1、MATHEMATICAL TERMS (Part 1)calculus 微積分definition 定義theorem 定理lemma 引理corollary推論prove 證明proof 證明show 證明solution 解formula 公式if and only if ( iff ) 當(dāng)且僅當(dāng) xX for all xX xX there exists an xXsuch that 使得given 已知set集合finite set有限集 infinite set 無限集interval區(qū)間open interval開區(qū)間 closed interval 閉區(qū)間neighborhood

2、 鄰域number 數(shù)natural number 自然數(shù)integer 整數(shù)odd number 奇數(shù)even number 偶數(shù)real number 實數(shù) rational number 有理數(shù)irrational number 無理數(shù)positive number 正數(shù)negative number 負(fù)數(shù)mapping 映射function 函數(shù)1 / 13monotone function 單調(diào)函數(shù) increasing function 增函數(shù)decreasing function 減函數(shù)bounded function 有界函數(shù)odd function 奇函數(shù)even funct

3、ion 偶函數(shù)periodic function 周期函數(shù)composite function 復(fù)合函數(shù)inverse function 反函數(shù)domain 定義域range 值域variable 變量independent variable自變量 dependent variable因變量sequence 數(shù)列convergent sequence收斂數(shù)列 divergent sequence 發(fā)散數(shù)列bounded sequence 有界數(shù)列decreasing sequence 遞減數(shù)列increasing sequence 遞增數(shù)列l(wèi)imit極限 one-sided limit 單側(cè)極

4、限left-hand limit 左極限r(nóng)ight-hand limit 右極限The Squeeze Theorem 夾擠定理infinity 無窮大infinitesimal 無窮小equivalent infinitesimal 等價無窮小 infinitesimal of higher order 高階無窮小order of infinitesimal 無窮小的階infinitesimals of the same order 同階無窮小increment 增量continuous function 連續(xù)函數(shù)continuity 連續(xù)性f(x) is continuous at x 在

5、x連續(xù)f(x) is discontinuous at x 在x間斷discontinuity 間斷點discontinuity of the first (second) kind 第一（二）類間斷點removable discontinuity 可去間斷點jump discontinuity 跳躍間斷點infinite discontinuity 無窮間斷點intermediate value 介值The Intermediate Value Theorem 介值定理zero point 零點The Zero Point Theorem 零點定理root 根equation 方程unifo

6、rm continuity 一致連續(xù)derivative 導(dǎo)數(shù)rate of change 變化率velocity 速度instantaneous velocity 瞬時速度tangent (line) 切線normal (line) 法線slope 斜率left-hand derivative 左導(dǎo)數(shù) right-hand derivative 右導(dǎo)數(shù)f(x) is differentiable at x f(x) 在x處可導(dǎo)(可微)differentiation 求導(dǎo)The Chain Rule 鏈?zhǔn)椒▌tdifferentiation formulas 求導(dǎo)公式implicit funct

7、ion 隱函數(shù)explicit function 顯函數(shù)implicit differentiation 隱函數(shù)求導(dǎo)logarithm 對數(shù)Logarithmic differentiation 對數(shù)求導(dǎo)法parameter 參數(shù)parametric equation 參數(shù)方程parametric curve 參數(shù)曲線hyperbolic function 雙曲函數(shù)hyperbolic sine 雙曲正弦hyperbolic cosine 雙曲余弦hyperbolic tangent 雙曲正切hyperbolic cotangent 雙曲余切differential 微分differentia

8、l quotient 微商approximate value 近似值error 誤差relative error 相對誤差absolute error 絕對誤差invariance of differential form 微分形式不變性higher derivative 高階導(dǎo)數(shù)first derivative 一階導(dǎo)數(shù)second derivative 二階導(dǎo)數(shù)third derivative 三階導(dǎo)數(shù)nth derivative n階導(dǎo)數(shù)twice differentiable 二階可導(dǎo)acceleration 加速度mean value 中值The Mean Value Theorem

9、 中值定理Rolles Theorem 羅爾定理Lagranges Mean Value Theorem 拉格朗日中值定理Cauchys Mean Value Theorem 柯西中值定理equality 等式inequality 不等式indeterminate form 不定型 (未定式)indeterminate form of type () ()型未定式LHospitals Rule 洛必達(dá)法則Taylors formula 泰勒公式polynomial 多項式nth-degree polynomial n次多項式remaind 余項Lagranges form of remaind

10、er 拉格朗日型余項Peanos form of remainder 皮亞諾型余項Maclaurin formula 麥克勞林公式Taylor polynomial 泰勒多項式Mauclaurin polynomial 麥克勞林多項式polynomial approximation 多項式逼近accuracy 精確度margin 邊際marginal cost 邊際成本marginal revenue 邊際收益elasticity 彈性density 密度mass 質(zhì)量extreme value 極值 local maximum value 極大值local minimum value 極小值

11、(absolute) maximum 最大值(absolute) minimum 最小值stationary point 駐點（穩(wěn)定點）critical point 臨界點The First (Second) Derivative Test （極值的）一（二）階判別法 convex 凸的convex curve 凸曲線concave 凹的concave curve 凹曲線convex function 凸函數(shù)point of inflection 拐點asymptote 漸近線horizontal asymptote 水平漸近線vertical asymptote 垂直漸近線slant asy

