数学专业英语第一章
高等数学名词（中英文）
高等数学名词（中英文）高等数学名词(中英文)第一章函数与极限Chapter1 Function and Limit 集合set元素element子集subset空集empty set并集union交集intersection差集difference of set基本集basic set补集complement set直积direct product笛卡儿积Cartesian product开区间open interval闭区间closed interval半开区间half open interval有限区间finite interval区间的长度length of an interval无限区间infinite interval领域neighborhood领域的中心centre of a neighborhood 领域的半径radius of a neighborhood 左领域left neighborhood右领域right neighborhood映射mappingX到Y的映射mapping of X ontoY 满射surjection单射injection一一映射one-to-one mapping双射bijection算子operator变化transformation函数function逆映射inverse mapping复合映射composite mapping自变量independent variable 因变量dependent variable定义域domain函数值value of function函数关系function relation值域range自然定义域natural domain单值函数single valued function多值函数multiple valued function单值分支one-valued branch函数图形graph of a function绝对值函数absolute value符号函数sigh function整数部分integral part阶梯曲线step curve当且仅当if and only if(iff)分段函数piecewise function上界upper bound下界lower bound有界boundedness无界unbounded函数的单调性monotonicity of a function 单调增加的increasing单调减少的decreasing单调函数monotone function函数的奇偶性parity(odevity) of a function 对称symmetry偶函数even function奇函数odd function函数的周期性periodicity of a function周期period反函数inverse function直接函数direct function复合函数composite function中间变量intermediate variable函数的运算operation of function基本初等函数basic elementary function 初等函数elementary function幂函数power function指数函数exponential function对数函数logarithmic function三角函数trigonometric function反三角函数inverse trigonometric function 常数函数constant function双曲函数hyperbolic function双曲正弦hyperbolic sine双曲余弦hyperbolic cosine双曲正切hyperbolic tangent反双曲正弦inverse hyperbolic sine反双曲余弦inverse hyperbolic cosine反双曲正切inverse hyperbolic tangent极限limit数列sequence of number收敛convergence收敛于a converge to a发散divergent极限的唯一性uniqueness of limits收敛数列的有界性boundedness of a convergent sequence子列subsequence函数的极限limits of functions函数当x趋于x0时的极限limit of functions as x approaches x0左极限left limit右极限right limit单侧极限one-sided limits水平渐近线horizontal asymptote无穷小infinitesimal无穷大infinity铅直渐近线vertical asymptote夹逼准则squeeze rule单调数列monotonic sequence高阶无穷小infinitesimal of higher order低阶无穷小infinitesimal of lower order同阶无穷小infinitesimal of the same order等阶无穷小equivalent infinitesimal 函数的连续性continuity of a function增量increment函数在x0连续the function is continuous at x0左连续left continuous右连续right continuous区间上的连续函数continuous function函数在该区间上连续function is continuous on an interval不连续点discontinuity point第一类间断点discontinuity point of the first kind第二类间断点discontinuity point of the second kind初等函数的连续性continuity of the elementary functions定义区间defined interval最大值global maximum value (absolute maximum)最小值global minimum value (absolute minimum)零点定理the zero point theorem介值定理intermediate value theorem第二章导数与微分Chapter2 Derivative and Differential速度velocity匀速运动uniform motion平均速度average velocity瞬时速度instantaneous velocity圆的切线tangent line of a circle切线tangent line切线的斜率slope of the tangent line位置函数position function导数derivative可导derivable函数的变化率问题problem of the change rate of a function 导函数derived function左导数left-hand derivative右导数right-hand derivative单侧导数one-sided derivatives在闭区间[a, b]上可导is derivable on the closed interval [a,b] 切线方程tangent equation角速度angular velocity成本函数cost function边际成本marginal cost链式法则chain rule隐函数implicit function显函数explicit function二阶函数second derivative三阶导数third derivative高阶导数nth derivative莱布尼茨公式Leibniz formula对数求导法log- derivative参数方程parametric equation相关变化率correlative change rata微分differential可微的differentiable函数的微分differential of function自变量的微分differential of independent variable微商differential quotient间接测量误差indirect measurement error绝对误差absolute error相对误差relative error第三章微分中值定理与导数的应用Chapter3 Mean Value Theorem ofDifferentials and the Application ofDerivatives罗马定理Rolle’s theorem费马引理Fermat’s lemma拉格朗日中值定理Lagrange’s mean value theorem驻点stationary point稳定点stable point 临界点critical point辅助函数auxiliary function拉格朗日中值公式Lagrange’s mean value formula柯西中值定理Cauchy’s mean value theorem 洛必达法则L’Hospital’s Rule0/0型不定式indeterminate form of type 0/0 不定式indeterminate form泰勒中值定理Taylor’s mean value theorem 泰勒公式T aylorformula余项remainder term拉格朗日余项Lagrange remainder term麦克劳林公式Ma claurin’s formula佩亚诺公式Peano remainder term凹凸性concavity凹向上的concave upward, cancave up凹向下的，向上凸的concave downward’ concave down 拐点inflection point函数的极值extremum of function极大值local(relative) maximum最大值global(absolute) mximum极小值local(relative) minimum最小值global(absolute) minimum目标函数objective function曲率curvature弧微分arc differential平均曲率average curvature曲率园circle of curvature曲率中心center of curvature曲率半径radius of curvature渐屈线evolute渐伸线involute根的隔离isolation of root隔离区间isolation interval切线法tangent line method第四章不定积分Chapter4 Indefinite Integrals原函数primitive function(anti-derivative)积分号sign of integration被积函数integrand积分变量integral variable积分曲线integral curve积分表table of integrals换元积分法integration by substitution分部积分法integration by parts分部积分公式formula of integration by parts 有理函数rational function真分式proper fraction假分式improper fraction第五章定积分Chapter5 Definite Integrals曲边梯形trapezoid with曲边curve edge窄矩形narrow rectangle曲边梯形的面积area of trapezoid with curved edge积分下限lower limit of integral积分上限upper limit of integral积分区间integral interval分割partition积分和integral sum可积integrable矩形法rectangle method积分中值定理mean value theorem of integrals函数在区间上的平均值average value of a function on an integvals牛顿－莱布尼茨公式Newton-Leibniz formula微积分基本公式fundamental formula of calculus换元公式formula for integration by substitution递推公式recurrence formula反常积分improper integral 反常积分发散the improper integral is divergent反常积分收敛the improper integral is convergent无穷限的反常积分improper integral on an infinite interval无界函数的反常积分improper integral of unbounded functions绝对收敛absolutely convergent第六章定积分的应用Chapter6 Applications of the DefiniteIntegrals元素法the element method面积元素element of area平面图形的面积area of a luane figure直角坐标又称“笛卡儿坐标(Cartesian coordinates)”极坐标polar coordinates抛物线parabola椭圆ellipse旋转体的面积volume of a solid of rotation旋转椭球体ellipsoid of revolution, ellipsoid of rotation曲线的弧长arc length of acurve可求长的rectifiable光滑smooth功work水压力water pressure引力gravitation变力variable force第七章空间解析几何与向量代数Chapter7 Space Analytic Geometry andVector Algebra向量vector自由向量free vector单位向量unit vector零向量zero vector相等equal平行parallel向量的线性运算linear poeration of vector三角法则triangle rule平行四边形法则parallelogram rule交换律commutative law结合律associative law负向量negative vector差difference分配律distributive law空间直角坐标系space rectangular coordinates坐标面coordinate plane卦限octant向量的模modulus of vector向量a与b的夹角angle between vector a and b方向余弦direction cosine方向角direction angle向量在轴上的投影projection of a vector onto an axis数量积，外积，叉积scalar product,dot product,inner product 曲面方程equation for a surface球面sphere旋转曲面surface of revolution母线generating line轴axis圆锥面cone顶点vertex旋转单叶双曲面revolution hyperboloids of one sheet旋转双叶双曲面revolution hyperboloids of two sheets柱面cylindrical surface ,cylinder圆柱面cylindrical surface准线directrix抛物柱面parabolic cylinder二次曲面quadric surface 椭圆锥面dlliptic cone椭球面ellipsoid单叶双曲面hyperboloid of one sheet双叶双曲面hyperboloid of two sheets旋转椭球面ellipsoid of revolution椭圆抛物面elliptic paraboloid旋转抛物面paraboloid of revolution双曲抛物面hyperbolic paraboloid马鞍面saddle surface椭圆柱面elliptic cylinder双曲柱面hyperbolic cylinder抛物柱面parabolic cylinder空间曲线space curve空间曲线的一般方程general form equations of a space curve 空间曲线的参数方程parametric equations of a space curve 螺转线spiral螺矩pitch投影柱面projecting cylinder投影projection平面的点法式方程pointnorm form eqyation of a plane法向量normal vector平面的一般方程general form equation of a plane两平面的夹角angle between two planes点到平面的距离distance from a point to a plane空间直线的一般方程general equation of a line in space方向向量direction vector直线的点向式方程pointdirection form equations of a line方向数direction number直线的参数方程parametric equations of a line两直线的夹角angle between two lines垂直perpendicular直线与平面的夹角angle between a line and a planes平面束pencil of planes平面束的方程equation of a pencil of planes 行列式determinant系数行列式coefficient determinant第八章多元函数微分法及其应用Chapter8 Differentiation of Functions ofSeveral Variables and Its Application一元函数function of one variable多元函数function of several variables内点interior point外点exterior point边界点frontier point,boundary point聚点point of accumulation开集openset闭集closed set连通集connected set开区域open region闭区域closed region有界集bounded set无界集unbounded setn维空间n-dimentional space二重极限double limit多元函数的连续性continuity of function of seveal连续函数continuous function不连续点discontinuity point一致连续uniformly continuous偏导数partial derivative对自变量x的偏导数partial derivative with respect to independent variable x高阶偏导数partial derivative of higher order 二阶偏导数second order partial derivative混合偏导数hybrid partial derivative全微分total differential偏增量oartial increment偏微分partial differential 全增量total increment可微分differentiable必要条件necessary condition充分条件sufficient condition叠加原理superpostition principle全导数total