数学专业英语(6)
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 第六讲 论文写作 常见错误分析
不要过于烦琐.
• 不要用On the Study of ... 或On the Existence of... 之类作为 标题.
• 标题通常是一个名词性短语, 不是一个完整的句子,
• 无需主、谓、宾齐全, 其中的动词一般以分词或动名词的 形式出现, 有时冠词可以省略. • 偶尔有以疑问句为标题的论文, 但不应以陈述句为标题. • 为方便二次检索, 标题尽量避免使用复杂的数学式子.
• [2] name(s) of author(s), title of the book (in italics),
publishinghouse, city of publication, year of publication.
16
Mathematical English
• [3] name(s) of author(s), title of the paper, in: title of the book (in italics), publishing house, city of publication, year of publication, numbers of the first and last pages of the paper.
improvements for you.
9
Mathematical English
数学论文对英语的要求其实并不高. 而且我们有功能强大的 LATEX(WinEdt) 编辑软件, 可以轻而易举地编排出有复
杂公式、美观漂亮的版面.
但是, 有些作者在论文写作中存在不少问题.
下面从论文的各个组成部分对一些常见的错误进行分析并举
发表论文的目的, 一是公开自己的成果并得到学术界的承认, 二是为了同行之间的交流.
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Lesson 1Mathematics as a Language of Scienceassert vt. 断言；坚持主张；维护表明qualitative adj. 性质的；定性的quantitative adj. 量的；数量的；定量的；与数量有关的astronomy n. 天文学postulate n. 假定, 基本条件, 基本原理 vt. 要求, 假定 vi. 要求hypothetical adj. 假设的, 假定的,爱猜想的Lesson 2deduction n. 减除, 扣除, 减除额, 推论, 演绎induction n. 归纳；归纳法；归纳所得之结论verification n. 验证；证实correlate vt. 使相互关联 vi. 和...相关discard vt. 丢弃, 抛弃 v. 放弃discredit n. 不信任；失信consistent adj. 一致的, 调和的, 坚固的, [数、统]相容的inadequacy n. 不充分 ,不适当,不适合,不足额conic, conical adj 圆锥的；圆锥形的ellipse n. 椭圆, 椭圆形 ellipt (n.)hyperbolic adj. 双曲线的 hyperbola (n.)parabolic adj. 用寓言表达的: 抛物线的，像抛物线的 parabola (n.) algebraic adj. 代数的, 关于代数学的mineralogy n. 矿物学refraction n. 折光, 折射stimulus n. 刺激物, 促进因素, 刺激, 刺激impetus n. 冲力推动力；刺激Lesson 3Axioms, definitions and Theoremsaxiom n. [数]公理definition n. 阐明；确定定义；界说extravagant adj. 奢侈的, 浪费的, 过分的, 放纵的collinear adj. 在同一直线上的, 同线的convex adj. 凸出的；凸面的segment n. 部分；片段；节, 弓形；圆缺；弧形, 线段conswquently adv. 从而, 因此in terms of adv. 根据, 按照, 用...的话, 在...方面pretense n. 主张, 要求, 伪称, 借口, 自称Lesson 4Geometry and Geometrical termsterm n. 学期, 期限, 期间, 条款, 条件, 术语triangle n. [数]三角形, 三人一组, 三角关系parallelogram n. 平行四边形straight angle n. [数]平角right angle n. 直角acute angle n. 锐角obtuse angle n. 钝角reflex angle n. 优角rectilinear adj 直线的；由直线组成的；循直线进行的isosceles triangle n. 等腰三角形equilateral triangle n. 等边三角形right triangle n. 直角三角形obtuse triangle n. 钝角三角形acute triangle n. 锐角三角形equiangular triangle n. 正三角形,等角三角形hypotenuse n. （直角三角形的）斜边circle 圆center 中心；中央；圆心diameter n. 直径radius n. 半径, 范围, 辐射光线, 有效航程, 范围, 界限circumference n. 圆周, 周围Lesson 5The Method of Limitslimit n. 限度,极限,极点infinite adj. 无限的；无穷的infinitesimal adj. 无穷小的, 极小的, 无限小的calculus n. 微积分学, 结石exemplify vt. 例证, 例示, 作为...例子inscribe v. 记下polygon n. [数]多角形, 多边形diminish v. (使)减少, (使)变小curvilinear adj 曲线的, 由曲线组成的intuition n. 直觉, 直觉的知识integral n. [数学] 积分, 完整, 部分defective adj. 有缺陷的, (智商或行为有)欠缺的differential coefficient 微分系数arithmetical adj. 算术的, 算术上的convergence n. 集中, 收敛criterion n. (批评判断的)标准, 准据, 规范sequence n. 