數(shù)學(xué)分析中的英文單詞和短語

1、數(shù)學(xué)分析中的英文單詞和短語第一章實數(shù)集與函數(shù)Chapter 1Set of Real Numbers and Functions實數(shù)及其性質(zhì)real numbe and its properties有理數(shù)rational number無理數(shù)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(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(shù)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(shù)的完備性全序域完備性公理戴德金分割戴德金性質(zhì)常量與變量函數(shù)的定義定義域值域自變量因變量中間變量單調(diào)性初等函數(shù)常量函數(shù)冪函數(shù)指數(shù)函數(shù)對數(shù)函數(shù)三角函數(shù)反三角函數(shù)反函數(shù)復(fù)合函數(shù)映射逆映射像原像分段函數(shù)符號函數(shù)狄利克雷函數(shù)黎曼函數(shù)有界函數(shù)單調(diào)函數(shù)單調(diào)增函數(shù)嚴格單調(diào)函數(shù)奇（偶）函數(shù)最小正周期絕對值函數(shù)恒等函數(shù)多項式函數(shù)線性函數(shù)二

4、次函數(shù)有理函數(shù)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(shù)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(shù)列極限Chapter 2Limits of Sequences數(shù)列極限limit of sequence發(fā)散數(shù)列divergent sequence無窮小數(shù)列infinitesimal sequence收斂數(shù)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(diào)遞增數(shù)列monotone increasing sequence單調(diào)遞減數(shù)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(shù)

9、極限Chapter 3Limits of Functions函數(shù)極限limit of function無窮極限infinite limit單側(cè)極限one-sided limit右（左）極限r(nóng)ight(left) limit/right(left) hand limit函數(shù)極限的性質(zhì)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(shù)的連續(xù)性Chapter 4Continuity of Functions自變量的增量increment of independent variable函數(shù)的增量increment of function右（左）連續(xù)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(xù)函數(shù)的局部性質(zhì)local properties of continuous function連續(xù)函數(shù)的復(fù)合性質(zhì)composition properties of continuous function閉區(qū)間上連續(xù)函數(shù)的properties of continuous f

12、unction over closed性質(zhì)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(shù)的連續(xù)性continuity of inverse function局部保號性local inheriting order property初等函數(shù)的連續(xù)性continuity of elementary function第五章導(dǎo)

13、數(shù)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(dǎo)數(shù)left(right) derivative導(dǎo)函數(shù)derivative function可導(dǎo)函數(shù)derivable function導(dǎo)數(shù)的集合意義geometric meaning of derivative費馬定理Fermat theorem達布定理Darboux theorem導(dǎo)函數(shù)的介值定intermediatevaluetheoremof理derivativ

14、e function導(dǎo)數(shù)的四則運算algebra of derivatives和的導(dǎo)數(shù)derivative of sum差的導(dǎo)數(shù)derivative of difference積的導(dǎo)數(shù)derivative of product商的導(dǎo)數(shù)derivative of quotient反函數(shù)的導(dǎo)數(shù)derivative of inverse function最大值maximum value最小值minimum value復(fù)合函數(shù)的導(dǎo)數(shù)derivative of composite function對數(shù)求導(dǎo)法logarithmic derivative圓的參數(shù)方程parametric equation o

15、f circle橢圓的參數(shù)方程parametric equation of ellipse擺線的參數(shù)方程parametric equation of cycloid星形線的參數(shù)方parametric equation of asteroid程二階導(dǎo)數(shù)second derivative三階導(dǎo)數(shù)third derivativen 階導(dǎo)數(shù)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(shù)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(yīng)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(shù)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(shù)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(shù)的凸性和拐點convexityand in

19、flectionpoint offunction凸函數(shù)convex function凹函數(shù)concave function嚴格凸函數(shù)strictly convex function嚴格凹函數(shù)strictly concave function部分分式分解decomposition into partial fractions琴生不等式Jenson inequality待定系數(shù)法method of undetermined coefficients第七章實數(shù)的完備性第九章定積分Chapter 7Completeness of Real Numbers實數(shù)集完備性的fundamentaltheote

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

21、al sign積分表達式expression of integrand積分常數(shù)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(shù)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(qū)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(shù)類integrable function class定積分的線性性質(zhì)linear property of defini

24、te integral積分區(qū)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(shù)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(yīng)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(zhuǎn)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(yīng)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(shù)項級數(shù)Chapter 12Series of Number Terms數(shù)項級數(shù)series with number terms無窮級數(shù)infinite series級數(shù)的收斂性convergence of series部分和序列sequence of partial sum幾何級數(shù)geometric series調(diào)和級數(shù)harmonic series級數(shù)收斂的柯西Cauchy onvergence criterion for series準則正項級數(shù)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(shù)p-series一般項級數(shù)series with arbitraty terms交錯級數(shù)alternating series萊布尼茨判別法Leibnitz test級數(shù)的重排rearrangement of series阿貝爾判別法Abel test反常積分的概念無窮區(qū)間上的反常積分無界函數(shù)

29、的反常積分無窮積分的性質(zhì)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(shù)列與函數(shù)項級數(shù)Chapter13Sequences ofFunctionsa

30、ndSeries of Functions在點 x0 收斂convergent at x0收斂域convergencedomain/regionofconvergence和函數(shù)sum function極限函數(shù)limit function函數(shù)項級數(shù)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(shù)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(shù)的傅里葉級數(shù)角頻率angular frequency分段光滑piecewise smooth傅里葉系數(shù)Fourier coefficient收斂定理convergence theorem奇函數(shù)與偶函數(shù)Fourierseriesofevenandodd的傅里葉級數(shù)func

32、tions振幅amplitude正弦級數(shù)sine series余弦級數(shù)cosine series周期延拓periodic extension第十四章冪級數(shù)Chapter 14Power Series冪級數(shù)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(shù)的運算operations of power series泰勒級數(shù)taylor series初等函數(shù)的冪級expansion of powerseries of數(shù)展開式elementary function復(fù)變量的指數(shù)函expo

33、nential function ofcomplex數(shù)variable歐拉公式Euler formula第十五章傅里葉級數(shù)Chapter 15Fourier Series傅里葉級數(shù)Fourier series三角級數(shù)trigonometric series第十六章多元函數(shù)的極限與連續(xù)Chapter16LimitsandContinuityofFunctions of Several Variavles多元函數(shù)functions of several variables平面點集plane point set坐標平面coordinate plane內(nèi)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(shù)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(xù)properties of continuous functions on函數(shù)的性質(zhì)bounded closed region累次極限r(nóng)epeated limits全增量total increment偏增量partial increment第十八章隱函數(shù)定理及其應(yīng)用Chapter 18 Implicit Funciton Theorems and their Applications第十七章多元函數(shù)微分學(xué)Chapter17DifferentialCalculusofFunctions of Several Variables全微分

36、total differential偏導(dǎo)數(shù)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(fù)合函數(shù)微分法differentiation of composite function隱函數(shù)存在唯一性定理隱函數(shù)組函數(shù)行列式（雅可比行列式）反函數(shù)組坐標變換幾何應(yīng)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(shù)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(shù)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?qū)?shù)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(dǎo)數(shù)partial derivative of higher order混合偏導(dǎo)數(shù)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