1、數學分析中的英文單詞和短語第一章實數集與函數Chapter 1Set 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 in

2、equality伯努利不等式Bernoulli inequality確界原理principles of supremum and infimum開區間closed interval閉區間closed interval半開區間semi-open interval半閉區間semi-closed interval有限區間finite interval無限區間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closenes

3、s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有序完備集實數的完備性全序域完備性公理戴德金分割戴德金性質常量與變量函數的定義定義域值域自變量因變量中間變量單調性初等函數常量函數冪函數指數函數對數函數三角函數反三角函數反函數復合函數映射逆映射像原像分段函數符號函數狄利克雷函數黎曼函數有界函數單調函數單調增函數嚴格單調函數奇（偶）函數最小正周期絕對值函數恒等函數多項式函數線性函數二

4、次函數有理函數order-complete set completeness of real numbers complete ordered field axiom of completeness Dedekind cutDedekind propertyconstant and variable quantities definition of function domainrangeindependent variabledependent variableintermediate variablemonotonicityelementary functionconstant funct

5、ionpower functionexponential functionlogarithmic functiontrigonometric functioninverse trigonometric functioninverse functioncompound functionmappinginverse mappingimageprimary imagepiecewise functionsign functionDirichlet functionRiemann functionbounded functionmonotone functionmonotone increasing

6、functionstrictly monotone functionodd(even) functionminimal positive periodabsolute value functionidentity functionpolynomial functionlinear functionquadratic functionrational 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

7、哥拉斯恒等式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 2Limits of Sequences數列極限limit of sequence發散數列divergent sequence無窮小數列infinitesimal sequence收斂數列convergent sequence唯一性定理uniqueness theorem有界性定理boundednes

8、s 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 critterion第三章函數

9、極限Chapter 3Limits 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) order同階無窮小量

10、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第四章函數的連續性Chapter 4Continuity of Functions自變量的增量increment of independent variable函數的增量increment of function右（左）連續right(left) continuous間斷點及其分類disco

11、ntinuity 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連續函數的局部性質local properties of continuous function連續函數的復合性質composition properties of continuous function閉區間上連續函數的properties of continuous f

12、unction over closed性質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一致連續性定理uniform continuity theorem反函數的連續性continuity of inverse function局部保號性local inheriting order property初等函數的連續性continuity of elementary function第五章導

13、數和微分Chapter 5Derivatives 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) derivative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intermediatevaluetheoremof理derivativ

14、e 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圓的參數方程parametric equation o

15、f 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parametric equation of asteroid程二階導數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可微函數differentiable function

16、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invariance of differential form性微分的幾何意義geometric meaning of differential高階微分higher-order differential第六章微分中值定理及其應用Chapter6MeanValueTheoremsofDifferentials and their Applications羅爾中值定理Rol

17、le 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l Hospittalrule0/0 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Taylor formula with Peano remainder的泰勒公式泰勒系數Taylor c

18、oefficient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TaylorformulawithLagrange項的泰勒公式remainder帶有拉格朗日型余MaclaurinformulawithLagrange項的麥克勞林公式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the first sufficient condition of extreme件value函數的凸性和拐點convexityand in

19、flectionpoint offunction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部分分式分解decomposition into partial fractions琴生不等式Jenson inequality待定系數法method of undetermined coefficients第七章實數的完備性第九章定積分Chapter 7Completeness of Real Numbers實數集完備性的fundamentaltheote

20、msof基本定理completeness inthe set ofrealnumbers區間套定理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 8Indefinite Integrals不定積分indefinite integral原函數primitive function被積函數integrand function積分符號integr

21、al sign積分表達式expression of integrand積分常數integral constant基本積分表table of basic integrals不定積分的幾何geometricmeaningofindefinite意義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indefinite integral

22、of rational function積分真分式proper fraction假分式imiproper fractionChapter 9Definite 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積分區間interval ofintegration上限和下限upper and lower limits定積分的幾何

23、意義geometric meaning ofdefiniteintegral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Newton-Leibniz formula式可積的必要條件necessary condition for integrability可積的充要條件necessary and sufficient conditionsfor integrability可積函數類integrable function class定積分的線性性質linear property of defini

