数学专业英语(Doc版).10
数学专业英语
数学专业英语课后答案2.1数学、方程与比例词组翻译1.数学分支branches of mathematics，算数arithmetics，几何学geometry，代数学algebra，三角学trigonometry，高等数学higher mathematics，初等数学elementary mathematics，高等代数higher algebra，数学分析mathematical analysis，函数论function theory，微分方程differential equation2.命题proposition，公理axiom，公设postulate，定义definition，定理theorem，引理lemma，推论deduction3.形form，数number，数字numeral，数值numerical value，图形figure，公式formula，符号notation(symbol)，记法/记号sign，图表chart4.概念conception，相等equality，成立/真true，不成立/不真untrue，等式equation，恒等式identity，条件等式equation of condition，项/术语term，集set，函数function，常数constant，方程equation，线性方程linear equation，二次方程quadratic equation5.运算operation，加法addition，减法subtraction，乘法multiplication，除法division，证明proof，推理deduction，逻辑推理logical deduction6.测量土地to measure land，推导定理to deduce theorems，指定的运算indicated operation，获得结论to obtain the conclusions，占据中心地位to occupy the centric place汉译英（1）数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。
数学专业英语词汇英汉对照
1 概率论与数理统计词汇英汉对照表A absolute value 绝对值accept 接受acceptable region 接受域additivity 可加性adjusted 调整的alternative hypothesis 对立假设analysis 分析analysis of covariance 协方差分析analysis of variance 方差分析arithmetic mean 算术平均值association 相关性assumption 假设assumption checking 假设检验availability 有效度average 均值B balanced 平衡的band 带宽bar chart 条形图beta-distribution 贝塔分布between groups 组间的bias 偏倚binomial distribution 二项分布binomial test 二项检验C calculate 计算case 个案category 类别center of gravity 重心central tendency 中心趋势chi-square distribution 卡方分布chi-square test 卡方检验classify 分类cluster analysis 聚类分析coefficient 系数coefficient of correlation 相关系数collinearity 共线性column 列compare 比较comparison 对照components 构成，分量compound 复合的confidence interval 置信区间consistency 一致性constant 常数continuous variable 连续变量control charts 控制图correlation 相关covariance 协方差covariance matrix 协方差矩阵critical point 临界点critical value 临界值crosstab 列联表cubic 三次的，立方的cubic term 三次项cumulative distribution function 累加分布函数curve estimation 曲线估计D data 数据default 默认的definition 定义deleted residual 剔除残差density function 密度函数dependent variable 因变量description 描述design of experiment 试验设计deviations 差异df.（degree of freedom）自由度diagnostic 诊断dimension 维discrete variable 离散变量discriminant function 判别函数discriminatory analysis 判别分析distance 距离distribution 分布D-optimal design D-优化设计E eaqual 相等effects of interaction 交互效应efficiency 有效性eigenvalue 特征值equal size 等含量equation 方程error 误差estimate 估计estimation of parameters 参数估计estimations 估计量evaluate 衡量exact value 精确值expectation 期望expected value 期望值exponential 指数的exponential distributon 指数分布extreme value 极值F factor 因素，因子factor analysis 因子分析factor score 因子得分factorial designs 析因设计factorial experiment 析因试验fit 拟合fitted line 拟合线fitted value 拟合值fixed model 固定模型fixed variable 固定变量fractional factorial design 部分析因设计frequency 频数F-test F检验full factorial design 完全析因设计function 函数G gamma distribution 伽玛分布geometric mean 几何均值group 组H harmomic mean 调和均值heterogeneity 不齐性histogram 直方图homogeneity 齐性homogeneity of variance 方差齐性hypothesis 假设hypothesis test 假设检验I independence 独立independent variable 自变量independent-samples 独立样本index 指数index of correlation 相关指数interaction 交互作用interclass correlation 组内相关interval estimate 区间估计intraclass correlation 组间相关inverse 倒数的iterate 迭代K kernal 核Kolmogorov-Smirnov test柯尔莫哥洛夫-斯米诺夫检验kurtosis 峰度L large sample problem 大样本问题layer 层least-significant difference 最小显著差数least-square estimation 最小二乘估计least-square method 最小二乘法level 水平level of significance 显著性水平leverage value 中心化杠杆值life 寿命life test 寿命试验likelihood function 似然函数likelihood ratio test 似然比检验linear 线性的linear estimator 线性估计linear model 线性模型linear regression 线性回归linear relation 线性关系linear term 线性项logarithmic 对数的logarithms 对数logistic 逻辑的lost function 损失函数M main effect 主效应matrix 矩阵maximum 最大值maximum likelihood estimation 极大似然估计mean squared deviation（MSD）均方差mean sum of square 均方和measure 衡量media 中位数M-estimator M估计minimum 最小值missing values 缺失值mixed model 混合模型mode 众数model 模型Monte Carle method 蒙特卡罗法moving average 移动平均值multicollinearity 多元共线性multiple comparison 