derivative中间变量intermediate variable隐函数存在定理theorem of the existence of implicit function 曲线的切向量tangent vector of a curve法平面normal plane向量方程vector equation向量值函数vector-valued function切平面tangent plane法线normal line方向导数directional derivative梯度gradient数量场scalar field梯度场gradient field向量场vector field势场potential field引力场gravitational field引力势gravitational potential曲面在一点的切平面tangent plane to a surface at a point曲线在一点的法线normal line to a surface at a point无条件极值unconditional extreme values条件极值conditional extreme values拉格朗日乘数法Lagrange multiplier method 拉格朗日乘子Lagrange multiplier经验公式empirical formula最小二乘法method of least squares均方误差mean square error第九章重积分Chapter9 Multiple Integrals二重积分double integral可加性additivity累次积分iterated integral体积元素volume element三重积分triple integral直角坐标系中的体积元素volume element in rectangular coordinate system柱面坐标cylindrical coordinates柱面坐标系中的体积元素volume element in cylindrical coordinate system球面坐标spherical coordinates球面坐标系中的体积元素volume element in spherical coordinate system反常二重积分improper double integral曲面的面积area of a surface质心centre of mass静矩static moment密度density形心centroid转动惯量moment of inertia参变量parametric variable第十章曲线积分与曲面积分Chapter10 Line (Curve) Integrals andSurface Integrals对弧长的曲线积分line integrals with respect to arc hength第一类曲线积分line integrals of the first type对坐标的曲线积分line integrals with respect to x,y,and z第二类曲线积分line integrals of the second type有向曲线弧directed arc单连通区域simple connected region复连通区域complex connected region格林公式Green formula第一类曲面积分surface integrals of the first type对面的曲面积分surface integrals with respect to area 有向曲面directed surface对坐标的曲面积分surface integrals with respect to coordinate elements第二类曲面积分surface integrals of the second type有向曲面元element of directed surface高斯公式gauss formula拉普拉斯算子Laplace operator格林第一公式Green’s first formula通量flux散度divergence斯托克斯公式Stokes formula环流量circulation旋度rotation,curl第十一章无穷级数Chapter11 Infinite Series一般项general term部分和partial sum余项remainder term等比级数geometric series几何级数geometric series公比common ratio调和级数harmonic series柯西收敛准则Cauchy convergence criteria, Cauchy criteria for convergence正项级数series of positive terms达朗贝尔判别法D’Alembert test柯西判别法Cauchy test交错级数alternating series绝对收敛absolutely convergent条件收敛conditionally convergent柯西乘积Cauchy product函数项级数series of functions发散点point of divergence收敛点point of convergence收敛域convergence domain和函数sum function幂级数power series幂级数的系数coeffcients of power series阿贝尔定理Abel Theorem收敛半径radius of convergence收敛区间interval of convergence泰勒级数Taylor series麦克劳林级数Maclaurin series二项展开式binomial expansion近似计算approximate calculation舍入误差round-off error,rounding error欧拉公式Euler’s formula魏尔斯特拉丝判别法Weierstrass test三角级数trigonometric series振幅amplitude角频率angular frequency初相initial phase矩形波square wave谐波分析harmonic analysis直流分量direct component基波fundamental wave二次谐波second harmonic三角函数系trigonometric function system傅立叶系数Fourier coefficient傅立叶级数Forrier series周期延拓periodic prolongation正弦级数sine series余弦级数cosine series奇延拓odd prolongation偶延拓even prolongation傅立叶级数的复数形式complex form of Fourier series第十二章微分方程Chapter12 Differential Equation解微分方程solve a differential equation常微分方程ordinary differential equation偏微分方程partial differential equation,PDE 微分方程的阶order of a differential equation 微分方程的解solution of adifferential equation 微分方程的通解general solution of a differential equation初始条件initial condition微分方程的特解particular solution of a differential equation 初值问题initial value problem微分方程的积分曲线integral curve of a differential equation 可分离变量的微分方程variable separable differential equation 隐式解implicit solution隐式通解inplicit general solution衰变系数decay coefficient衰变decay齐次方程homogeneous equation一阶线性方程linear differential equation of first order非齐次non-homogeneous齐次线性方程homogeneous linear equation 非齐次线性方程non-homogeneous linear equation常数变易法method of variation of constant 暂态电流transient state current稳态电流steady state current伯努利方程Bernoulli equation全微分方程total differential equation积分因子integrating factor高阶微分方程differential equation of higher order悬链线catenary高阶线性微分方程linear differential equation of higher order 自由振动的微分方程differential equation of free vibration强迫振动的微分方程differential equation of forced oscillation 串联电路的振荡方程oscillation equation of series circuit二阶线性微分方程second order lineardifferential equation线性相关linearly dependence线性无关linearly independence二阶常系数齐次线性微分方程second order homogeneous linear differential equation with constant coefficient二阶变系数齐次线性微分方程second order homogeneous linear differential equation with variable coefficient特征方程characteristic equation无阻尼自由振动的微分方程differential equation of free vibration with zero damping 固有频率natural frequency 简谐振动simple harmonic oscillation,simple harmonic vibration 微分算子differential operator待定系数法method of undetermined coefficient共振现象resonance phenomenon欧拉方程Euler equation幂级数解法power series solution数值解法numerial solution勒让德方程Legendre equation微分方程组system of differential equations常系数线性微分方程组system of linear differential equations with constant coefficientV、X、Z:Value of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线Volume ：体积X-axis ：x轴x-coordinate ：x坐标x-intercept ：x截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点T:Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分S:Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称R:Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution , solid of ：旋转体Revolution , surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根P、Q:Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程Partial fractions ：部分分式Partial integration ：部分积分Partiton ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function 分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积Quadrant ：象限Quotient Law of limit 极限的商定律Quotient Rule ：商定律M、N、O:Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function 自然指数函数Natural logarithm function 自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的L:Laplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule 洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Lobachevski geometry 罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum 局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数I:Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test 递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem 中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function反三角函数Iterated integral ：逐次积分H:Higher mathematics 高等数学/高数E、F、G、H:Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyperbola ：双曲线Hyper boloid ：双曲面D:Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数of a composite function ：复合函数之导数of a constant function ：常数函数之导数directional ：方向导数domain of ：导数之定义域of exponential function ：指数函数之导数higher ：高阶导数partial ：偏导数of a power function ：幂函数之导数of a power series ：羃级数之导数of a product ：积之导数of a quotient ：商之导数as a rate of change ：导数当作变率right-hand ：右导数second ：二阶导数as the slope of a tangent 导数看成切线之斜率Determinant ：行列式Differentiable function 可导函数Differential ：微分Differential equation ：微分方程partial ：偏微分方程Differentiation ：求导法implicit ：隐求导法partial ：偏微分法term by term ：逐项求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分change of variable in二重积分之变数变换in polar coordinates ：极坐标二重积分C:Calculus ：微积分differential ：微分学integral ：积分学Cartesian coordinates ：笛卡儿坐标一般指直角坐标Cartesian coordinates system 笛卡儿坐标系Cauch’s Mean Value Theorem 柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性at a point ：在一点处之连续性of a function ：函数之连续性on an interval ：在区间之连续性from the left ：左连续from the right ：右连续Continuous function ：连续函数Convergence ：收敛interval of ：收敛区间radius of ：收敛半径Convergent sequence ：收敛数列series ：收敛级数Coordinate：s：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标polar ：极坐标rectangular ：直角坐标spherical ：球面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder：圆柱Cylindrical Coordinates ：圆柱坐标A、B:Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum 绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法by differentials ：用微分逼近linear ：线性逼近法by Simpson’s Rule ：Simpson法则逼近法by the Trapezoidal Rule 梯形法则逼近法Arbitrary constant ：任意常数Arc length ：弧长Area ：面积under a curve ：曲线下方之面积between curves ：曲线间之面积in polar coordinates ：极坐标表示之面积of a sector of a circle ：扇形之面积of a surface of a revolution 旋转曲面之面积Asymptote ：渐近线horizontal ：水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴Binomial series ：二项级数微积分专有名词中英文对照absolutely convergent 绝对收敛absolute value 绝对值algebraic function 代数函数analytic geometry 解析几何antiderivative 不定积分approximate integration 近似积分approximation 近似法、逼近法arbitrary constant 任意常数arithmetic series/progression (AP)算数级数asymptotes (vertical and horizontal)(垂直/水平)渐近线average rate of change 平均变化率base 基数binomial theorem 二项式定理，二项展开式Cartesian coordinates 笛卡儿坐标(一般指直角坐标)Cartesian coordinates system 笛卡儿坐标系Cauch’s Mean Value Theorem 柯西均值定理chain rule 链式求导法则calculus 微积分学closed interval integral 闭区间积分coefficient 系数composite function 复合函数conchoid 蚌线continuity (函数的)连续性concavity (函数的)凹凸性conditionally convergent 有条件收敛continuity 连续性critical point 临界点cubic function 三次函数cylindrical coordinates 圆柱坐标decreasing function 递减函数decreasing sequence 递减数列definite integral 定积分derivative 导数determinant 行列式differential coefficient 微分系数differential equation 微分方程directional derivative 方向导数discontinuity 不连续性discriminant (二次函数)判别式disk method 圆盘法divergence 散度divergent 发散的domain 定义域dot product 点积double integral 二重积分ellipse 椭圆ellipsoid 椭圆体epicycloid 外摆线Euler's method (BC)欧拉法expected valued 期望值exponential function 指数函数extreme value heorem 极值定理factorial 阶乘finite series 有限级数fundamental theorem of calculus 微积分基本定理geometric series/progression (GP)几何级数gradient 梯度Green formula 格林公式half-angle formulas 半角公式harmonic series 调和级数helix 螺旋线higher derivative 高阶导数horizontal asymptote 水平渐近线horizontal line 水平线hyperbola 双曲线hyper boloid 双曲面implicit differentiation 隐函数求导implicit function 隐函数improper integral 广义积分、瑕积分increment 增量increasing function 增函数indefinite integral 不定积分independent variable 自变数inequality 不等式ndeterminate form 不定型infinite point 无穷极限infinite series 无穷级数infinite series 无限级数inflection point (POI) 拐点initial condition 初始条件instantaneous rate of change 瞬时变化率integrable 可积的integral 积分integrand 被积分式integration 积分integration by part 分部积分法intercept 截距intermediate value of Theorem ：中间值定理inverse function 反函数irrational function 无理函数iterated integral 逐次积分Laplace transform 拉普拉斯变换law of cosines 余弦定理least upper bound 最小上界left-hand derivative 左导数left-hand limit 左极限L'Hospital's rule 洛必达法则limacon 蚶线linear approximation 线性近似法linear equation 线性方程式linear function 线性函数linearity 线性linearization 线性化local maximum 极大值local minimum 极小值logarithmic function 对数函数MacLaurin series 麦克劳林级数maximum 最大值mean value theorem (MVT)中值定理minimum 最小值method of lagrange multipliers 拉格朗日乘数法modulus 绝对值multiple integral 多重积分multiple 倍数multiplier 乘子octant 卦限open interval integral 开区间积分optimization 优化法，极值法origin 原点orthogonal 正交parametric equation (BC)参数方程partial derivative 偏导数partial differential equation 偏微分方程partial fractions 部分分式piece-wise function 分段函数parabola 抛物线parabolic cylinder 抛物柱面paraboloid ：抛物面parallelepiped 平行六面体parallel lines 并行线parameter ：参数partial integration 部分积分partiton ：分割period ：周期periodic function 周期函数perpendicular lines 垂直线piecewise defined function 分段定义函数plane 平面point of inflection 反曲点point-slope form 点斜式polar axis 极轴polar coordinates 极坐标polar equation 极坐标方程pole 极点polynomial 多项式power series 幂级数product rule 积的求导法则quadrant 象限quadratic functions 二次函数quotient rule 商的求导法则radical 根式radius of convergence 收敛半径range 值域(related) rate of change with time (时间)变化率rational function 有理函数reciprocal 倒数remainder theorem 余数定理Riemann sum 黎曼和Riemannian geometry 黎曼几何right-hand limit 右极限Rolle's theorem 罗尔(中值)定理root 根rotation 旋转secant line 割线second derivative 二阶导数second derivative test 二阶导数试验法second partial derivative 二阶偏导数series 级数shell method (积分)圆筒法sine function 正弦函数singularity 奇点slant 母线slant asymptote 斜渐近线slope 斜率slope-intercept equation of a line 直线的斜截式smooth curve 平滑曲线smooth surface 平滑曲面solid of revolution 旋转体symmetry 对称性substitution 代入法、变量代换tangent function 正切函数tangent line 切线tangent plane 切(平)面tangent vector 切矢量taylor's series 泰勒级数three-dimensional analytic geometry 空间解析几何total differentiation 全微分trapezoid rule 梯形(积分)法则。
数学专业英语1
数学专业英语1学专业英语－How to Write Mathematics?How to Write Mathematics?------ Honesty is the Best PolicyThe purpose of using good mathematical language is, of course, to make the u nderstanding of the subject easy for the reader, and perhaps even pleasant. The style should be good not in the sense of flashy brilliance, but good in the se nse of perfect unobtrusiveness. The purpose is to smooth the reader’s wanted, not pedantry; understanding, not fuss.The emphasis in the preceding paragraph, while perhaps necessary, might see m to point in an undesirable direction, and I hasten to correct a possible misin terpretation. While avoiding pedantry and fuss, I do not want to avoid rigor an d precision; I believe that these aims are reconcilable. I do not mean to advise a young author to be very so slightly but very very cleverly dishonest and to gloss over difficulties. Sometimes, for instance, there may be no better way t o get a result than a cumbersome computation. In that case it is the author’s duty to carry it out, in public; the he can do to alleviate it is to extend his s ympathy to the reader by some phrase such as “unfortunately the only known proof is the following cumbersome comp utation.”Here is the sort of the thing I mean by less than complete honesty. At a certa in point, having proudly proved a proposition P, you feel moved to say: “Not e, however, that p does not imply q”, and then, thinking that you’ve done a good expositor y job, go happily on to other things. Your motives may be per fectly pure, but the reader may feel cheated just the same. If he knew all abo ut the subject, he wouldn’t be reading you; for him thenonimplication is, qui te likely, unsupported. Is it obvious? (Say so.) Will a counterexample be suppl ied later? (Promise it now.) Is it a standard present purposes irrelevant part of the literature? (Give a reference.) Or, horrible dictum, do you merely mean th at you have tried to derive q from p, you failed, a nd you don’t in fact know whether p implies q? (Confess immediately.) any event: take the reader into y our confidence.There is nothing wrong with often derided “obvious”and “easy to see”, b ut there are certain minimal rules to their use. Surely when you wrote that so mething was obvious, you thought it was. When, a month, or two months, or six months later, you picked up the manuscript and re-read it, did you still thi nk that something was obvious? (A few months’ripening always improves ma nuscripts.) When you explained it to a friend, or to a seminar, was the someth ing at issue accepted as obvious? (Or did someone question it and subside, mu ttering, when you reassured him? Did your assurance demonstration or intimida tion?) the obvious answers to these rhetorical questions are among the rules th at should control the use of “obvious”. There is the most frequent source o f mathematical error: make that the “obvious”is true.It should go without saying that you are not setting out to hide facts from the reader: you are writing to uncover them. What I am saying now is that you should not hide the status of your statements and your attitude toward them eit her. Whenever you tell him something, tell him where it stands: this has been proved, that hasn’t, this will be proved, that won’t. Emphasize the importan t and minimize the trivial. The reason saying that they are obvious is to put t hem in proper perspecti e for the uninitiated. Even if your saying so makes an occasional readerangry at you, a good purpose is served by your telling him how you view the matter. But, of course, you must obey the rules. Don’t le t the reader down; he wants to believe in you. Pretentiousness, bluff, and conc ealment may not get caught out immediately, but most readers will soon sense that there is something wrong, and they will blame neither the facts nor them selves, but quite properly, the author. Complete honesty makes for greatest clar ity.---------Paul R.Haqlmosvocabularyflashy 一闪的 counter-example 反例unobtrusiveness 谦虚dictum 断言；格言forestall 阻止，先下手deride嘲弄anticipate 预见 subside沉静pedantry 迂腐；卖弄学问 mutter出怨言，喃喃自语fuss 小题大做 intimidation威下reconcilable 使一致的 rhetorical合符修辞学的gloss 掩饰 pretentiousness自命不凡alleviate 减轻，缓和bluff 欺骗implication 包含，含意concealment隐匿notes1. 本课文选自美国数学学会出版的小册子How to write mathematics 中Paul R.Halmos. 的文章第9节2. The purpose is smooth the reader’way, to anticipates his difficulties and to forestall them. Clarit y is what’s wanted, not pedantry; understanding, not fuss.意思是：目的是为读者扫清阅读上的障碍，即预先设想读者会遇到什么困难，并力求避免出现这类困难。
数学专业英语之课件一
• 3. Analysis of language – implication
蕴含
• 4. Analysis of language – equivalence
等价
• 5. Analysis of language – quantifiers
量词
• 6. Working with quantifiers
• 课程目标：帮助我们形成一种有益 的心智能力——我们祖先三千年前 形成的一种强大的思考方式.
Introduction to mathematical thingking
• 数学思维不等同于做数
• Mathematical thinking is not the same as doing mathematics – at
• 校园数学成功的关键是了解数学内部世界.相比而言， 数学思维的主要特征是考虑数学外部世界--种必要的 技能在当今世界.这门课帮助我们形成思维方式的关 键.
Introduction to mathematical thingking
• The primary audience is firstyear students at college or university who are thinking of majoring in mathematics or a mathematically-dependent subject, or high school seniors who have such a college career in mind.
学运算-至少不像数学那
least not as mathematics is
样是我们教育系统的主
typically presented in our school system. School math typically
数学专业英语第一章
特点二：科学内容的完整性与表达形式的精 炼性要求
1、长句较多 2、非限定动词使用频率高
Eg1: By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions.
