數學分析中的英文單詞和短語

1、數學分析中的英文單詞和短語 第一章 實數集與函數Chapter 1 Set of Real Numbers and Functions實數及其性質real numbe and its properties有理數rational number無理數irrational number定義definition命題proposition加plus減minus乘multiplied by/times除over/is to/divided by絕對值與不等式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確界原理principles of supremum and infimum開區(qū)間closed interval閉區(qū)間closed interval半開區(qū)間semi-open interval半閉區(qū)間semi-closed interval有限區(qū)間finite interval無限區(qū)間infinite interval鄰域neighborhood去心鄰域deleted neighborhood和sum差difference積product商quotient數軸number axis/number line封閉性close

3、ness阿基米德性Archimedean property稠密性density上界與下屆upper and lower bounds有界集bounded set無界集unbounded set存在域existence domain上確界supremum下確界infimum有序完備集order-complete set實數的完備性completeness of real numbers全序域complete ordered field完備性公理axiom of completeness戴德金分割Dedekind cut戴德金性質Dedekind property常量與變量constant and

4、 variable quantities 函數的定義definition of function定義域domain值域range自變量independent variable因變量dependent variable中間變量intermediate variable 單調性monotonicity初等函數elementary function常量函數constant function冪函數power function指數函數exponential function對數函數logarithmic function三角函數trigonometric function反三角函數inverse tri

5、gonometric function反函數inverse function復合函數compound function映射mapping逆映射inverse mapping像image原像primary image分段函數piecewise function符號函數sign function狄利克雷函數Dirichlet function黎曼函數Riemann function有界函數bounded function單調函數monotone function單調增函數monotone increasing function嚴格單調函數strictly monotone function奇（偶）

6、函數odd(even) function最小正周期minimal positive period絕對值函數absolute value function恒等函數identity function多項式函數polynomial function線性函數linear function二次函數quadratic function有理函數rational function雙曲正弦hyperbolic sine雙曲余弦hyperbolic cosine三角恒等式trigonometric identity奇偶恒等式odd-even identity余函數恒等式cofunction identity畢達哥

7、拉斯恒等式Pythagorean identity半角恒等式half-angle identity積恒等式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 數列極限Chapter 2 Limits of Sequences數列極限limit of sequence發(fā)散數列divergent sequence無窮小數列infinitesimal nfntesml sequence收斂數列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嚴格遞增strictly increasing單調遞增數列monotone increasing sequence單調遞減數列monotone decreasing sequence嚴格遞增strictly decreasing必要條件necessary condition充分條件sufficient condition夾逼定理squeeze principle柯西收斂準則Cauchy convergence critt

9、erion第三章 函數極限Chapter 3 Limits of Functions函數極限limit of function無窮極限infinite limit單側極限one-sided limit右（左）極限right(left) limit/right(left) hand limit函數極限的性質property of limit of function局部有界性local boundedness海涅定理Heine theorem無窮大量infinity無窮小量的階order of infinitesimal高（低）階無窮小量infinitesimal of higher(lower)

10、 order同階無窮小量infinitesimal of same order等價無窮小量equivalent infinitesimalk-階無窮小量k-order infinitesimal垂直漸近線vertical asymptote斜漸近線oblique asymptote水平漸近線horizontal asymptote第四章 函數的連續(xù)性Chapter 4 Continuity of Functions自變量的增量increment of independent variable函數的增量increment of function右（左）連續(xù)right(left) continuo

11、us間斷點及其分類discontinuity point and its classification可去間斷點removable discontinuity跳躍間斷點jump discontinuity第一類間斷點discontinuity of the first kind第二類間斷點discontinuity of the second kind連續(xù)函數的局部性質local properties of continuous function連續(xù)函數的復合性質composition properties of continuous function閉區(qū)間上連續(xù)函數的性質properties

12、 of continuous function over closed interval極值定理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零點定理zero-point theorem一致連續(xù)性定理uniform continuity theorem反函數的連續(xù)性continuity of inverse function局部保號性local inheriting order property初等函數的連續(xù)性continuity of elementa

13、ry function第五章 導數和微分Chapter 5 Derivatives and Differentials有限增量公式finite increment formula變化率rate of change差商difference quotient左（右）導數left(right) derivativedrvtv導函數derivative function可導函數derivable function導數的集合意義geometric meaning of derivative費馬定理Fermat theorem達布定理Darboux theorem導函數的介值定理intermediate