次序, 顺序, 序列irrational numbers n. [数]无理数domain ，定义域contradiction 矛盾reversal n. 颠倒, 反转, 反向, 逆转, 撤销Lesson 6Functioncontinuous variable 连续变量;［连续变数］variation 变分, 变化interval 区间independent variable 自变量dependent variable 应变量rectangular coordinate 直角坐标abscissa n. 〈数〉横坐标ordinate n. [数]纵线, 纵座标gradient adj. 倾斜的n. 梯度, 倾斜度, 坡度slope n. 斜坡, 斜面, 倾斜 v. (使)顺斜Lesson 7Differential and Integral calculusdifferential adj. 微分的n. 微分 (differentiation)Integral n. [数学] 积分, 完整, 部分 (integration)calculus n. 微积分学, 结石interrelation n. 相互关系trigonometry n. 三角法exponential adj. 指数的, 幂数的logarithm n. [数] 对数derivative n. 导数；微商tangent n. 切线, [数]正切counterclockwise adj. 反时针方向的adv. 反时针方向 (clockwise) definite integral 定积分approximation n. 接近, 走近, [数]近似值culminate v. 达到顶点mean n. 平均数, 中间, 中庸differential equation 微分方程extreme value n. 极值multiple integral 多重积分double integralline integralfunctional analysis 泛函分析Lesson 8 The Concept of Cardinal Number (I)cardinal number n. 基数（如： 1, 2, 3, ... 有别于序数）denumerable adj. 可数的aggregate n. 合计, 总计, 集合体adj. 合计的, 集合的, 聚合的v. 聚集, 集合, 合计purport n. 主旨 v. 声称fancier n. 空想家, 培育动物(或植物)的行家, 爱好者sniff v. 用力吸, 嗅, 闻到, 发觉, 轻视, 用力吸气n. 吸, 闻, 吸气声, 嗤之以鼻scheme n. 安排, 配置, 计划, 阴谋, 方案, 图解, 摘要v. 计划, 设计, 图谋, 策划, * n.（计算数学）方法，格式superior n. 长者, 高手, 上级adj. 较高的, 上级的, 上好的, 出众的, 高傲的cumbersome adj. 讨厌的, 麻烦的, 笨重的instruction n. 指示, 用法说明(书), 教育, 指导, 指令drastically adv. 激烈地, 彻底地conservation 守衡律quadrature n. 求积, 求积分interpolation n. 插值extrapolation n. [数]外推法, 推断internal point 内点identical adj. 同一的, 同样的generalized solution 广义解functional 泛函hydrodynamics 流体力学，水动力学divergence 发散（性），梯度，发散量play an important (fundamental ... ) role 起着重要的（...）作用integro-interpolation method 积分插值法Variational method 变分方法comparatively adv. 比较地, 相当地deficiency n. 缺乏, 不足fictive adj. 虚构的, 想象上的, 虚伪的self-adjoint (nonself-adjoint) 自治的，自伴的，自共轭的finite element method 有限元法spline approximation 样条逼近Particles-in-the-Cell 网格质点法herald n. 使者, 传令官, 通报者, 先驱, 预兆vt. 预报, 宣布, 传达, 欢呼advection n. 水平对流phenomenological adj. 现象学的, 现象的fluctuation n. 波动, 起伏optimism n. 乐观, 乐观主义pessimism n. 悲观, 悲观主义unjustified adj 未被证明其正确的mean-square 均方dispersion n. [数] 离差, 差量Polynomial n adj. [数]多项式的interpolation 插值arithmetic n. 算术, 算法rounding errors 舍入误差multiple n. 倍数, 若干subjective adj. 主观的, 个人的objective adj. 客观的,outcome n. 结果, 成果pattern n. 样品toss v. 投, 掷exhaust vt. 用尽, 耗尽, 抽完, 使精疲力尽divisible adj. 可分的dice, die n. 骰子assign vt. 分配, 指派attach vt. 缚上, 系上, 贴上v. 配属, 隶属于pitfall n. 缺陷chairperson 主席mechanics n. (用作单数)机械学、力学, (用作复数)技巧, 结构statics n. [物]静力学dynamics n. 动力学adequately adv. 充分地celestial adj. 天上的macroscopic adj. 肉眼可见的, 巨观的classical field theory 经典场理论rigit adj. 刚硬的, 刚性的, 严格的elastic adj. 弹性的plastic n. 可塑的,塑性的,塑料的quantum n. 量, 额, [物] 量子, 量子论inception n. 起初, 获得学位pertain v. 适合, 属于gravitation n. 地心吸力, 引力作用tide n. 潮, 潮汐, 潮流, 趋势monumental adj. 纪念碑的, 纪念物的, 不朽的, 非常的encompass v. 包围, 环绕, 包含或包括某事物ingredient n. 成分, 因素acquainted adj. 有知识的, 知晓的synonymous adj. 