多重比较multiple correlation 多重相关multiple correlation coefficient 复相关系数multiple correlation coefficient 多元相关系数multiple regression analysis 多元回归分析multiple regression equation 多元回归方程multiple response 多响应multivariate analysis 多元分析N negative relationship 负相关nonadditively 不可加性nonlinear 非线性nonlinear regression 非线性回归noparametric tests 非参数检验normal distribution 正态分布null hypothesis 零假设number of cases 个案数O one-sample 单样本one-tailed test 单侧检验one-way ANOVA 单向方差分析one-way classification 单向分类optimal 优化的optimum allocation 最优配制order 排序order statistics 次序统计量origin 原点orthogonal 正交的outliers 异常值P paired observations 成对观测数据paired-sample 成对样本parameter 参数parameter estimation 参数估计partial correlation 偏相关partial correlation coefficient 偏相关系数partial regression coefficient 偏回归系数percent 百分数percentiles 百分位数pie chart 饼图point estimate 点估计poisson distribution 泊松分布polynomial curve 多项式曲线polynomial regression 多项式回归polynomials 多项式positive relationship 正相关power 幂P-P plot P-P概率图predict 预测predicted value 预测值prediction intervals 预测区间principal component analysis 主成分分析proability 概率probability density function 概率密度函数probit analysis 概率分析proportion 比例Q qadratic 二次的Q-Q plot Q-Q概率图quadratic term 二次项quality control 质量控制quantitative 数量的，度量的quartiles 四分位数R random 随机的random number 随机数random number 随机数random sampling 随机取样random seed 随机数种子random variable 随机变量randomization 随机化range 极差rank 秩rank correlation 秩相关rank statistic 秩统计量regression analysis 回归分析regression coefficient 回归系数regression line 回归线reject 拒绝rejection region 拒绝域relationship 关系reliability 可靠性repeated 重复的report 报告，报表residual 残差residual sum of squares 剩余平方和response 响应risk function 风险函数robustness 稳健性root mean square 标准差row 行run 游程run test 游程检验S sample 样本sample size 样本容量sample space 样本空间sampling 取样sampling inspection 抽样检验scatter chart 散点图S-curve S形曲线separately 单独地sets 集合sign test 符号检验significance 显著性significance level 显著性水平significance testing 显著性检验significant 显著的，有效的significant digits 有效数字skewed distribution 偏态分布skewness 偏度small sample problem 小样本问题smooth 平滑sort 排序soruces of variation 方差来源space 空间spread 扩展square 平方standard deviation 标准离差standard error of mean 均值的标准误差standardization 标准化standardize 标准化statistic 统计量statistical quality control 统计质量控制std. residual 标准残差stepwise regression analysis 逐步回归stimulus 刺激strong assumption 强假设stud. deleted residual 学生化剔除残差stud. residual 学生化残差subsamples 次级样本sufficient statistic 充分统计量sum 和sum of squares 平方和summary 概括，综述T table 表t-distribution t分布test 检验test criterion 检验判据test for linearity 线性检验test of goodness of fit 拟合优度检验test of homogeneity 齐性检验test of independence 独立性检验test rules 检验法则test statistics 检验统计量testing function 检验函数time series 时间序列tolerance limits 容许限total 总共，和transformation 转换treatment 处理trimmed mean 截尾均值true value 真值t-test t检验two-tailed test 双侧检验U unbalanced 不平衡的unbiased estimation 无偏估计unbiasedness 无偏性uniform distribution 均匀分布V value of estimator 估计值variable 变量variance 方差variance components 方差分量variance ratio 方差比various 不同的vector 向量W weight 加权，权重weighted average 加权平均值within groups 组内的Z Z score Z分数2. 最优化方法词汇英汉对照表A active constraint 活动约束active set method 活动集法analytic gradient 解析梯度approximate 近似arbitrary 强制性的argument 变量attainment factor 达到因子B bandwidth 带宽be equivalent to 等价于best-fit 最佳拟合bound 边界C coefficient 系数complex-value 复数值component 分量constant 常数constrained 有约束的constraint 约束constraint function 约束函数continuous 连续的converge 收敛cubic polynomial interpolation method 三次多项式插值法curve-fitting 曲线拟合D data-fitting 数据拟合default 默认的，默认的define 定义diagonal 对角的direct search method 直接搜索法direction of search 搜索方向discontinuous 不连续E eigenvalue 特征值empty matrix 空矩阵equality 等式exceeded 溢出的F feasible 可行的feasible solution 可行解finite-difference 有限差分first-order 一阶G Gauss-Newton method 高斯-牛顿法goal attainment problem 目标达到问题gradient 梯度gradient method 梯度法Hhandle 句柄Hessian matrix 海色矩阵I independent variables 独立变量inequality 不等式infeasibility 不可行性infeasible 不可行的initial feasible solution 初始可行解initialize 初始化inverse 逆invoke 激活iteration 迭代iteration 迭代J