1、數(shù)學分析語碼轉(zhuǎn)換1實數(shù)及其性質(zhì) real number and its properties有理數(shù) rational numbers 無理數(shù) irrational numbers 定義 definition 命題 proposition加 plus 減 minus 乘 multiplied by； times 除 over; is to; divided by2確界原理 principles of supremum and infimum數(shù)集 set of the numbers區(qū)間 interval 開區(qū)間 open interval 閉區(qū)間 closed interval 半開區(qū)間 sem

2、i-open interval半閉區(qū)間 semi-closed interval有限區(qū)間 finite interval3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 closeness 稠密性 the density無限區(qū)間 infinite interval 4有界集 bounded set無界集 unbounded set 下確界 infimum (or greatest lower bound)函數(shù)的定義 defi

3、nition of function定義域 domain of definition 值域 region 5ri:dVEn5數(shù)列極限 the limits of sequences發(fā)散序列 divergent sequence 無窮小序列 infinitesimal sequence 收斂序列 convergent sequence6唯一性定理 uniqueness theorem有界性定理 boundedness theorem保號性 inheriting order properties保不等式性 inheriting inequality子列 subsequence 7嚴格遞增 stri

4、ctly increasing單調(diào)遞增序列 monotone increasing sequence單調(diào)遞減序列 monotone decreasing sequence 嚴格遞減 strictly decreasing必要條件 necessary condition充分條件 sufficient condition夾逼定理 squeeze principle 8函數(shù)極限 the limits of functions無窮極限 infinite limits 單側(cè)極限 one-sided limit 右（左）極限 right (left) limit ; right(left) hand li

5、mit函數(shù)極限的性質(zhì) properties of limits of functions9局部有界性 local boundedness 海涅定理 Heine theorem 無窮大量 infinities無窮小量的階 the order of infinitesimal高（低）階無窮小量 infinitesimal of higher (lower) order10同階無窮小量 infinitesimal of the same order 等價無窮小量 equivalent infinitesimals 垂直漸近線 vertical asymptote 斜漸近線 oblique asymp

6、tote水平漸近線 horizontal asymptote11f在點x連續(xù) f is continuous at a point x函數(shù)的增量 increment of function 自變量的增量 increment of independent variable 12右（左）連續(xù) right(left) continuous 有界性定理 boundedness theorem 間斷點及其分類 discontinuity point and classification可去間斷點 removable discontinuity 跳躍間斷點 jump discontinuity 第一類間斷

7、點 discontinuity of the first kind 第二類間斷點 discontinuity of the second kind13局部有界性 local boundedness 連續(xù)函數(shù)的局部性質(zhì) local properties of continuous functions 連續(xù)函數(shù)的復合性質(zhì) composition property for continuous functions閉區(qū)間上連續(xù)函數(shù)的性質(zhì) properties of continuous functions over closed interval 14最大值和最小值定理 maximum and min

8、imum 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初等函數(shù)的連續(xù)性 continuity of elementary function15導數(shù)與微分Derivatives and Differentials有限增量公式 finite increment formula 左導數(shù) left derivative 右導數(shù) right derivative單側(cè)導

9、數(shù) one-sided derivative導函數(shù) derivative function可導函數(shù) derivable function 16導數(shù)的幾何意義 geometric meaning of derivative 費馬定理 Fermat theorem導函數(shù)的介值定理 intermediate value theorem of derivative function導數(shù)的四則運算 the algebra of derivatives 17和的導數(shù) derivative of a sum差的導數(shù) derivative of a difference 積的導數(shù) derivative of

10、a product 商的導數(shù) derivative of a quotient rule 反函數(shù)的導數(shù) derivatives of inverse functions復合函數(shù)的導數(shù) derivatives of composite functions18圓的參數(shù)方程 parametric equation of circle橢圓的參數(shù)方程 parametric equation of ellipse 擺線，旋輪線的參數(shù)方程 parametric equation of cycloid 星形線的參數(shù)方程 parametric equation of asteroid 19對數(shù)求導法 logar

11、ithmic derivative基本初等導數(shù)公式 derivative formulas of basic elementary function 二階導數(shù) second derivative 三階導數(shù) third derivative 四階導數(shù) fourth derivative n階導數(shù) nth derivative 20微分的概念 concept of differential 線性主部 linear principle part 自變量的微分 differential of the independent variable加速度 acceleration微分的運算法則 operati

12、onal rules of differentials物理解釋 physical interpretation 21和的極限等于極限的和. The limit of a sum is equal to the sum of the limits.冪的極限等于極限的冪. The limit of a power is equal to the power of the limits.收斂數(shù)列的和收斂于極限的和. The sum of convergent sequences convergences to the sum of the limits.22達布定理 Darboux theorem

13、高階微分 higher differential 微分形式的不變性 preservation of the form of differential微分的幾何意義 geometric meaning of a differential 23微分中值定理及其應(yīng)用 Mean Value Theorems of Differentials and Its Applications羅爾中值定理 Rolle mean value theorem拉格朗日中值定理 Lagrange mean value theorem 24柯西中值定理 Cauchy mean value theorem 洛必達法則 Lho