3、能用英语书写文章摘要、学术会议通知、 学术交流信件等。同时培养简单的英语会 话能力。 4、为部分优秀学生攻读研究生奠定数学专 业英语的基础，同时让大部分同学了解数 学专业英语与生活英语的区别，为今后走 上工作岗位，特别是服务于IT业或外资企 业有独当一面的能力。
本课程分四部分讲解：
精选课文：1、理解数学专业文章和一般英语文章写作的不同；
对于学习数学的学生和准备从事数学研究的人员在掌握了公共英语的基本知识的基础上如果希望较快地掌握阅读英文版数学教程和科研资料的基本方法进一步学习数学专业英语是必须的
数学专业英语 第一讲
为什么要学习数学专业英语？
❖ 数学学科发展需要：20世纪90年代计算机科学技术 的迅速发展宣告了人类信息时代的到来。数学，这 个古老而又优雅的学科获得了新的发展动力和发挥 作用的舞台。
Eg2: A right angle is a 90 angle. 见P31
2. 被动语态出现频率高，应用范围广
Eg1:The Fermat Conjecture has been proved to be true.
Eg2: The function idea may be illustrated schematically in many ways. 见P54
数学专业英语(Doc版).Word6
数学专业英语－First Order Differential EquationsA differential equation is an equation between specified derivatives of a functio n, itsvalves,and known quantities.Many laws of physics are most simply and naturall y formu-lated as differential equations (or DE’s, as we shall write for short).For this r eason,DE’shave been studies by the greatest mathematicians and mathematical physicists si nce thetime of Newton..Ordinary differential equations are DE’s whose unknowns are functions of a s ingle va-riable;they arise most commonly in the study of dynamic systems and electric networks.They are much easier to treat than partial differential equations,whose unknown functionsdepend on two or more independent variables.Ordinary DE’s are classified according to their order. The order of a DE is d efined asthe largest positive integer, n, for which an n-th derivative occurs in the equati on. Thischapter will be restricted to real first order DE’s of the formΦ(x, y, y′)=0 (1)Given the function Φof three real variables, the problem is to determine all re al functions y=f(x) which satisfy the DE, that is ,all solutions of(1)in the follo wing sense.DEFINITION A solution of (1)is a differentiable function f(x) such thatΦ(x. f(x),f′(x))=0 for all x in the interval where f(x) is defined.EXAMPLE 1. In the first-other DEthe function Φis a polynomial function Φ(x, y, z)=x+ yz of three variables i n-volved. The solutions of (2) can be found by considering the identityd(x²+y²)/d x=2(x+yyˊ).From this identity,one sees that x²+y²is a con-stant if y=f(x) is any solution of (2).The equation x²+y²=c defines y implicitly as a two-valued function of x,for any positive constant c.Solving for y,we get two solutions,the(single-valued) functions y=±(c-x²)0.5,for each positive constant c.The graphs of these so-lutions,the so-called solution curves,form two families of scmicircles,which fill t he upper half-plane y>0 and the lower half-plane y>0,respectively.On the x-axis,where y=0,the DE(2) implies that x=0.Hence the DE has no solu tionswhich cross the x-axis,except possibly at the origin.This fact is easily overlook ed,because the solution curves appear to cross the x-axis;hence yˊdoes not exist, and the DE (2) is not satisfied there.The preceding difficulty also arises if one tries to solve the DE(2)for yˊ. Div iding through by y,one gets yˊ=-x/y,an equation which cannot be satisfied if y=0.The preceding difficulty is thus avoided if one restricts attention to regions where the DE(1) is normal,in the following sense.DEFINITION. A normal first-order DE is one of the formyˊ=F(x,y) (3)In the normal form yˊ=-x/y of the DE (2),the function F(x,y) is continuous i n the upper half-plane y>0 and in the lower half-plane where y<0;it is undefin ed on the x-axis.Fundamental Theorem of the Calculus.The most familiar class of differential equations consists of the first-order DE’s of the formSuch DE’s are normal and their solutions are descried by the fundamental tho rem of the calculus,which reads as follows.FUNDAMENTAL THEOREM OF THE CALCULUS. Let the function g(x)i n DE(4) be continuous in the interval a<x<b.Given a number c,there is one an d only one solution f(x) of the DE(4) in the interval such that f(a)=c. This sol ution is given by the definite integralf(x)=c+∫a x g(t)dt , c=f(a) (5)This basic result serves as a model of rigorous formulation in several respects. First,it specifies the region under consideration,as a vertical strip a<x<b in the xy-plane.Second,it describes in precise terms the class of functions g(x) consid ered.And third, it asserts the existence and uniqueness of a solution,given the “initial condition”f(a)=c.We recall that the definite integral∫a x g(t)dt=lim(maxΔt k->0)Σg(t k)Δt k , Δt k=t k-t k-1 (5ˊ)is defined for each fixed x as a limit of Ricmann sums; it is not necessary to find a formal expression for the indefinite integral ∫g(x) dx to give meanin g to the definite integral ∫a x g(t)dt,provided only that g(t) is continuous.Such f unctions as the error function crf x =(2/(π)0.5)∫0x e-t²dt and the sine integral f unction SI(x)=∫x∞[(sin t )/t]dt are indeed commonly defined as definite int egrals.Solutions and IntegralsAccording to the definition given above a solution of a DE is always a functi on. For example, the solutions of the DE x+yyˊ=0 in Example I are the func tions y=±(c-x²)0.5,whose graphs are semicircles of arbitrary diameter,centered at the origin.The graph of the solution curves are ,however,more easily describ ed by the equation x²+y²=c,describing a family of circles centered at the origi n.In what sense can such a family of curves be considered as a solution of th e DE ?To answer this question,we require a new notion.DEFINITION. An integral of DE(1)is a function of two variables,u(x,y),whic h assumes a constant value whenever the variable y is replaced by a solution y=f(x) of the DE.In the above example, the function u(x,y)=x²+y²is an integral of the DE x +yyˊ=0,because,upon replacing the variable y by any function ±( c-x²)0.5,we obtain u(x,y)=c.The second-order DEd²x/dt²=-x (2ˊ)becomes a first-order DE equivalent to (2) after setting dx/dx=y:y ( dy/dx )=-x (2)As we have seen, the curves u(x,y)=x²+y²=c are integrals of this DE.When th e DE (2ˊ)is interpreted as equation of motion under Newton’s second law,the integrals c=x²+y²represent curves of constant energy c.This illustrates an important prin ciple:an integral of a DE representing some kind of motion is a quantity that r emains unchanged through the motion.Vocabularydifferential equation 微分方程 error function 误差函数ordinary differential equation 常微分方程 sine integral function 正弦积分函数order 阶,序 diameter 直径derivative 导数 curve 曲线known quantities 已知量replace 替代unknown 未知量substitute 代入single variable 单变量strip 带形dynamic system 动力系统 exact differential 恰当微分electric network 电子网络line integral 线积分partial differential equation 偏微分方程path of integral 积分路径classify 分类 endpoints 端点polynomial 多项式 general solution 通解several variables 多变量parameter 参数family 族rigorous 严格的semicircle 半圆 existence 存在性half-plane 半平面 initial condition 初始条件region 区域uniqueness 唯一性normal 正规,正常Riemann sum 犁曼加identity 恒等(式)Notes1. The order of a DE is defined as the largest positive integral n,for which an nth derivative occurs i n the question.这是另一种定义句型,请参看附录IV.