Jacobian 雅可比矩阵L Lagrange multiplier 拉格朗日乘子large-scale 大型的least square 最小二乘least squares sense 最小二乘意义上的Levenberg-Marquardt method列文伯格-马夸尔特法line search 一维搜索linear 线性的linear equality constraints 线性等式约束linear programming problem 线性规划问题local solution 局部解M medium-scale 中型的minimize 最小化mixed quadratic and cubic polynomial interpolation and extrapolation method 混合二次、三次多项式内插、外插法multiobjective 多目标的N nonlinear 非线性的norm 范数O objective function 目标函数observed data 测量数据optimization routine 优化过程optimize 优化optimizer 求解器over-determined system 超定系统P parameter 参数partial derivatives 偏导数polynomial interpolation method多项式插值法Q quadratic 二次的quadratic interpolation method 二次内插法quadratic programming 二次规划R real-value 实数值residuals 残差robust 稳健的robustness 稳健性，鲁棒性S scalar 标量semi-infinitely problem 半无限问题Sequential Quadratic Programming method序列二次规划法simplex search method 单纯形法solution 解sparse matrix 稀疏矩阵sparsity pattern 稀疏模式sparsity structure 稀疏结构starting point 初始点step length 步长subspace trust region method 子空间置信域法sum-of-squares 平方和symmetric matrix 对称矩阵T termination message 终止信息termination tolerance 终止容限the exit condition 退出条件the method of steepest descent 最速下降法transpose 转置U unconstrained 无约束的under-determined system 负定系统V variable 变量vector 矢量W weighting matrix 加权矩阵3 样条词汇英汉对照表A approximation 逼近array 数组a spline in b-form/b-spline b样条a spline of polynomial piece /ppform spline 分段多项式样条B bivariate spline function 二元样条函数break/breaks 断点C coefficient/coefficients 系数cubic interpolation 三次插值/三次内插cubic polynomial 三次多项式cubic smoothing spline 三次平滑样条cubic spline 三次样条cubic spline interpolation三次样条插值/三次样条内插curve 曲线D degree of freedom 自由度dimension 维数E end conditions 约束条件I input argument 输入参数interpolation 插值/内插interval 取值区间K knot/knots 节点L least-squares approximation 最小二乘拟合M multiplicity 重次multivariate function 多元函数O optional argument 可选参数order 阶次output argument 输出参数P point/points 数据点R rational spline 有理样条rounding error 舍入误差（相对误差）S scalar 标量sequence 数列（数组）spline 样条spline approximation 样条逼近/样条拟合spline function 样条函数spline curve 样条曲线spline interpolation 样条插值/样条内插spline surface 样条曲面smoothing spline 平滑样条T tolerance 允许精度U univariate function 一元函数V vector 向量W weight/weights 权重4 偏微分方程数值解词汇英汉对照表A absolute error 绝对误差absolute tolerance 绝对容限adaptive mesh 适应性网格B boundary condition 边界条件C contour plot 等值线图converge 收敛coordinate 坐标系D decomposed 分解的decomposed geometry matrix 分解几何矩阵diagonal matrix 对角矩阵Dirichlet boundary conditionsDirichlet边界条件E eigenvalue 特征值elliptic 椭圆形的error estimate 误差估计exact solution 精确解G generalized Neumann boundary condition 推广的Neumann边界条件geometry 几何形状geometry description matrix 几何描述矩阵geometry matrix 几何矩阵graphical user interface（GUI）图形用户界面H hyperbolic 双曲线的I initial mesh 初始网格J jiggle 微调L Lagrange multipliers 拉格朗日乘子Laplace equation 拉普拉斯方程linear interpolation 线性插值loop 循环M machine precision 机器精度mixed boundary condition 混合边界条件N Neuman boundary condition Neuman边界条件node point 节点nonlinear solver 非线性求解器normal vector 法向量P Parabolic 抛物线型的partial differential equation 偏微分方程plane strain 平面应变plane stress 平面应力Poisson's equation 泊松方程polygon 多边形positive definite 正定Q quality 质量R refined triangular mesh 加密的三角形网格relative tolerance 相对容限relative tolerance 相对容限residual 残差residual norm 残差范数S singular 奇异的。
数学专业英语
举一个简单的例子，当常微分方程满足f‟(x) = f(x) 的关系时，最特殊的情况是指数函数 f(x) = ex
We shall see presently that every solution of (9.1) f‟(x) = f(x) must be of the form f(x) = Cex , where C may be any constant.
The study of differential equations is one part of mathematics that, perhaps more than any other , has been directly inspired by mechanics, astronomy, and mathematical physics .Its history began in the 17th century when Newton, Leibniz, and the Bernoullis solved some simple differential equations arising from problems in geometry and mechanics .
displacement Bernoulli
n.位移
n. (人名) 伯努利
The Bernoullis 伯努利(家族) mathematical physics 数学物理
A large variety of scientific problems arise in which one tries to determine something from its rate of change.