14、spitals Rule0/0 型不定式極限 limits of indeterminate forms of the type 0/0 / 型不定式 indeterminate forms of the type / 25 泰勒公式的余項 Remainder of Taylor formula帶有佩亞諾型余項的泰勒公式 Taylor formula with Peano Remainder帶有拉格朗日型余項的麥克勞林公式Maclaurin formula with Lagrange Remainder 帶有拉格朗日型余項的泰勒公式 Taylor formula with Lagrange R

15、emainder 26函數(shù)的極值 extreme values of function極值判別 test of extreme value凸函數(shù) convex function 凹函數(shù) concave function 極值的第一充分條件 the first sufficient condition of extreme values 極值的第二充分條件 the second sufficient condition of extreme values27函數(shù)的凸點與拐點 convexity of a function and inflection嚴格凸函數(shù) strictly convex f

16、unction嚴格凹函數(shù) strictly concave function 28換元積分法 integration by substitution 第一換元公式 the formula of the first kind of substitution 第二換元公式 the formula of the second kind of substitution 分部積分法 integration by parts29 有理函數(shù)的不定積分 indefinite integrals of rational functions 真分式 proper fraction 假分式 improper fra

17、ction 部分分式分解 decomposition into partial fractions 待定系數(shù)法 method of the undetermined coefficients 30被積函數(shù) integrand function 積分符號 integral sign被積表達式 element of integration積分變量 integral variable積分常數(shù) integral constant31定積分 definite integrals曲邊梯形 curvilinear trapezoid 分割 dividing; partition 近似求和 summing a

18、pproximately 取極限 taking the limit 32黎曼和 Riemann sums 黎曼積分 Riemann integrals 黎曼可積 integrable in the sense of Riemann 積分區(qū)間 interval of integration上限 upper limit 下限 lower limit 33可積的必要條件 a necessary condition for the integrability 可積的充要條件 necessary and sufficient conditions for the integrability 達布小和 D

19、arboux lower sum達布大和 Darboux upper sum 可積函數(shù)類 class of integrable functions34 定積分的線性性質(zhì) linearity of the definite integral 關(guān)于積分區(qū)間的可加性 additive with respect to integral intervals (additivity over integrals) 積分不等式性 monotonicity of the integral(comparison property) 積分中值定理 mean value theorems of integrals

20、 積分第一中值定理 first mean value theorem of integrals 積分第一中值定理的推廣 generalized first mean value theorem of integrals 35變限積分 integral with variant upper limit 原函數(shù)的存在性 the existence of the primitive 原函數(shù)的存在定理 existence theorem of the primitive 定積分換元積分法 the substitution method for definite integrals 定積分分部積分法 i

21、ntegration by parts for definite integrals泰勒公式的積分型余項 integral form of remainder of Taylor formula 36定積分的應(yīng)用Applications of Definite Integrals平面圖形的面積 areas of plane figures 由平行截面面積求體積 volume of a solid with known parallel sections37平面曲線的弧長與曲率 arclength of plane curves and curvature光滑曲線 smoothed curve

22、弧微分 differential of arc length 曲率圓 circle of curvature曲率半徑 radius of curvature 曲率中心 center of curvature 38旋轉(zhuǎn)曲面的面積 area of a surface of revolution 定積分在物理中的某些應(yīng)用 applications of definite integral in physics 功與平均功率 work and average power 定積分的近似計算 approximate computation of definite integral 梯形法 trapezoi

23、dal method 拋物線法 parabola method 引力 gravitation 39反常積分的概念 notion of improper integrals 無窮限反常積分 improper integrals with infinite limits無界函數(shù)的反常積分 improper integrals with an unbounded integrand收斂 convergent 40無窮積分的性質(zhì)與收斂判別 Properties of in finite integral and tests of convergence比較判別法 comparison test 柯西判

24、別法 Cauchys test 狄利克雷判別法 Dirichlets test 阿貝爾判別法 Abels test 絕對收斂 absolutely convergent 條件收斂 conditional convergent41關(guān)于實數(shù)集完備性的基本定理 fundamental theorems of completeness in the set of real numbers區(qū)間套定理 the nested interval theorem 柯西收斂準則 Cauchys general principle of convergence聚點定理 accumulation theorem 42

25、有限覆蓋定理 finite covering theorem 實數(shù)完備性基本定理的等價性 equivalence of the completeness of real numbers 43級數(shù)的收斂性 convergence of series部分和序列 sequence of partial sums 幾何級數(shù) geometric series 調(diào)和級數(shù) harmonic series 級數(shù)收斂的柯西準則 Cauchy convergence criterion for series 收斂的必要條件 a necessary condition for convergence 44正項級數(shù)

26、positive term series比較原則 comparison test 比式判別法(達朗貝爾判別法) DAlemberts ratio test 根式判別法 root test 正項級數(shù)收斂的一般判別原則 general criterion of convergence for positive terms series 45比式判別法的極限形式 limit form of ratio test 柯西判別法（根式判別法） Cauchys root test 根式判別法的極限形式 limit form of root test 高斯判別法 Gausss test 積分判別法 integral test 拉貝判別法 Raabes test 46一致收斂性 uniform convergence收斂域 convergence domain; region of convergence 一致收斂于f converge uniformly to f函數(shù)項級數(shù)及其一