此外要注意nth derivative 之前用an 不用a .2. This chapter will be restricted to real first order differential equations of the formΦ(x,y,yˊ)=0意思是;文章限于讨论形如Φ(x,y,yˊ)=0的实一阶微分方程.有时可以用of the type代替of the form 的用法.The equation can be rewritten in the form yˊ=F(x,y).3. Dividing through by y,one gets yˊ=-x/y,…划线短语意思是:全式除以y4. As we have seen, the curves u(x,y)=x²+y²=c are integrals of this DE这里x²+y²=c 因c是参数,故此方程代表一族曲线,由此”曲线”这一词要用复数curves.5. Their solutions are described by the fundamental theorem of the calculus,which reads as follows.意思是:它们的解由微积分基本定理所描述,(基本定理)可写出如下.句中reads as follows 就是”写成(读成)下面的样子”的意思.注意follows一词中的”s”不能省略.ExerciseⅠ.Translate the following passages into Chinese:1.A differential M(x,y) dx +N(x,y) dy ,where M, N are real functions of two variables x and y, is called exact in a domain D when the line integral ∫c M(x,y) dx +N(x,y) dy is the same for all paths of int egration c in D, which have the same endpoints.Mdx+Ndy is exact if and only if there exists a continuously differentiable function u(x,y) such that M= u/ x, N=u/ y.2. For any normal first order DE yˊ=F(x,y) and any initial x0 , the initial valve problem consists of finding the solution or solutions of the DE ,for x>x0 which assumes a given initial valve f(x0)=c.3. To show that the initial valve problem is well-set requires proving theorems of existence (there isa solution), uniqueness (there is only one solution) and continuity (the solution depends continuously on t he initial value).Ⅱ. Translate the following sentences into English:1) 因为y=ч(x) 是微分方程dy/ dx=f(x,y)的解,故有dч(x)/dx=f (x,ч(x))2) 两边从x0到x取定积分得ч(x)-ч(x0)=∫x0x f(x,ч(x)) dx x0<x<x0+h3) 把y0=ч(x0)代入上式, 即有ч(x)=y0+∫x0x f(x,ч(x)) dx x0<x<x0+h4) 因此y=ч(x) 是积分方程y=y0+∫x0x f (x,y) dx定义于x0<x<x0+h 的连续解.Ⅲ. Translate the following sentences into English:1) 现在讨论型如 y=f (x,yˊ) 的微分方程的解,这里假设函数f (x, dy/dx) 有连续的偏导数.2) 引入参数dy/dx=p, 则已给方程变为y=f (x,p).3) 在y=f (x,p) x p=dy/dx p= f/ x+f/ p dp/dx4) 这是一个关于x和p的一阶微分方程,它的解法我们已经知道.5) 若(A)的通解的形式为p=ч(x,c) ,则原方程的通解为y=f (x,ч(x,c)).6) 若(A) 有型如x=ψ(x,c)的通解,则原方程有参数形式的通解 x=ψ(p,c)y=f(ψ(p,c)p)其中p是参数,c是任意常数.。
数学专业英语报告参考模板
读书报告院系：数学与统计学院专业：数学与应用数学班级：09级本一班姓名：陈明英读书报告之数学专业英语书名：《数学专业英语》作者：吴炯圻出版社：高等教育出版社本书以数学文献（主要是教材）的阅读理解为重点，讲授掌握数学专业英语的基本方法。
全书分六章。
第一章介绍数学英语的特点和阅读翻译的基本方法；第二章为精读课程，分为12课，每课含3篇短文，附有生词与词组、预习要求、注释与说明和课外作业；第三章是阅读提高课程，根据内容分为6节，共含30篇短文，取材于各个数学分支英文版的本科、研究生教材和参考书；第四章是英语数学论文写作基础；第五章是查阅（包括上网查阅），英语数学文献的基本知识；第六章是数学文献常用词汇。
本书的科学性和实用性强，适应面较广且富有时代感。
第二版对第一版做了局部修改和完善，特别在第二章增加了大量练习、扩充了词汇表并给独立单词附上了国际音标。
本书可作为数学学科各专业本科生和研究生的教材或参考书，也适用于其他相关学科领域的师生和科研人员阅读和参考。
-第一章数学专业英语的阅读和翻译初阶1.1 数学专业英语的基本特点1.2 数学专业英语的阅读与翻译第二章精读课文——入门必修2.1 数学、方程与比例2.2 几何与三角2.3 集合论的基本概念2.4 整数、有理数与实数 2.5 笛卡儿几何学的基本概念2.6 函数的概念与函数思想2.7 序列及其极限2.8 函数的导数和它的几何意义2.9微分方程简介2.10 线性空间中的相关与无关集2.11 数理逻辑入门2.12 概率论与数理统计附录1 基本运算符号与算式的读法第三章专业文选——进阶需读3.1 科技图书的序言3.2 数学基础与数学方法3.3 代数、几何与函数论3.4 数学的应用与应用数学 3.5 计算数学与计算机科学 3.6 新数学分支简介第四章英语数学论文写作基础4.1 英语数学论文的组成部分及书写要求 4.2 英语数学论文中的语法与习惯用法4.3 英语数学论文的精练要求4.4 英语标点和数学符号的正确使用附录2 参考论文附录3 来稿须知(英文，附译文)附录4 美国英语和英国英语对数学表述的影响第五章查阅英语数学文献的基本知识5.1 英语数学文献简介 5.2 英语数学文献的著录(编排)格式 5.3 英语数学文摘杂志 5.4 上网查阅数学文献和有关信息一、课程的性质和目的本课程是数理试点专业和数学与应用数学专业本科生的专业基础课程。
第二讲数学专业英语的阅读与翻译(1)案例
(1)找出句子的主要成分和较明显的语法特征. (2) 做大意群的划分. (3) 然后做小意群的划分. (4) 最后进行综合校对. 对数的使用把计算的劳动强度减少到这样的程度， 即许多计算当未采用对数时需要耗费几个小时，而 现在由于借助于对数只需要用原来的很少一部分时 间就能完成.
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7. 翻译的若干技巧
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（4）词序的变动
英语中，倒序的句子，翻译时，需要按自然词序译 出.英语中一般短语和定语从句，单个分词或形容词 做定语都后置，翻译时一般需要译成正常词序. The square root of a negative number is a pure imaginary .
负数的平方根是纯虚数.
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2
（4）归纳段落大意和全文主题. 2. 提高阅读理解的方法 （1）掌握基本语法，习惯用法和常用单词的基本用 法，逐步扩大常用数学单词和词组的识记和使用范围; (2)多读多练，在阅读专业英语的过程中，逐步掌握 数学专业英语的特点，不断总结提高.
3. 正确的阅读方法
3
（1）speed:60/min ； （2）理解具体信息（main idea,best title) ； （3）理解文中概念性涵义（concept) ； （4）相关推理判断和引申（infer) （5）根据上下文推测生词的涵义； （6）理解文章总体结构； （7）理解作者的意图，观点和态度； （8）区分论点和论据（例证题）.
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（6）句子成分的转换
状语译成主语，定语译成谓语.
The statement of the Gauss theorem is as follows:.
高斯定理叙述如下：
主语statement 译成了谓语.
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（7）语态和人称的改变
《数学专业英语》课件
2 三角恒等式和方程 4 三角学在几何和物理中的应用
IV. Calculus
1 极限和连续性 4 定积分及其性质
2 导数及其性质
3 导数在优化和相关速
率中的应用
5 定积分在面积和体积计算中的应用
V. Linear Algebra
1 向量和向量运算 3 线性方程组
《数学专业英语》PPT课 件
探索数学专业英语的精髓，为您呈现一场精彩的数学之旅。
I. Introduction
- 数学的定义 - 数学在现代社会中的重要性 - 课程目标
II. Algebra
1 基础代数表达式和方程 3 多项式和因式分解
2 根式和指数 4 二次方程和函数
III. Trigonom etry
2 矩阵及其运算 4 特征值和特征向量
VI. Probability and Statistics
1 概率的基本概念 3 数据的统计度量
2 离散和连续概率分布 4 假设检验和置信区间
VII. Conclusion
1 课程内容回顾
2 数学在不同领域的未 3 继
高中数学双语 第一章 集合
Contents Chapter 1 setSET1.1The Definition of SetBasically, anything with a collection of objects is a set. You could be surprised Finite sets are sets that have a finite number of members. If the elements of a finite set are listed one after another, the process will eventually “run out” ofelements to list.An infinite set is a set which is not finite. It is not possible to explicitly list out all the elements of an infinite set.All sets described above are finite because you can list or count all their elements . However, among the four sets, one set can be turned into an infinite set with one small change. That set is set number 3. If we get rid of the word positive and say instead "the set of all numbers less than 10" the set in now infinite because you cannot count all those numbers less than 101.1集合的概念Beijing Concord College Of Canada-Sina北京中加学校Positive 正的Negative 负的Integer 整数Rational number 有理数Natural number 自然数Irrational number 无理数定义：一般地，我们把研究对象放在一起，就构成了集合。
数学专业英语课文翻译(吴炯圻)
第一章：数学方程与比例1-A 什么是数学数学来自于人的社会实践，例如，工业和农业生产、商业活动、军事行动和科研工作。
与数学反过来，为实践服务和所有字段中的伟大作用。
没有现代的科学和技术分支机构可以定期制定中的数学，应用无早有需要的人来了数字和形式的概念。
然后，开发出的几何土地和三角测量的问题来自测量的问题。
若要对付一些更复杂的实际问题，男子成立，然后解决方程未知号码，因此代数发生。
17 世纪前, 男子向自己限于小学数学，即几何、三角和代数，只有常量被认为在其中。
17 世纪产业的快速发展促进了经济和技术的进展和所需变量的数量、处理从常量到带来两个分支的数学-解析几何和微积分，属于高等数学，现在有很多分支机构，其中有数学分析、高等代数、微分方程的高等数学中的可变数量的飞跃函数理论等。
数学家研究理念和主张。
所有命题公理、假设、定义和定理都。
符号是一种特殊和功能强大的数学工具，用于表示很多时候的理念和主张。
公式、数字和图表是阿拉伯数字1,2,3,4,5,6,7,8,9,0 与另外的符号"+"、减法"-"，乘"*"，除"\"和平等"="。
数学中的结论得到主要由逻辑推理和计算。
长期的数学史上，以中心地点的数学方法被占领逻辑扣除。
现在，由于电子计算机是迅速发展和广泛应用，计算的作用变得越来越多重要。
在我们这个时代计算不只用于处理大量的信息和数据，而且还进行一些只是可以做的工作较早前的逻辑推理，例如，大部分的几何定理的证明。
1--B 方程方程是平等的语句的两个相等的数字或数字符号之间。
因此(a-5）= 一5a 和x 3 = 5 是方程。
方程的两种——身份和方程的条件。
方程的算术或代数的身份。
这种方程中两名成员是相似的或成为相似的指示操作的性能。
因此12-2=2+8,(m+n)(m-n) = m n 是身份。
1—c 比与测量今天的思想沟通往往根据编号和数量的比较。
数学专业英语(只是部分,不是很完整)
Mathematicians study conceptions and propositions, Axioms, postulates, definitions and theorems are all propositions. Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often. Formulas ,figures and charts are full of different symbols. Some of the best known symbols of mathematics are the Arabic numerals 1,2,3,4,5,6,7,8,9,0 and the signs of addition “+”, subtraction “-” , multiplication “×”, division “÷” and equality “=”. 数学家研究的是概念和命题，公理，公设，定义和定理都 是命题。符号是数学中一个特殊而有用的工具，常用于表 达概念和命题。 公式，图形和图表都是不同的符号……..