这些方程叫做微分方程，他们的研究方式 是最难的数学分ations are classified under two main headings: ordinary and partial, depending on whether the unknown is a function of just one variable or of two or more variables. classified 分类 variable 变量
数学专业英语(Doc版).Word4
数学专业英语－Continuous Functions of One Real VariableThis lesson deals with the concept of continuity, one of the most important an d also one of the most fascinating ideas in all of mathematics. Before we give a preeise technical definition of continuity, we shall briefly discuss the concep t in an informal and intuitive way to give the reader a feeling for its meanin g.Roughly speaking the situation is this: Suppose a function f has the value f ( p )at a certain point p. Then f is said to be continuous at p if at every ne arby point x the function value f ( x )is close to f ( p ). Another way of pu tting it is as follows: If we let x move toward p, we want the corresponding f unction value f ( x )to become arbitrarily close to f ( p ), regardless of the manner in which x approaches p. We do not want sudden jumps in the values of a continuous function.Consider the graph of the function f defined by the equation f ( x ) = x –[ x ], where [ x ] denotes the greatest integer < x. At each integer we have what is known ad a jump discontinuity. For example, f ( 2 ) = 0 ,but as x a pproaches 2 from the left, f ( x )approaches the value 1, which is not equal to f ( 2 ).Therefore we have a discontinuity at 2. Note that f ( x )does appro ach f ( 2 )if we let x approach 2 from the right, but this by itself is not en ough to establish continuity at 2. In case like this, the function is called conti nuous from the right at 2 and discontinuous from the left at 2. Continuity at a point requires both continuity from the left and from the right.In the early development of calculus almost all functions that were dealt with were continuous and there was no real need at that time for a penetrating loo k into the exact meaning of continuity. It was not until late in the 18th centur y that discontinuous functions began appearing in connection with various kind s of physical problems. In particular, the work of J.B.J. Fourier(1758-1830) on the theory of heat forced mathematicians the early 19th century to examine m ore carefully the exact meaning of the word “continuity”.A satisfactory mathematical definition of continuity, expressed entirely in term s of properties of the real number system, was first formulated in 1821 by the French mathematician, Augustin-Louis Cauchy (1789-1857). His definition, whi ch is still used today, is most easily explained in terms of the limit concept to which we turn now.The definition of the limit of a function.Let f be a function defined in some open interval containing a point p, altho ugh we do not insist that f be defined at the point p itself. Let A be a real n umber.The equationf ( x ) = Ais read “The limit of f ( x ), as x approached p, is equal to A”, or “f ( x )approached A as x approached p.”It is also written without the limit symb ol, as follows:f ( x )→A as x →pThis symbolism is intended to convey the idea that we can make f ( x )as close to A as we please, provided we choose x sufficiently close to p.Our first task is to explain the meaning of these symbols entirely in terms of real numbers. We shall do this in two stages. First we introduce the concept of a neighborhood of a point, the we define limits in terms of neighborhoods.Definition of neighborhood of a point.Any open interval containing a point p as its midpoint is called a neighborho od of p.NOTATION. We denote neighborhoods by N ( p ), N1( p ), N2( p )etc. S ince a neighborhood N( p )is an open interval symmetric about p, it consists of all real x satisfying p-r < x < p+r for some r > 0. The positive number r is called the radius of the neighborhood. We designate N ( p )by N ( p; r )if we wish to specify its radius. The inequalities p-r < x < p+r are equiv alent to –r<x-p<r,and to ∣x-p∣< r. Thus N ( p; r )consists of all points x whose distance from p is less than r.In the next definition, we assume that A is a real number and that f is a fun ction defined on some neighborhood of a point p(except possibly at p) . Th e function may also be defined at p but this is irrelevant in the definition.Definition of limit of a function.The symbolismf ( x ) = A or [ f ( x )→A as x→p ]means that for every neighborhood N1( A )there is some neighborhood N2( p)such thatf ( x )∈N1( A ) whenever x ∈N2( p ) and x ≠p (*)The first thing to note about this definition is that it involves two neighborho ods,N1( A) andN2( p). The neighborhood N1( A)is specified first; it tells us how close we wish f ( x )to be to the limit A. The second neighborhood, N2( p ),tells u s how close x should be to p so that f ( x ) will be within the first neighbor hood N1( A). The essential part of the definition is that, for every N1( A),n o matter how small, there is some neighborhood N2(p)to satisfy (*). In gener al, the neighborhood N2( p)will depend on the choice of N1( A). A neighbo rhood N2( p )that works for one particular N1( A) will also work, of course, for every larger N1( A), but it may not be suitable for any smaller N1( A).The definition of limit can also be formulated in terms of the radii of the n eighborhoodsN1( A)and N2( p ). It is customary to denote the radius of N1( A) byεan d the radius of N2( p)by δ.The statement f ( x )∈N1( A ) is equivalent to the inequality ∣f ( x ) –A∣<ε,and the statement x ∈N1( A) ,x ≠p ,is equivalent to the inequalities 0<∣x-p∣<δ. Therefore, the definition of limit can also be expressed as follows:The symbol f ( x ) = A means that for everyε> 0, there is aδ> 0 such th at∣f ( x ) –A∣<εwhenever 