14、value theorem of derivative function導數的四則運算algebra of derivatives和的導數derivative of sum差的導數derivative of difference積的導數derivative of product商的導數derivative of quotient反函數的導數derivative of inverse function最大值maximum value最小值minimum value復合函數的導數derivative of composite function對數求導法logarithmic derivative圓

15、的參數方程parametric equation of circle橢圓的參數方程parametric equation of ellipse擺線的參數方程parametric equation of cycloid星形線的參數方程parametric equation of asteroid二階導數second derivative三階導數third derivativen階導數n-th derivative萊布尼茨公式Leibniz formula加速度acceleration物理解釋physical interpretation微分的概念concept of differential可微

16、函數differentiable function線性主部linear principal part自變量的微分differential of independent variable微分的運算法則operational rules of differential微分形式的不變性invariance of differential form微分的幾何意義geometric meaning of differential高階微分higher-order differential第六章 微分中值定理及其應用Chapter 6 Mean Value Theorems of Differentials

17、 and their Applications羅爾中值定理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洛必達法則lHospittal rule0/0型不定式極限limit of indeterminate form of type 0/0/型不定式indeterminate form of type /其它類型不定式other indeterminate forms帶有皮亞諾型余項的泰勒公式Taylor formula wi

18、th Peano remainder泰勒系數Taylor coefficient泰勒多項式Taylor polynomial泰勒公式的余項remainder of Taylor formula帶有拉格朗日型余項的泰勒公式Taylor formula with Lagrange remainder帶有拉格朗日型余項的麥克勞林公式Maclaurin formula with Lagrange remainder函數的極值extreme value of function極值判別test of extreme value極值的第一充分條件the first sufficient condition

19、of extreme value函數的凸性和拐點convexity and inflection point of function凸函數convex function凹函數concave function嚴格凸函數strictly convex function嚴格凹函數strictly concave function琴生不等式Jenson inequality第七章 實數的完備性Chapter 7 Completeness of Real Numbers實數集完備性的基本定理fundamental theotems of completeness in the set of real n

20、umbers區(qū)間套定理nested interval theorem柯西收斂準則Cauchy convergence criterion聚點定理accumulation theorem有限覆蓋定理finite covering theorem上極限和下極限upper and lower limits第八章 不定積分Chapter 8 Indefinite Integrals不定積分indefinite integral原函數primitive function被積函數integrand function積分符號integral sign積分表達式expression of integrand積

21、分常數integral constant基本積分表table of basic integrals不定積分的幾何意義geometric meaning of indefinite integral積分曲線integral curve初始條件initial condition分部積分法integration by parts換元積分法integration by substitution第一換元公式formula of substitution of the first kind有理函數的不定積分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待定系數法method of undetermined coefficients第九章 定積分Chapter 9 Definite Integrals定積分definite integral曲邊梯形curvilinear trapezoid分割dividing/partition模norm/modulus黎曼和Riemann sum黎曼積分Riemann integral黎曼可積integrability in the sense of Riemann積

23、分區(qū)間interval ofintegration上限和下限upper and lower limits定積分的幾何意義geometric meaning of definite integral上和upper sum下和lower sum達布上和Darboux upper sum達布下和Darboux lower sum牛頓-萊布尼茨公式Newton-Leibniz formula可積的必要條件necessary condition for integrability可積的充要條件necessary and sufficient conditions for integrability可積函

24、數類integrable function class定積分的線性性質linear property of definite integral積分區(qū)間的可加性additive with respect to integral interval/additivity over integralinterval積分中值定理mean value theorem of integral平均值average value變上限積分integral with variant upper limit原函數的存在性existence of primitive function換元積分法integration b

25、y substitution分部積分法integration by parts泰勒公式的積分型余項integral form of remainder of Taylor formula第十章 定積分的應用Chapter 10 Applications of Definite Integrals平面圖形的面積area of plane figure由平行截面面積求體積的方法method of finding volume of a solid from the known area of parallel sections平面曲線的弧長與曲率arc length and curvature o

26、f plane curve光滑曲線smooth curve弧微分differential of arc曲率圓circle of curvature曲率半徑radius of curvature曲率中心center of curvature旋轉曲面的面積area of revolution surface定積分在物理中的某些應用some applications of definite integral in physics引力gravitation功work定積分的近似計算approximate computation of definite integral梯形法trapezoidal me

27、thod拋物線法parabola method第十一章 反常積分Chapter 11 Improper Integrals反常積分的概念notion of improper integral無窮區(qū)間上的反常積分improper integral on infinite interval無界函數的反常積分improper integral of unbounded function無窮積分的性質和收斂判別property of infinite integral and test of convergence絕對收斂absolutely convergent比較判別法comparison tes