1－B Equation
An equation is a statement of the equality between two equal numbers or number symbols. Equations are of two kinds---- identities and equations of condition. An arithmetic or an algebraic identity is an equation. In such an equation either the two members are alike, or become alike on the performance of the indicated operation. 等式是关于两个数或者数的符号相等的一种描述。 等式有两种－恒等式和条件等式。算术或者代数恒等式都是 等式。这种等式的两端要么一样，要么经过执行指定的运算 后变成一样。
最新Rcj数学专业英语第一讲基础知识与基本特点[PPT课件]讲学课件
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English for Mathematics
常用标点符号 ：
‖ parallel 双线号 ／ virgule 斜线号 ＆ ampersand = and ['æmpəsænd] ～ swung dash 代字号 § section ; division 分节号 → arrow 箭号；参见号 ＋ plus ；positive 加号；正号 － minus；negative 减号；负号 ± plus or minus 正负号
English for Mathematics
常用标点符号 ：
． period 句号 ， comma 逗号 ： colon 冒号 ； semicolon 分号 ！ exclamation 惊叹号 ？ question mark 问号 ... ellipsis 省略号 ¨ tandem colon 双点号 ['tændəm] " ditto 同上 ['dɪtəʊ]
English for Mathematics
English for Mathematics
基本数学符号的表示法： 1．加：用and ,plus , added to 等词表示。
2 ＋ 4 ＝ 6 可译为：
Two and four is six . Two plus four is equal to six . Two added to four equals six . Two and four makes six . Two plus four will be six . If you add two to four , you get six .
Rcj数学专业英语第一讲基础知 识与基本特点［PPT课件］
为什么要学习数学专业英语？
高等数学专业名词中英文对照(全面)
微积分英文词汇,高数名词中英文对照,高等数学术语英语翻译一览关键词：微积分英文,高等数学英文翻译,高数英语词汇来源：上海外教网 | 发布日期：2008-05-16 17:12V、X、Z:Value of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线Volume ：体积X-axis ：x轴x-coordinate ：x坐标x-intercept ：x截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点T:Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分S:Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称R:Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution , solid of ：旋转体Revolution , surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根P、Q:Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程Partial fractions ：部分分式Partial integration ：部分积分Partiton ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律M、N、O:Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的L:Laplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数I:Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分H:Higher mathematics 高等数学/高数E、F、G、H:Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyperbola ：双曲线Hyper boloid ：双曲面D:Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数of a composite function ：复合函数之导数of a constant function ：常数函数之导数directional ：方向导数domain of ：导数之定义域of exponential function ：指数函数之导数higher ：高阶导数partial ：偏导数of a power function ：幂函数之导数of a power series ：羃级数之导数of a product ：积之导数of a quotient ：商之导数as a rate of change ：导数当作变率right-hand ：右导数second ：二阶导数as the slope of a tangent ：导数看成切线之斜率Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differential equation ：微分方程partial ：偏微分方程Differentiation ：求导法implicit ：隐求导法partial ：偏微分法term by term ：逐项求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分change of variable in ：二重积分之变数变换in polar coordinates ：极坐标二重积分C:Calculus ：微积分differential ：微分学integral ：积分学Cartesian coordinates ：笛卡儿坐标一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cauch’s Mean Value Theor em ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性at a point ：在一点处之连续性of a function ：函数之连续性on an interval ：在区间之连续性from the left ：左连续from the right ：右连续Continuous function ：连续函数Convergence ：收敛interval of ：收敛区间radius of ：收敛半径Convergent sequence ：收敛数列series ：收敛级数Coordinate：s：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标polar ：极坐标rectangular ：直角坐标spherical ：球面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder：圆柱Cylindrical Coordinates ：圆柱坐标A、B:Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法by differentials ：用微分逼近linear ：线性逼近法by Simpson’s Rule ：Simpson法则逼近法by the Trapezoidal Rule ：梯形法则逼近法Arbitrary constant ：任意常数Arc length ：弧长Area ：面积under a curve ：曲线下方之面积between curves ：曲线间之面积in polar coordinates ：极坐标表示之面积of a sector of a circle ：扇形之面积of a surface of a revolution ：旋转曲面之面积Asymptote ：渐近线horizontal ：水平渐近线slant ：斜渐近线vertical ：垂直渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴Binomial series ：二项级数微积分词汇第一章函数与极限Chapter1 Function and Limit集合 set元素 element子集 subset空集 empty set并集 union交集 intersection差集 difference of set基本集 basic set补集 complement set直积 direct product笛卡儿积 Cartesian product开区间 open interval闭区间 closed interval半开区间 half open interval有限区间 finite interval区间的长度 length of an interval无限区间 infinite interval领域 neighborhood领域的中心 centre of a neighborhood 领域的半径 radius of a neighborhood 左领域 left neighborhood右领域 right neighborhood映射 mappingX到Y的映射 mapping of X ontoY满射 surjection单射 injection一一映射 one-to-one mapping双射 bijection算子 operator变化 transformation函数 function逆映射 inverse mapping复合映射 composite mapping自变量 independent variable因变量 dependent variable定义域 domain函数值 value of function函数关系 function relation值域 range自然定义域 natural domain单值函数 single valued function多值函数 multiple valued function单值分支 one-valued branch函数图形 graph of a function绝对值函数 absolute value符号函数 sigh function整数部分 integral part阶梯曲线 step curve当且仅当 if and only if(iff)分段函数 piecewise function上界 upper bound下界 lower bound有界 boundedness无界 unbounded函数的单调性 monotonicity of a function单调增加的 increasing单调减少的 decreasing单调函数 monotone function函数的奇偶性 parity(odevity) of a function 对称 symmetry偶函数 even function奇函数 odd function函数的周期性 periodicity of a function周期 period反函数 inverse function直接函数 direct function复合函数 composite function中间变量 intermediate variable函数的运算 operation of function基本初等函数 basic elementary function初等函数 elementary function幂函数 power function指数函数 exponential function对数函数 logarithmic function三角函数 trigonometric function反三角函数 inverse trigonometric function 常数函数 constant function双曲函数 hyperbolic function双曲正弦 hyperbolic sine双曲余弦 hyperbolic cosine双曲正切 hyperbolic tangent反双曲正弦 inverse hyperbolic sine反双曲余弦 inverse hyperbolic cosine反双曲正切 inverse hyperbolic tangent极限 limit数列 sequence of number收敛 convergence收敛于 a converge to a发散 divergent极限的唯一性 uniqueness of limits收敛数列的有界性 boundedness of a convergent sequence子列 subsequence函数的极限 limits of functions函数当x趋于x0时的极限 limit of functions as x approaches x0左极限 left limit右极限 right limit单侧极限 one-sided limits水平渐近线 horizontal asymptote无穷小 infinitesimal无穷大 infinity铅直渐近线 vertical asymptote夹逼准则 squeeze rule单调数列 monotonic sequence高阶无穷小 infinitesimal of higher order低阶无穷小 infinitesimal of lower order同阶无穷小 infinitesimal of the same order作者：新少年特工 2007-10-8 18:37 回复此发言--------------------------------------------------------------------------------2 高等数学-翻译等阶无穷小 equivalent infinitesimal函数的连续性 continuity of a function增量 increment函数在x0连续 the function is continuous at x0左连续 left continuous右连续 right continuous区间上的连续函数 continuous function函数在该区间上连续 function is continuous on an interval不连续点 discontinuity point第一类间断点 discontinuity point of the first kind第二类间断点 discontinuity point of the second kind初等函数的连续性 continuity of the elementary functions定义区间 defined interval最大值 global maximum value (absolute maximum)最小值 global minimum value (absolute minimum)零点定理 the zero point theorem介值定理 intermediate value theorem第二章导数与微分Chapter2 Derivative and Differential速度 velocity匀速运动 uniform motion平均速度 average velocity瞬时速度 instantaneous velocity圆的切线 tangent line of a circle切线 tangent line切线的斜率 slope of the tangent line位置函数 position function导数 derivative可导 derivable函数的变化率问题 problem of the change rate of a function导函数 derived function左导数 left-hand derivative右导数 right-hand derivative单侧导数 one-sided derivatives在闭区间【a,b】上可导 is derivable on the closed interval [a,b] 切线方程 tangent equation角速度 angular velocity成本函数 cost function边际成本 marginal cost链式法则 chain rule隐函数 implicit function显函数 explicit function二阶函数 second derivative三阶导数 third derivative高阶导数 nth derivative莱布尼茨公式 Leibniz formula对数求导法 log- derivative参数方程 parametric equation相关变化率 correlative change rata微分 differential可微的 differentiable函数的微分 differential of function自变量的微分 differential of independent variable微商 differential quotient间接测量误差 indirect measurement error绝对误差 absolute error相对误差 relative error第三章微分中值定理与导数的应用Chapter3 MeanValue Theorem of Differentials and the Application of Derivatives罗马定理Rolle’s theorem费马引理Fermat’s lemma拉格朗日中值定理Lagrange’s mean value theorem驻点 stationary point稳定点 stable point临界点 critical point辅助函数 auxiliary function拉格朗日中值公式Lagrange’s mean value formula柯西中值定理Cauchy’s mean value theorem洛必达法则L’Hospital’s Rule0/0型不定式 indeterminate form of type 0/0不定式 indeterminate form泰勒中值定理Taylor’s mean value theorem泰勒公式 Taylor formula余项 remainder term拉格朗日余项 Lagrange remainder term麦克劳林公式Maclaurin’s formula佩亚诺公式 Peano remainder term凹凸性 concavity凹向上的 concave upward, cancave up凹向下的，向上凸的concave downward’ concave down拐点 inflection point函数的极值 extremum of function极大值 local(relative) maximum最大值 global(absolute) mximum极小值 local(relative) minimum最小值 global(absolute) minimum目标函数 objective function曲率 curvature弧微分 arc differential平均曲率 average curvature曲率园 circle of curvature曲率中心 center of curvature曲率半径 radius of curvature渐屈线 evolute渐伸线 involute根的隔离 isolation of root隔离区间 isolation interval切线法 tangent line method第四章不定积分Chapter4 Indefinite Integrals原函数 primitive function(antiderivative)积分号 sign of integration被积函数 integrand积分变量 integral variable积分曲线 integral curve积分表 table of integrals换元积分法 integration by substitution分部积分法 integration by parts分部积分公式 formula of integration by parts有理函数 rational function真分式 proper fraction假分式 improper fraction第五章定积分Chapter5 Definite Integrals曲边梯形 trapezoid with曲边 curve edge窄矩形 narrow rectangle曲边梯形的面积 area of trapezoid with curved edge积分下限 lower limit of integral积分上限 upper limit of integral积分区间 integral interval分割 partition积分和 integral sum可积 integrable矩形法 rectangle method积分中值定理 mean value theorem of integrals函数在区间上的平均值 average value of a function on an integvals 牛顿－莱布尼茨公式 Newton-Leibniz formula微积分基本公式 fundamental