0 <∣x –p∣<δ“One-sided”limits may be defined in a similar way. For example, if f ( x )→A as x→p through values greater than p, we say that A is right-hand limi t of f at p, and we indicate this by writingf ( x ) = AIn neighborhood terminology this means that for every neighborhood N1( A) ,t here is some neighborhood N2( p) such thatf ( x )∈N1( A) wheneve r x ∈N1( A) and x > pLeft-hand limits, denoted by writing x→p-, are similarly defined by restricti ng x to values less than p.If f has a limit A at p, then it also has a right-hand limit and a left-hand li mit at p, both of these being equal to A. But a function can have a right-hand limit at p different from the left-hand limit.The definition of continuity of a function.In the definition of limit we made no assertion about the behaviour of f at the point p itself. Moreover, even if f is defined at p, its value there need not b e equal to the limit A. However, if it happens that f is defined at p and if it also happens that f ( p ) = A, then we say the function f is continuous at p. In other words, we have the following definition.Definition of continuity of a function at a point.A function f is said to be continuous at a point p if( a ) f is defined at p, and ( b ) f ( x ) = f ( p )This definition can also be formulated in term of neighborhoods. A function f is continuous at p if for every neighborhood N1( f(p))there is a neighborhood N2(p)such thatf ( x ) ∈N1( f (p)) whenever x ∈N2( p).In theε-δterminology , where we specify the radii of the neighborhoods, the definition of continuity can be restated ad follows:Function f is continuous at p if for every ε> 0 ,there is aδ> 0 such that ∣f( x ) –f ( p )∣< εwhenever ∣x –p∣< δIn the rest of this lesson we shall list certain special properties of continuou s functions that are used quite frequently. Most of these properties appear obvi ous when interpreted geometrically ; consequently many people are inclined to accept them ad self-evident. However, it is important to realize that these state ments are no more self-evident than the definition of continuity itself, and ther efore they require proof if they are to be used with any degree of generality. The proofs of most of these properties make use of the least-upper bound axio m for the real number system.THEOREM 1. (Bolzano’s theorem) Let f be continuous at each point of a cl osed interval [a, b] and assume that f ( a )an f ( b )have opposite signs. T hen there is at least one c in the open interval (a ,b) such that f ( c )= 0.THEOREM 2. Sign-preserving property of continuous functions. Let f be conti nuious at c and suppose that f ( c )≠0. Then there is an interval (c-δ,c +δ) about c in which f has the same sign as f ( c ).THEOREM 3. Let f be continuous at each point of a closed interval [a, b]. Choose two arbitrary points x1<x2in [a, b] such that f ( x1 ) ≠f ( x2 ). T hen f takes every value between f ( x1) and f(x2)somewhere in the interval ( x1,x2 ).THEOREM 4. Boundedness theorem for continuous functions. Let f be contin uous on a closed interval [a, b]. Then f is bounded on [a, b]. That is , there is a number M > 0, such that∣f ( x )∣≤M for all x in [a, b].THEOREM 5. (extreme value theorem) Assume f is continuous on a closed i nterval [a, b]. Then there exist points c and d in [a, b] such that f ( c ) = su p f and f ( d ) = inf f .Note. This theorem shows that if f is continuous on [a, b], then sup f is its absolute maximum, and inf f is its absolute minimum.Vocabularycontinuity 连续性 assume 假定,取continuous 连续的 specify 指定, 详细说明continuous function 连续函数statement 陈述,语句intuitive 直观的 right-hand limit 右极限corresponding 对应的left-hand limit 左极限correspondence 对应 restrict 限制于graph 图形 assertion 断定approach 趋近,探索,入门consequently 因而,所以tend to 趋向 prove 证明regardless 不管,不顾 proof 证明discontinuous 不连续的 bound 限界jump discontinuity 限跳跃不连续least upper bound 上确界mathematician 科学家 greatest lower bound 下确界formulate 用公式表示,阐述boundedness 有界性limit 极限maximum 最大值Interval 区间 minimum 最小值open interval 开区间 extreme value 极值equation 方程extremum 极值neighborhood 邻域increasing function 增函数midpoint 中点decreasing function 减函数symmetric 对称的strict 严格的radius 半径(单数) uniformly continuous 一致连续radii 半径(复数) monotonic 单调的inequality 不等式monotonic function 单调函数equivalent 等价的Notes1. It wad not until late in the 18th century that discontinuous functions began appearing in connection with various kinds of physical problems.意思是:直到十八世纪末,不连续函数才开始出现于与物理学有关的各类问题中.这里It was not until …that译为“直到……才”2. The symbol f ( x ) = A means that for every ε> 0 ,there is a δ> 0, such that|f( x ) - A|<εwhenever 0 <|x –p |<δ注意此种句型.凡涉及极限的其它定义,如本课中定义函数在点P连续及往后出现的关于收敛的定义等,都有完全类似的句型,参看附录IV.有时句中there is可换为there exists; such that可换为satisfying; whenever换成if或for.3. Let…and assume (suppose)…Then…这一句型是定理叙述的一种最常见的形式;参看附录IV.一般而语文课Let假设条件的大前提,assume (suppo se)是小前提(即进一步的假设条件),而if是对具体而关键的条件的使用语.4. Approach在这里是“趋于”,“趋近”的意思,是及物动词.如:f ( x ) approaches A as x approaches p. Approach有时可代以tend to. 如f ( x )tends to A as x tends t o p.值得留意的是approach后不加to而tend之后应加to.5. as close to A as we please = arbitrarily close to A..ExerciseI. Fill in each blank with a suitable word to be chosen from the words given below:independent domain correspondenceassociates variable range(a) Let y = f ( x )be a function defined on [a, b]. Then(i) x is called the ____________variable.(ii) y is called the dependent ___________.(iii) The interval [a, b] is called the ___________ of the function.(b) In set terminology, the definition of a function may be given as follows:Given two sets X and Y, a function f : X →Y is a __________which ___________with each element of X one and only one element of Y.II. a) Which function, the exponential function or the logarithmic function, has the property that it satisfi es the functional equationf ( xy ) = f ( x ) + f ( v )b) Give the functional equation which will be satisfied by the function which you do not choose in (a).III. Let f be a real-valued function defined on a set S of real numbers. Then we have the following two definitions:i) f is said to be increasing on the set S if f ( x ) < f ( y )for every pair of points x and y with x < y.ii) f is said to have and absolute maximum on the set S if there is a point c in S such that f ( x ) < f ( c )for all x∈S.Now definea) a strictly increasing function;b) a monotonic function;c) the relative (or local ) minimum of f .IV. Translate theorems 1-3 into Chinese.V. Translate the following definition into English:定义:设E 是定义在实数集E上的函数,那么, 当且仅当对应于每一ε>0(ε不依赖于E上的点)存在一个正数δ使得当p和q属于E且|p –q| <δ时有|f ( p ) – f ( q )|<ε,则称f在E上一致连续.。