28、t柯西判別法Cauchy test狄利克雷判別法Dirichlet test阿貝爾判別法Abel test第十二章 數項級數Chapter 12 Series of Number Terms數項級數series with number terms無窮級數infinite series級數的收斂性convergence of series部分和序列sequence of partial sum幾何級數geometric series調和級數harmonic series級數收斂的柯西準則Cauchy onvergence criterion for series正項級數series with p

29、ositive terms收斂的必要條件necessary condition for convergence根式判別法root test達朗貝爾判別法（比式判別法）Dalembert(ratio) test比式判別法的極限形式limit form of ratio test高斯判別法Gauss test積分判別法integral test拉阿伯判別法Raabe testp級數p-series一般項級數series with arbitraty terms交錯級數alternating series萊布尼茨判別法Leibnitz test級數的重排rearrangement of series

30、阿貝爾判別法Abel test狄利克雷判別法Dirichlet test第十三章 函數列與函數項級數Chapter 13 Sequences of Functions and Series of Functions在點x0收斂convergent at x0收斂域convergence domain/region of convergence和函數sum function極限函數limit function函數項級數series of functions一致收斂性uniform convergence一致收斂判別法test of uniform convergence維爾斯特拉斯判別法Weie

31、rstrasss test/Weierstrass uniform convergence criterion/Weierstrass m-test for uniform convergence一致有界uniform boundedness狄利克雷判別法Dirichlet test第十四章 冪級數Chapter 14 Power Series冪級數power series收斂區(qū)間interval of convergence收斂半徑radius of convergence阿貝爾定理Abel theorem冪級數的運算operations of power series泰勒級數taylor

32、series初等函數的冪級數展開式expansion of power series of elementary function復變量的指數函數exponential function of complex variable歐拉公式Euler formula第十五章 傅里葉級數Chapter 15 Fourier Series傅里葉級數Fourier series三角級數trigonometric series正交函數系system of orthogonal functions簡諧振動simple harmonic vibration以2為周期的函數的傅里葉級數Fourier series

33、 for functions of period 2角頻率angular frequency分段光滑piecewise smooth傅里葉系數Fourier coefficient收斂定理convergence theorem奇函數與偶函數的傅里葉級數Fourier series of even and odd functions振幅amplitude正弦級數sine series余弦級數cosine series周期延拓periodic extension第十六章 多元函數的極限與連續(xù)Chapter 16 Limits and Continuity of Functions of Sever

34、al Variavles多元函數functions of several variables平面點集plane point set坐標平面coordinate plane內點interior point外點outer point界點boundary point邊界boundary孤立點isolated point開集open set閉集closed set連通性connectedness連通開集connected open set開域open domain(region)閉域closed domain(region)有界點集bounded point set無界點集unbounded poin

35、t set閉域套定理nested closed domain theorem二元函數function of two variablesn-維向量空間n-dimensional vector space非正常極限improper limit二重極限double limit有界閉域上連續(xù)函數的性質properties of continuous functions on bounded closed region累次極限repeated limits全增量total increment偏增量partial increment第十七章 多元函數微分學Chapter 17 Differential C

36、alculus of Functions of Several Variables全微分total differential偏導數partial derivative連續(xù)可微continuously differentiable曲面的切平面tangent plane of surface曲線的法平面normal plane of curve復合函數微分法differentiation of composite function多元函數的鏈式法則chain rule for fucntions of several variables一階微分形式不變性invariance of differen

37、tial form of first order方向導數與梯度directional derivative and gradient可微性differentialbility泰勒公式Taylor formula極值問題problem of extreme value高階偏導數partial derivative of higher order混合偏導數mixed partial derivative第十八章 隱函數定理及其應用Chapter 18 Implicit Funciton Theorems and their Applications隱函數存在唯一性定理existence and u

38、niqueness theorem of implicit fucntions隱函數組system of implicit functions函數行列式（雅可比行列式）functional determinant (Jacobian determinant)反函數組system of inverse functions坐標變換coordinate transformation幾何應用geometrical application平面曲線的切線與法線tangent line and normal line of plane curve空間曲線的切線與法平面tangent line and normal plane of space curve條件極值conditional extremum拉格朗日乘數法Lagrange multiplier method第十九章 含參量積分Chapter 19 Integrals with Parameters含參量的正常積分proper integral with parameter含參量的反常積分improper integral with parameter歐拉積分Euler integral維爾斯特拉斯（M）判別法Weierstrass (m) test狄利克雷判別法Dirichlet