formula of calculus换元公式 formula for integration by substitution递推公式 recurrence formula反常积分 improper integral反常积分发散 the improper integral is divergent反常积分收敛 the improper integral is convergent无穷限的反常积分 improper integral on an infinite interval无界函数的反常积分 improper integral of unbounded functions绝对收敛 absolutely convergent第六章定积分的应用Chapter6 Applications of the Definite Integrals元素法 the element method面积元素 element of area平面图形的面积 area of a luane figure直角坐标又称“笛卡儿坐标(Cartesian coordinates)”极坐标 polar coordinates抛物线 parabola椭圆 ellipse旋转体的面积 volume of a solid of rotation旋转椭球体 ellipsoid of revolution, ellipsoid of rotation 曲线的弧长 arc length of acurve可求长的 rectifiable光滑 smooth功 work水压力 water pressure引力 gravitation变力 variable force第七章空间解析几何与向量代数Chapter7 Space Analytic Geometry and Vector Algebra向量 vector自由向量 free vector单位向量 unit vector零向量 zero vector相等 equal平行 parallel向量的线性运算 linear poeration of vector三角法则 triangle rule平行四边形法则 parallelogram rule交换律 commutative law结合律 associative law负向量 negative vector差 difference分配律 distributive law空间直角坐标系 space rectangular coordinates坐标面 coordinate plane卦限 octant向量的模 modulus of vector向量a与b的夹角 angle between vector a and b方向余弦 direction cosine方向角 direction angle向量在轴上的投影 projection of a vector onto an axis数量积，外积，叉积 scalar product,dot product,inner product 曲面方程 equation for a surface球面 sphere旋转曲面 surface of revolution母线 generating line轴 axis圆锥面 cone顶点 vertex旋转单叶双曲面 revolution hyperboloids of one sheet旋转双叶双曲面 revolution hyperboloids of two sheets柱面 cylindrical surface ,cylinder圆柱面 cylindrical surface准线 directrix抛物柱面 parabolic cylinder二次曲面 quadric surface椭圆锥面 dlliptic cone椭球面 ellipsoid单叶双曲面 hyperboloid of one sheet双叶双曲面 hyperboloid of two sheets旋转椭球面 ellipsoid of revolution椭圆抛物面 elliptic paraboloid旋转抛物面 paraboloid of revolution双曲抛物面 hyperbolic paraboloid马鞍面 saddle surface椭圆柱面 elliptic cylinder双曲柱面 hyperbolic cylinder抛物柱面 parabolic cylinder空间曲线 space curve空间曲线的一般方程 general form equations of a space curve 空间曲线的参数方程 parametric equations of a space curve 螺转线 spiral螺矩 pitch投影柱面 projecting cylinder投影 projection平面的点法式方程 pointnorm form eqyation of a plane法向量 normal vector平面的一般方程 general form equation of a plane两平面的夹角 angle between two planes点到平面的距离 distance from a point to a plane空间直线的一般方程 general equation of a line in space方向向量 direction vector直线的点向式方程 pointdirection form equations of a line 方向数 direction number直线的参数方程 parametric equations of a line两直线的夹角 angle between two lines垂直 perpendicular直线与平面的夹角 angle between a line and a planes平面束 pencil of planes平面束的方程 equation of a pencil of planes行列式 determinant系数行列式 coefficient determinant第八章多元函数微分法及其应用Chapter8 Differentiation of Functions of Several Variables and Its Application一元函数 function of one variable多元函数 function of several variables内点 interior point外点 exterior point边界点 frontier point,boundary point聚点 point of accumulation开集 openset闭集 closed set连通集 connected set开区域 open region闭区域 closed region有界集 bounded set无界集 unbounded setn维空间 n-dimentional space二重极限 double limit多元函数的连续性 continuity of function of seveal连续函数 continuous function不连续点 discontinuity point一致连续 uniformly continuous偏导数 partial derivative对自变量x的偏导数 partial derivative with respect to independent variable x高阶偏导数 partial derivative of higher order二阶偏导数 second order partial derivative混合偏导数 hybrid partial derivative全微分 total differential偏增量 oartial increment偏微分 partial differential全增量 total increment可微分 differentiable必要条件 necessary condition充分条件 sufficient condition叠加原理 superpostition principle全导数 total derivative中间变量 intermediate variable隐函数存在定理 theorem of the existence of implicit function曲线的切向量 tangent vector of a curve法平面 normal plane向量方程 vector equation向量值函数 vector-valued function切平面 tangent plane法线 normal line方向导数 directional derivative梯度 gradient数量场 scalar field梯度场 gradient field向量场 vector field势场 potential field引力场 gravitational field引力势 gravitational potential曲面在一点的切平面 tangent plane to a surface at a point曲线在一点的法线 normal line to a surface at a point无条件极值 unconditional extreme values条件极值 conditional extreme values拉格朗日乘数法 Lagrange multiplier method拉格朗日乘子 Lagrange multiplier经验公式 empirical formula最小二乘法 method of least squares均方误差 mean square error第九章重积分Chapter9 Multiple Integrals二重积分 double integral可加性 additivity累次积分 iterated integral体积元素 volume element三重积分 triple integral直角坐标系中的体积元素 volume element in rectangular coordinate system 柱面坐标 cylindrical coordinates柱面坐标系中的体积元素 volume element in cylindrical coordinate system 球面坐标 spherical coordinates球面坐标系中的体积元素 volume element in spherical coordinate system 反常二重积分 improper double integral曲面的面积 area of a surface质心 centre of mass静矩 static moment密度 density形心 centroid转动惯量 moment of inertia参变量 parametric variable第十章曲线积分与曲面积分Chapter10 Line(Curve)Integrals and Surface Integrals对弧长的曲线积分 line integrals with respect to arc hength第一类曲线积分 line integrals of the first type对坐标的曲线积分 line integrals with respect to x,y,and z第二类曲线积分 line integrals of the second type有向曲线弧 directed arc单连通区域 simple connected region复连通区域 complex connected region格林公式 Green formula第一类曲面积分 surface integrals of the first type对面的曲面积分 surface integrals with respect to area有向曲面 directed surface对坐标的曲面积分 surface integrals with respect to coordinate elements 第二类曲面积分 surface integrals of the second type有向曲面元 element of directed surface高斯公式 gauss formula拉普拉斯算子 Laplace operator格林第一公式Green’s first formula通量 flux散度 divergence斯托克斯公式 Stokes formula环流量 circulation旋度 rotation,curl第十一章无穷级数Chapter11 Infinite Series一般项 general term部分和 partial sum余项 remainder term等比级数 geometric series几何级数 geometric series公比 common ratio调和级数 harmonic series柯西收敛准则Cauchy convergence criteria, Cauchy criteria for convergence正项级数 series of positive terms达朗贝尔判别法D’Alembert test柯西判别法 Cauchy test交错级数 alternating series绝对收敛 absolutely convergent条件收敛 conditionally convergent柯西乘积 Cauchy product函数项级数 series of functions发散点 point of divergence收敛点 point of convergence收敛域 convergence domain和函数 sum function幂级数 power series幂级数的系数 coeffcients of power series阿贝尔定理 Abel Theorem收敛半径 radius of convergence收敛区间 interval of convergence泰勒级数 Taylor series麦克劳林级数 Maclaurin series二项展开式 binomial expansion近似计算 approximate calculation舍入误差 round-off error,rounding error欧拉公式Euler’s formula魏尔斯特拉丝判别法 Weierstrass test三角级数 trigonometric series振幅 amplitude角频率 angular frequency初相 initial phase矩形波 square wave谐波分析 harmonic analysis直流分量 direct component基波 fundamental wave二次谐波 second harmonic三角函数系 trigonometric function system傅立叶系数 Fourier coefficient傅立叶级数 Forrier series周期延拓 periodic prolongation正弦级数 sine series余弦级数 cosine series奇延拓 odd prolongation偶延拓 even prolongation傅立叶级数的复数形式 complex form of Fourier series第十二章微分方程Chapter12 Differential Equation解微分方程 solve a dirrerential equation常微分方程 ordinary differential equation偏微分方程 partial differential equation,PDE微分方程的阶 order of a differential equation微分方程的解 solution of a differential equation微分方程的通解 general solution of a differential equation初始条件 initial condition微分方程的特解 particular solution of a differential equation初值问题 initial value problem微分方程的积分曲线 integral curve of a differential equation可分离变量的微分方程 variable separable differential equation隐式解 implicit solution隐式通解 inplicit general solution衰变系数 decay coefficient衰变 decay齐次方程 homogeneous equation一阶线性方程 linear differential equation of first order非齐次 non-homogeneous齐次线性方程 homogeneous linear equation非齐次线性方程 non-homogeneous linear equation常数变易法 method of variation of constant暂态电流 transient stata current稳态电流 steady state current伯努利方程 Bernoulli equation全微分方程 total differential equation积分因子 integrating factor高阶微分方程 differential equation of higher order悬链线 catenary高阶线性微分方程 linera differential equation of higher order自由振动的微分方程 differential equation of free vibration强迫振动的微分方程 differential equation of forced oscillation串联电路的振荡方程 oscillation equation of series circuit二阶线性微分方程 second order linera differential equation线性相关 linearly dependence线性无关 linearly independce二阶常系数齐次线性微分方程 second order homogeneour linear differential equation with constant coefficient二阶变系数齐次线性微分方程 second order homogeneous linear differential equation with variable coefficient特征方程 characteristic equation无阻尼自由振动的微分方程 differential equation of free vibration with zero damping固有频率 natural frequency简谐振动 simple harmonic oscillation,simple harmonic vibration微分算子 differential operator待定系数法 method of undetermined coefficient共振现象 resonance phenomenon欧拉方程 Euler equation幂级数解法 power series solution数值解法 numerial solution勒让德方程 Legendre equation微分方程组 system of differential equations常系数线性微分方程组system of linera differential equations with constant coefficientV、X、Z:Value of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线Volume ：体积X-axis ：x轴x-coordinate ：x坐标x-intercept ：x截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点T:Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分S:Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称R:Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution , solid of ：旋转体Revolution , surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根P、Q:Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程Partial fractions ：部分分式Partial integration ：部分积分Partiton ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律M、N、O:Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程Origin ：原点Orthogonal ：正交的L:Laplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数I:Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分H:Higher mathematics 高等数学/高数E、F、G、H:Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Horizontal asymptote ：水平渐近线。
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B(a,r), we have h(x)d (x) h(a) . B
特点二：科学内容的完整性与表达形式的精 炼性要求
1、长句较多 2、非限定动词使用频率高
Eg1: By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions.