数学专业英语（Doc版）.12
数学专业英语（Doc版）.12数学专业英语－Linear ProgrammingLinear Programming is a relatively new branch of mathematics.The cornerstone of this exciting field was laid independently bu Leonid V. Kantorovich,a Russ ian mathematician,and by Tjalling C,Koopmans, a Yale economist,and George D. Dantzig,a Stanford mathematician. Kantorovich’s pioneering work was moti vated by a production-scheduling problem suggested by the Central Laboratory of the Len ingrad Plywood Trust in the late 1930’s. The development in the U nited States was influenced by the scientific need in World War II to solve lo gistic military problems, such as deploying aircraft and submarines at strategic positions and airlifting supplies and personnel.The following is a typical linear programming problem:A manufacturing company makes two types of television sets: one is black and white and the other is color. The company has resources to make at most 300 sets a week. It takes $180 to make a black and white set and $270 to make a color set. The company does not want to spend more than $64,800 a wee k to make television sets. If they make a profit of $170 per black and white set and $225 per color set, how many sets of each type should the company make to have a maximum profit?This problem is discussed in detail in Supplementary Reading Material Lesson 14.Since mathematical models in linear programming problems consist of linear in equalities, the next section is devoted to suchinequalities.Recall that the linear equation lx+my+n=0represents a straight line in a plane. Every solution (x,y) of the equation lx+my+n=0is a point on this line, and vice versa.An inequality that is obtained from the linear equation lx+my+n=0by replacin g the equality sign “=”by an inequality sign < (less than), ≤(less than or equal to), > (greater than), or ≥(greater than or equal to) is called a linear i nequality in two variables x and y. Thus lx+my+n≤0, lx+my+n≥0are all lin ear liequalities. A solution of a linear inequality is an ordered pair (x,y) of nu mbers x and y for which the inequality is true.EXAMPLE 1 Graph the solution set of the pair of inequalities SOLUTION Let A be the solution set of the inequality x+y-7≤0 and B be th at of the inequalit y x-3y +6 ≥0 .Then A∩B is the solution set of the given pair of inequalities. Set A is represented by the region shaded with horizontal lines and set B by the region shaded with vertical lines in Fig.1. Therefore thecrossed-hatched region represents the solution set of the given pair of inequali ties. Observe that the point of intersection (3.4) of the two lines is in the solu tion set.Generally speaking, linear programming problems consist of finding the maxim um value or minimum value of a linear function, called the objective function, subject to some linear conditions, called constraints. For example, we may wa nt to maximize the production or profit of a company or to maximize the num ber of airplanes that can land at or take off from an airport during peak hours; or we may want to minimize the cost of production or of transportation or to minimize grocery expenses while still meeting the recommended nutritional re quirements, all subject to certain restrictions. Linearprogramming is a very use ful tool that can effectively be applied to solve problems of this kind, as illust rated by the following example.EXAMPLE 2 Maximize the function f(x,y)=5x+7y subject to the constraintsx≥0 y≥0x+y-7≤02x-3y+6≥0SOLUTION First we find the set of all possible pairs(x,y) of numbers that s atisfy all four inequalities. Such a solution is called a feasible sulution of the problem. For example, (0,0) is a feasible solution since (0,0) satisf ies the giv en conditions; so are (1,2) and (4,3).Secondly, we want to pick the feasible solution for which the giv en function f (x,y) is a maximum or minimum (maximum in this case). S uch a feasible solution is called an optimal solution.Since the constraints x ≥0 and y ≥0 restrict us to the first quadrant, it follows from example 1 that the given constraints define the polygonal regi on bounded by the lines x=0, y=0,x+y-7=0, and 2x-3y+6=0, as shown in Fig.2.Fig.2.Observe that if there are no conditions on the values of x and y, then the f unction f can take on any desired value. But recall that our goal is to determi ne the largest value of f (x,y)=5x+7y where the values of x and y are restrict ed by the given constraints: that is, we must locate that point (x,y) in the pol ygonal region OABC at which the expression 5x+7y has the maximum possibl e value.With this in mind, let us consider the equation 5x+7y=C, where C is any n umber. This equation represents a family ofparallel lines. Several members of this family, corresponding to different values of C, are exhibited in Fig.3. Noti ce that as the line 5x+7y=C moves up through the polygonal region OABC, th e value of C increases steadily. It follows from the figure that the line 5x+7y =43 has a singular position in the family of lines 5x+7y=C. It is the line farth est from the origin that still passes through the set of feasible solutions. It yiel ds the largest value of C: 43.