24、te integral積分區間的可加性additive with respect tointegralinterval/additivityoverintegralinterval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 by substitution分部積分法integration by parts泰勒公式的積分型integral form of

25、remainder ofTaylor余項formula第十章定積分的應用Chapter 10Applications of Definite Integrals平面圖形的面積area of plane figure由平行截面面積method of finding volume of a solid求體積的方法fromtheknownareaofparallelsections平面曲線的弧長arc length and curvature of plane curve與曲率光滑曲線smooth curve弧微分differential of arc曲率圓circle of curvature曲率

26、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some applications of definite integral的某些應用in physics引力gravitation功work定積分的近似計approximate computation of definite算integral梯形法trapezoidal method拋物線法parabola method第十一章反常積分Chapter 11Improper Integrals狄利克雷判別法Dirichlet t

27、est阿貝爾判別法Abel test第十二章數項級數Chapter 12Series 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Cauchy onvergence criterion for series準則正項級數series with positive terms收斂的必要條件necessary conditi

28、on for convergence根式判別法root test達朗貝爾判別法Dalembert(ratio) test（比式判別法）比式判別法的極limit form of ratio test限形式高斯判別法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阿貝爾判別法Abel test反常積分的概念無窮區間上的反常積分無界函數

29、的反常積分無窮積分的性質和收斂判別絕對收斂比較判別法柯西判別法notion of improper integralimproper integral on infinite intervalimproper integral of unbounded functionproperty of infinite integral and test of convergenceabsolutely convergentcomparison testCauchy test狄利克雷判別法Dirichlet test第十三章函數列與函數項級數Chapter13Sequences ofFunctionsa

30、ndSeries of Functions在點 x0 收斂convergent at x0收斂域convergencedomain/regionofconvergence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Weierstrasstest/sWeierstrass uniform別法convergencecriterion/Weierstrassm-test for uniform conv

31、ergence一致有界uniform boundedness狄利克雷判別法Dirichlet test正交函數系system of orthogonal functions簡諧振動simple harmonic vibration以 2 為周期的Fourier series for functions of period 2函數的傅里葉級數角頻率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Fourierseriesofevenandodd的傅里葉級數func

32、tions振幅amplitude正弦級數sine series余弦級數cosine series周期延拓periodic extension第十四章冪級數Chapter 14Power Series冪級數power series收斂區間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 series初等函數的冪級expansion of powerseries of數展開式elementary function復變量的指數函expo

33、nential function ofcomplex數variable歐拉公式Euler formula第十五章傅里葉級數Chapter 15Fourier Series傅里葉級數Fourier series三角級數trigonometric series第十六章多元函數的極限與連續Chapter16LimitsandContinuityofFunctions of Several 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 p

34、oint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 point 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impro

35、per limit二重極限double limit有界閉域上連續properties of continuous functions on函數的性質bounded closed region累次極限repeated limits全增量total increment偏增量partial increment第十八章隱函數定理及其應用Chapter 18 Implicit Funciton Theorems and their Applications第十七章多元函數微分學Chapter17DifferentialCalculusofFunctions of Several Variables全微分

36、total differential偏導數partial derivative連續可微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隱函數存在唯一性定理隱函數組函數行列式（雅可比行列式）反函數組坐標變換幾何應用平面曲線的切線與法線空間曲線的切線與法平面條件極值拉格朗日乘數法existence and uniqueness theorem of implicit fucntionssyst

37、em of implicit functionsfunctional determinant (Jacobian determinant)system of inverse functionscoordinate transformationgeometrical applicationtangent line and normal line of plane curvetangent line and normal plane of space curveconditional extremumLagrange multiplier method多元函數的鏈式chain rule for f

38、ucntions of several法則variables一階微分形式不invariance of differential form of first變性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第十九章含參量積分Chapter 19Integrals with Parameters含參量的正常積proper integral with parameter分含參量的反常積improper integral with parameter分歐拉積分Euler integral維爾斯特拉斯Weierstrass (m) test（M）判別法狄利