2.提高阅读能力的训练
❖ 练好英语基本功，逐步扩大常用数学单词和词组的 识记和使用范围。
❖ 多读多练，在阅读中总结提高。 ❖ 努力掌握正确的阅读方法。 （1）以意群为单位阅读； （2）既要会精读，又要学会泛读和查阅； （3）三个步骤，即先粗、后细、再加深。
先粗:尽快抓住主旨和大意。可以抓住大小标题、 关键词，根据已有专业知识来理解并做大体估计。
数学专业英语 第一讲
为什么要学习数学专业英语？
❖ 数学学科发展需要：20世纪90年代计算机科学技术 的迅速发展宣告了人类信息时代的到来。数学，这 个古老而又优雅的学科获得了新的发展动力和发挥 作用的舞台。
❖ 现在国际数学教育与研究交流日益频繁，很多最新 的研究成果是以英文形式呈现的。专业人员需要有 阅读翻译英语，甚至用英文写作论文的能力。对于 学习数学的学生和准备从事数学研究的人员，在掌 握了公共英语的基本
4、数学词汇的形式多样、数量庞大 Icosahedral，heptagon(七边形） heliciods
Algebra(代数)，geometry(几何)，trigonometry （三角学），isosceles triangles（等腰三角形）等
合成词：right-handed(system)右手系 jump-function(跳跃函数)
3. 主动语态句型也多数用于强调事实，而 不是强调行为发出者及其情感
Eg1:Given 0 ，there exists a number N>0 such that an nce h(x) is harmonic on neighborhood of
学习本课程的基本要求：
1、认真自学好教材，并勾画出学习重点。 /中国知网翻译助手 2、积极参与本课程的讨论，做好听课笔记。 3、按时独立完成平时作业。
1.1 数学专业英语的基本特点
特点一：注意对客观事实与真理的描述
1. 时态常用一般现在时
Eg1: Equations are of two kinds-identities and equations of condition.见P19
Eg2: A right angle is a 90 angle. 见P31
2. 被动语态出现频率高，应用范围广
Eg1:The Fermat Conjecture has been proved to be true.
Eg2: The function idea may be illustrated schematically in many ways. 见P54
(8)采用反证法论述时，开头常用虚拟语态给出假设， 结束时用“This contradicts the hypothesis” 之类句子 表示推导的结果与原假设矛盾。
6、形成了一批数学专业性很强的特殊记号和表达 方式 (1) 表示充分必要条件
The sufficient and necessary condition for the
3、能用英语书写文章摘要、学术会议通知、 学术交流信件等。同时培养简单的英语会 话能力。 4、为部分优秀学生攻读研究生奠定数学专 业英语的基础，同时让大部分同学了解数 学专业英语与生活英语的区别，为今后走 上工作岗位，特别是服务于IT业或外资企 业有独当一面的能力。
本课程分四部分讲解：
精选课文：1、理解数学专业文章和一般英语文章写作的不同；
❖ 例 “To be more accurate,we must measure what fractional part of the book the desk exceeds 5 book-lengths.” 这句话要意译：为了得到更准确的值，我们 要测量出课桌超过5倍书长的部分是（那本） 书的几分之一。
(2)用if、when、as为连接词的条件从句
Suppose D is an open set with its closure in G.
(3) 用such that 为连接词的从句表示条件或不种条 件 Suppose f(x) is a function on a domain D such that
派生词：irregularity（非正则性）、inhomogeneous （非齐次的）、idempotence（幂等性）
以数学家名字命名的术语：Jacobian(雅克比行列 式），Laplacian(拉普拉斯算符)，
5、表示条件、推理根据的句型相对固定 (1)用if、when、as为连接词的条件从句 The function f(x) approaches infinity as x tends to zero.
后细:进一步阐述主旨论据的具体事实和细节；根 据英语知识做句子的语法分析，按上下文的数学 内容判断词义，辨明意群与句子的意思及它们之 间的关系。
再加深:抓住长句主要成分，适当分解，弄清意群 的含义与相互关系，综合成全句的意思。尽可能 深刻理解作者的观点、意图，准确了解全文的意 义。
三、专业英语翻译的要求
❖ (1)抓住主句，了解大意：对数的应用在一定程度上减少了 计算的劳动量。
❖ (2)抓住从句来理清出从关系：进一步说明计算量减少产生 的结果。
❖ (3)逐步细化：that从句中， with their aid 是条件， in a small fraction of that time表示时间范围；which引导 的定语从句用虚拟语气说明对calculations的假定(without the use of logarithms,can 是条件)；fraction在 “a small fraction of that time”不直译为分数(部分)
(6) 表示推理的根据常用“by短语”，有时也用 “according to”.
By Lemma 2 we have x y .
(7)有时用现在分词表示“经过……而得到……”（推论
Integrating the above inequality twice，we see that
y'(t) c0t log t .
四、专业英语翻译的基本方法
❖ 1. 翻译的步骤
正确理解原文，恰当表达成中文，认真校对 ❖ 2.长句翻译的实例分析
*一般说来，短的句子的句型和成分相对简单，单词之间 的关系较容易搞清楚，难点在于理解和翻译中对词义必要 的转换和引申。 *翻译长句的一般程序是“总-分-总”。
❖ 例 The use of logarithms(对数) has decreased the
labour of computing to such an extent that many calculations,which would require hours without the use of logarithms,can be performed with their aid in a small fraction of that time. 这是一个主从复合句。
知识的基础上，如果希望较快地掌握阅读英文 版数学教程和科研资料的基本方法，进一步学 习数学专业英语是必须的。
❖由于数学内容的英语表达有其特殊性，阅读 时不仅要靠个人在公共英语上的基本功，而且 还要靠数学基本知识的掌握及其运用的能力。
开设本课程目的
1、为数学专业本科大学生阅读英文数学文章 和发表数学英文文章做培训，同时扩大学生英 语知识面，为相关大学生的研究生考试做英语 保温和一定强化。 2、掌握数学专业词汇，培养快速阅读数学专业 英语的能力。了解数学的学科分类，英文学术 论文摘要的检索，数学文章的英译汉和汉译英 的技巧。
3、名词化结构及其他简化表达的形式也较常出现 Eg1: The line rotates on x-axis, which forms a conicoid.
The rotation of the line on x-axis forms a conicoid.
特点三：数学的专业性十分典型
1、数学符号、公式和图表到处可见 2、专业术语是构建数学大厦的砖瓦 Eg: integer, diameter, differential, triangle 3、半专业性术语穿插频繁、词义多变 Eg: function, power, set,right
Given 0，there exists 0 Such that f (x) f (x0) for all x with x x0 .
(4) 逻辑符号的使用.
0 0( x x0 f (x) f (x0) ).
数学专业英语 第二讲
2、掌握数学一般的主要不同专业文章中的基本术语和普遍常用 词汇，英文习作的一般习惯。
专业文选：1、掌握数学英语专业文献和普通英语文章的不同；
2、了解专业文献方面知识。
英文数学论文写作基础：掌握英文论文的基本写作格式、
用词和符号等方面的规范。