(Remember, we are not interested in what happen s outside the region OABC) Thus the largest value of the function f(x,y)=5x+7 y subject to the condition that the point (x,y) must belong to the region OAB C is 43; clearly this maximum value occurs at the point B(3,4).Fig.3.Consider the polygonal region OABC in Fig.3. This shaded region has the p roperty that the line segment PQ joining any two points P and Q in the regio n lies entirely within the region. Such a set of points in a plane is called a c onvex set. An interesting observation about example 2 is that the maximum va lue of the objective function f occurs at a corner point of the polygonal conve x set OABC, the point B(3,4).The following celebrated theorem indicates that it was not accidental.THEOREM (Fundamental theorem of linear programming) A linear objective function f defined over a polygonal convex set attains a maximum (or minim um) value at a corner point of the set.We now summarize the procedure for solving a linear programming problem:1.Graph the polygonal region determined by the constraints.2.Find the coordinates of the corner points of the polygon.3.Evaluate the objective function at the corner points.4.Identify the corner point at which the function has an optimal value.Vocabularylinear programming 线形规划 quadrant 象限objective function 目标函数 convex 凸的constraints 限制条件，约束条件 convex set 凸集feaseble solution 容许解，可行解corner point 偶角点optimal solution 最优解simplex method 单纯形法Notes1. A Yale economist, a Stanford mathematician 这里Yale Stanford 是指美国两间著名的私立大学：耶鲁大学和斯坦福大学，这两间大学分别位于康涅狄格州（Connecticut）和加里福尼亚州(California)2. subject to some lincar conditions 解作“在某些线形条件的限制下”。
数学专业英语(Doc版).Word5
数学专业英语－Differential CalculusHistorical IntroductionNewton and Leibniz,quite independently of one another,were largely responsible for developing the ideas of integral calculus to the point where hitherto insur mountable problems could be solved by more or less routine methods.The succ essful accomplishments of these men were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch o f calculus,differential calculus.The central idea of differential calculus is the notion of derivative.Like the inte gral,the derivative originated from a problem in geometry—the problem of find ing the tangent line at a point of a curve.Unlile the integral,however,the deriva tive evolved very late in the history of mathematics.The concept was not form ulated until early in the 17th century when the French mathematician Pierre de Fermat,attempted to determine the maxima and minima of certain special func tions.Fermat’s idea,basically very simple,can be understood if we refer to a curve a nd assume that at each of its points this curve has a definite direction that ca n be described by a tangent line.Fermat noticed that at certain points where th e curve has a maximum or minimum,the tangent line must be horizontal.Thus t he problem of locating such extreme values is seen to depend on the solution of another problem,that of locating the horizontal tangents.This raises the more general question of determining the direction of the tange nt line at an arbitrary point of the curve.It was the attempt to solve this gener al problem that led Fermat to discover some of the rudimentary ideas underlyi ng the notion of derivative.At first sight there seems to be no connection whatever between the problem of finding the area of a region lying under a curve and the problem of findin g the tangent line at a point of a curve.The first person to realize that these t wo seemingly remote ideas are,in fact, rather intimately related appears to have been Newton’s teacher,Isaac Barrow(1630-1677).However,Newton and Leibniz were the first to understand the real importance of this relation and they explo ited it to the fullest,thus inaugurating an unprecedented era in the development of mathematics.Although the derivative was originally formulated to study the problem of tang ents,it was soon found that it also provides a way to calculate velocity and,mo re generally,the rate of change of a function.In the next section we shall consi der a special problem involving the calculation of a velocity.The solution of this problem contains all the essential fcatures of the derivative concept and may help to motivate the general definition of derivative which is given below.A Problem Involving VelocitySuppose a projectile is fired straight up from the ground with initial velocity o f 144 feet persecond.Neglect friction,and assume the projectile is influenced onl y by gravity so that it moves up and back along a straight line.Let f(t) denote the height in feet that the projectile attains t seconds after firing.If the force of gravity were not acting on it,the projectile would continue to move upward with a constant velocity,traveling a distance of 144 feet every second,and at ti me t we woule have f(t)=144 t.In actual practice,gravity causes the projectile t o slow down until its velocity decreases to zero and then it drops back to eart h.Physical experiments suggest that as the projectile is aloft,its height f(t) is gi ven by the formula(1)f(t)=144t –16 t2The term –16t2is due to the influence of gravity.Note that f(t)=0 when t=0 a nd when t=9.This means that the projectile returns to earth after 9 seconds and it is to be understood that formula (1) is valid only for 0<t<9.The problem we wish to consider is this:To determine the velocity of the proj ectile at each instant