12、mptote 斜漸近線curve sketching 作圖sketch a curve 作圖curvature 曲率The bisection method 二分法The secant method 弦位法Newtons method 牛頓（切線）法The tangent method 切線法differential calculus 微分學(xué)integral 積分integral calculus 積分學(xué)definite integral 定積分indefinite integral 不定積分partition 分割Riemann sum 黎曼和integral sign 積分符號integr

13、and 被積函數(shù)upper (lower) limit of integration 積分上（下）限integration 積分（求積）integrable 可積的f(x) is integrable on a, b integrable function 可積函數(shù)integrability 可積性sufficient condition 充分條件necessary condition 必要條件piecewise continuous 分段連續(xù)property 性質(zhì)The mean value theorem of integral 積分中值定理The fundamental theorem

14、of calculus 微積分基本定理Newton-Leibniz formulaprimitive function (anti-derivative) 原函數(shù)（反導(dǎo)數(shù)）The substitution rule for integration 換元積分法The inverse of the chain rule 反鏈?zhǔn)椒ǎ愇⒎址ǎ﹊ntegration by parts 分部積分法rational function 有理函數(shù)fraction 分式irreducible fraction 最簡分式partial fraction 部分分式partial fraction decomposi

15、tion 部分分式分解vector 矢量free vector 自由矢量zero vector 零矢量magnitude of a vector 矢量的模 unit vector 單位矢量scalar product 數(shù)量積dot product 點積vector product 矢量積cross product 叉積a is perpendicular (orthogonal) to b a與b 垂直coordinate 坐標(biāo)coordinate system 坐標(biāo)系coordinate axis 坐標(biāo)軸x-axis x軸coordinate plane 坐標(biāo)面direction angle

16、 方向角direction cosine 方向余弦rectangular coordinate system 直角坐標(biāo)系octant 卦限the first octant 第一卦限 variable 變量function of two (three) variables 二（三）元函數(shù)function of several variables 多元函數(shù)independent variable 自變量dependent variable 因變量domain 定義域range 值域set of points 點集neighborhood 鄰域interior point 內(nèi)點boundary po

17、int 邊界點bound 邊界open set 開集closed set 閉集connected set 連通集region 區(qū)域open region 開區(qū)域closed region 閉區(qū)域bounded region 有界區(qū)域unbounded region 無界區(qū)域cluster point 聚點double limit 二重極限iterated limit 累次極限continuity 連續(xù)性increment 增量total increment 全增量partial increment 偏增量partial derivative 偏導(dǎo)數(shù)partial derivative of f

18、(x,y) with respect to x ( y ) f(x,y)關(guān)于x（y）的偏導(dǎo)數(shù)higher partial derivative 高階偏導(dǎo)數(shù)mixed partial derivative 混合偏導(dǎo)數(shù)Laplace equation 拉普拉斯方程total differential 全微分differentiable 可微chain rule 鏈?zhǔn)椒▌timplicit function 隱函數(shù)implicit differentiation 隱函數(shù)微分法Jacobian determinant 雅可比行列式curve 曲線space curve 空間曲線 tangent vect

19、or 切矢tangent line 切線normal plane 法平面surface 曲面normal vector 法矢normal line 法線tangent plane 切平面 sphere 球面cylinder 柱面cone 錐面directional derivative 方向?qū)?shù)gradient 梯度gradient vector 梯度矢量Ñf delflevel curve 等值線level surface 等值面local extremum 極值local maximum 極大值local minimum 極小值extreme value 最值absolute m

20、aximum (minimum) 最大（最?。┲祍tationary point (critical point) 駐點（臨界點）conditional extremum 條件極值Lagrange multiplier 拉格朗日乘數(shù)method of Lagrange multiplier 拉格朗日乘數(shù)法 objective function 目標(biāo)函數(shù) constraint 約束條件method of least square 最小二乘法field 場scalar field 數(shù)量場vector field 矢量場gradient field 梯度場potential field 勢場pote

21、ntial function 勢函數(shù)conservative field 保守場gravitational field 引力場force field 力場velocity field 速度場multiple integral 重積分double integral 二重積分iterated integral 累次積分region 區(qū)域region of integration 積分區(qū)域type X (Y) region X（Y）型區(qū)域order of integration 積分秩序reverse the order of integration 交換積分秩序polar coordinates

22、極坐標(biāo)double integrals in polar coordinates 極坐標(biāo)下的二重積分volume 體積lamina 平面薄片mass 質(zhì)量density 密度moment about x-axis 關(guān)于x軸的（靜）力矩center of mass 重心moment of inertia 轉(zhuǎn)動慣量surface 曲面area of a surface 曲面的面積triple integral 三重積分rectangle 矩形rectangular coordinates 直角坐標(biāo)系cylinder 柱面cylindrical coordinates 柱面坐標(biāo)系sphere 球面s

23、pherical coordinates 球面坐標(biāo)系change of variables in multiple integrals 重積分的變量替換Jacobian determinant 雅可比行列式line integral 曲線積分line integral with respect to arc length 對弧長的曲線積分（第一型）line integral with respect to x ( y ) 對坐標(biāo)x（y）曲線積分（第二型）line integral of a vector field 向量場的曲線積分smooth curve 光滑曲線piecewise smooth curve 逐段光滑曲線oriented curve 有向曲線orientation of a curve 曲線的方向work