions坐标变换coordinate transformation几何应用geometrical application平面曲线的切线与法线tangent line and normal line of plane curve空间曲线的切线与法平面tangent line and normal plane of space curve条件极值conditional extremum拉格朗日乘数法Lagrange multiplier method第十九章 含参量积分Chapter 19 Integrals with Parameters含参量的正常积分proper integral with parameter含参量的反常积分improper integral with parameter欧拉积分Euler integral维尔斯特拉斯（M）判别法Weierstrass (m) test狄利克雷判别法Dirichlet