of its motion.Before we can understand this problem,we must decide on what is meant by the velocity at each instant.To do this,we int roduce first the notion of average velocity during a time interval,say from time t to time t+h.This is defined to be the quotient.Change in distance during time interval =f(t+h)-f(t)/hThis quotient,called a difference quotient,is a number which may be calculated whenever both t and t+h are in the interval[0,9].The number h may be positiv e or negative,but not zero.We shall keep t fixed and see what happens to the difference quotient as we take values of h with smaller and smaller absolute v alue.The limit process by which v(t) is obtained from the difference quotient is wri tten symbolically as follows:V(t)=lim(h→0)[f(t+h)-f(t)]/hThe equation is used to define velocity not only for this particular example bu t,more generally,for any particle moving along a straight line,provided the position function f is such that the differerce quotient tends to a definite limit as h approaches zero.The example describe in the foregoing section points the way to the introducti on of the concept of derivative.We begin with a function f defined at least on some open interval(a,b) on the x axis.Then we choose a fixed point in this in terval and introduce the difference quotient[f(x+h)-f(x)]/hwhere the number h,which may be positive or negative(but not zero),is such th at x+h also lies in(a,b).The numerator of this quotient measures the change in the function when x changes from x to x+h.The quotient itself is referred to a s the average rate of change of f in the interval joining x to x+h.Now we let h approach zero and see what happens to this quotient.If the quot ient.If the quotient approaches some definite values as a limit(which implies th at the limit is the same whether h approaches zero through positive values or through negative values),then this limit is called the derivative of f at x and is denoted by the symbol f’(x) (read as “f prime of x”).Thus the formal defi nition of f’(x) may be stated as follows:Definition of derivative.The derivative f’(x)is defined by the equationf’(x)=lim(h→o)[f(x+h)-f(x)]/hprovided the limit exists.The number f’(x) is also called the rate of change of f at x.In general,the limit process which produces f’(x) from f(x) gives a way of ob taining a new function f’from a given function f.This process is called differ entiation,and f’is called the first derivative of f.If f’,in turn,is defined on an interval,we can try to compute its first derivative,denoted by f’’,and is calle d the second derivative of f.Similarly,the nth derivative of f denoted by f^(n),is defined to be the first derivative of f^(n-1).We make the convention that f^(0) =f,that is,the zeroth derivative is the function itself.Vocabularydifferential calculus微积分differentiable可微的intergral calculus 积分学differentiate 求微分hither to 迄今 integration 积分法insurmountable 不能超越 integral 积分routine 惯常的integrable 可积的fuse 融合integrate 求积分originate 起源于sign-preserving保号evolve 发展，引出 axis 轴（单数）tangent line 切线 axes 轴（复数）direction 方向 contradict 矛盾horizontal 水平的contradiction 矛盾vertical 垂直的 contrary 相反的rudimentary 初步的，未成熟的composite function 合成函数，复合函数area 面积composition 复合函数intimately 紧密地interior 内部exploit 开拓，开发 interior point 内点inaugurate 开始 imply 推出，蕴含projectile 弹丸 aloft 高入云霄friction摩擦initial 初始的gravity 引力 instant 瞬时rate of change 变化率integration by parts分部积分attain 达到definite integral 定积分defferential 微分indefinite integral 不定积分differentiation 微分法 average 平均Notes1. Newton and Leibniz,quite independently of one another,were largely responsible for developing…by more or less routine methods.意思是：在很大程度上是牛顿和莱伯尼，他们相互独立地把积分学的思想发展到这样一种程度，使得迄今一些难于超越的问题可以或多或少地用通常的方法加以解决。
《数学专业英语》常用英文表达
7
max, min maximum value, minimum value Eample: max f ( x) Maximum value of f(x)
max(a1 ,an ) Maximum value of the series
a1 to an Powers and roots Example: x 2 x squared
10
Function f
2 Example: f ( x ) ax bx c , a 0. The function of x
equals a times the square of x plus b times x plus c, where a is not equal to zero.
Imaginary number and two common irrational numbers
, ratio of the circumference of a circle to its
diameter,approximate value 3.1415926
9
1 x i 1, e lim(1 ) x x
Example:
f ( x ) f double-prime x, f double-dashed x f ( x ) f triple-prime x, f triple-dashed x (4) f ( x ) f four x
8
xn x 1
Example:
3
x x
4 n
x x
fourth root (of) x
nth root (of) x
Example: 24 16 Two to the fourth power is sixteen
数学专业英语
数学专业英语数学，作为一门自然科学，对于任何一个国家和民族来说都具有非常重要的意义。
作为一种研究物质与现象本质的学科，数学通过一系列公理和推理，进行精确的描述和分析，从而刻画出了丰富的数学结构和表现形式，成为自然科学中应用最广泛的基础学科之一。
为了更好地传承和发展数学学科，我国数学教育体系逐渐完善，大量优秀的数学教师和研究人员涌现出来，为数学事业不断贡献力量。
同时，我国的数学研究不仅在量上有了大幅度的提升，更在质上得到了越来越多的认可和尊重。
在数学研究领域，英语作为国际性的语言，具有不可替代的地位。
数学专业英语，是由数学领域专业人士使用的一种专门的术语和表达方式。
正如同其他学科一样，数学专业英语也有其独特的语言规范、术语和语法，并且赋予了数学的某些具体含义。
因此，掌握数学专业英语，不仅是进入国际学术圈的重要标志，同时也是增进数学研究深度和广度的关键所在。
为此，本文将从数学专业英语的基本要素、相关研究和未来发展趋势等方面进行探讨和解析。
一、数学专业英语的基本要素1. 语法语法是数学专业英语的基本要素之一，它是制定语言规范的重要基础。
英语语法主要包括名词、动词、形容词和副词等四个基本部分。
而在数学专业英语中，这些基本部分的构成都会涉及到特定的数学概念和符号。
此外，数学专业英语还包括数学符号、公式等。
例如，对于一个数学论文中的一个公式，“f(x)=ax+b”，这个公式中的"f(x)"，代表一个函数；"a"和"b"分别代表函数的系数；"x"则代表自变量。
在这个公式中，如果说我们把"f(x)"看作数学专业英语中的名词，那么"a"和"b"就是形容词，而"x"可以看作动词。
因此，理解数学专业英语中的语法结构和语法规则，对于正确地理解和应用数学专业英语极为重要。
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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 computation.”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 expository 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 the nonimplication 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, and 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 intimidation?) 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 reader angry 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。