同义的configuration n. 构造, 结构, 配置, 外形reference n. 提及, 涉及, 参考, 参考书目inertia n. 惯性, 惯量attribute 特性momentum n. 动量proportional adj. 比例的, 成比例的, 相称的, 均衡的designate 指明negligible adj. 可以忽略的, 不予重视的projectile n. 射弹 adj. 发射的ballistics n. 弹道学, 发射学intractable adj. 难处理的{Mechanics of a Particlein consequence of adv. 由于的...缘故exert vt. 尽(力), 施加(压力等), 努力v. 发挥, 竭尽全力, 尽galaxy n. 星系, 银河, 一群显赫的人, 一系列光彩夺目的东furnish vt. 供应, 提供, 装备, 布置v. 供给torque n. 扭矩, 转矩moment 力矩的friction 摩擦dissipation n. 消散, 分散, 挥霍, 浪费, 消遣, 放荡, 狂饮infer v. 推断Hooke s Law and Its Consequenceselasticity n. 弹力, 弹性constitutive adj. 构成的, 制定的atomistic adj. 原子论的crack n. 裂缝, 噼啪声v. (使)破裂, 裂纹, (使)爆裂continuum mechanics n. 连续介质力学superposition n. 重叠, 重合, 叠合strain n. 过度的疲劳, 紧张, 张力, 应变vt. 扭伤, 损伤v. 拉紧, 扯紧, (使)紧张, 尽力thermodynamics n. [物] 热力学reckon vt. 计算, 总计, 估计, 猜想vi. 数, 计算, 估计, 依赖, 料想lesson 20strength 强度load 载荷empirical 以经验为依据的member 构件isolated 孤立的segment 部分、段、节stress 应力strain 应变tension 拉伸shear 剪切bend 弯曲torsion 扭转、扭力insofar 在……范围cohesive 内聚性的tensile 拉力、张力stiffness 硬度furnish 供给Lesson 23 Fluid Mechanicseruption 喷发、爆发turbulent 湍流laminar 层流isothermal 等温isotropic 各向同性prevalent 普遍的、流行的tornado 旋风、飓风eddy 旋涡viscosity 粘性、粘度nonviscous 无粘性的rotation 旋转adiabatic 绝热的reversible 可逆的isentropic 等熵的instant 瞬时的streamline 流线stream tube 流管tangential 切线的incompressible 不可压缩的resultant 合成的,组合的downstream 下游的,顺流的elbow 弯管,肘similitude 相似性hydraulic 水力的,水力学的predominante 占主导地位spillway (河或水坝的)放水道,泄洪道prototype 原型,样板Lesson 24 Mechanical Vibration repetitive 重复的,反复的periodic 周期的,定期的tidal 潮的,像潮的stationary 固定的,不动的vibratory 振动的,摆动的propagation 传播couple v .连接,连合acoustic 听觉的,声学的annoyance 烦恼,困惑adjacent 接近的,邻近的damp 阻尼,衰减restore 复职,归还neutral 平衡exciting force 激励力resonant adj. 共振的,谐振的stiffness 刚度,刚性proportionality 成比例地inclusion 包含,包括magnitude 数值,大小substantially adv. 实质上的perturb 干扰,扰乱resonance n. 共振vibratory adj. 振动的, 可知的perceptible 可见的,可知的adudible 听得见的,可闻的foregoing 前述的impulsive 冲击的shock 冲击Fourier series 傅里叶级数excitation 激发,激励discrete 分离,离散的contend with 向…作斗争compressor 压气机fatigue 疲劳perceptible 可见的,可知觉的shredder 切菜器disposal 处理urban 都市的metropolitan 大都市的at-grade 在同一水平面上elevated 高架的guideway 导轨Lesson 25 A prefect to the Continuum Mechanics preface 序言continuum连续 pl. continuua rigid body 刚体contemporary 当代的,同时期的widespread 分布广的, 普及的accommodate 容纳,使适应medium 介质plasticity 塑性residual 剩余的,残留的creep 蠕变,爬行,塑性变形aging 老化polymeric聚合(物)的sandy 沙的,沙质的aubterranean 地下的,隐藏的essence 精髓,本质thermodynamics 热力学self-similar 自相似expedient 方便的sonsolidate 把…联合为一体,统一justify 证明…有理radically 根本地,本质上deliberate 从容不迫的,深思熟虑Lesson 33 what is a computer Attribute v. 赋予medieval 中世纪的astronomer 天文学家Mars 火星resemble vt. 像,相似tedious adj. 冗长乏味的pulp 浆状物,果肉filter vt.过滤underlying adj. 潜在的, 基本的ore n. 矿沙,矿石perceive v. 察觉,看见intervention n. 干涉,插入intelligent adj. 有智力的,聪明的Lesson 34 A computer system manipulate vt. 操纵,使用chip n. 芯片etch vt. 蚀刻,蚀镂fingernail 指甲mount vt. 安装,安置assemble vt. 集合,聚集cabinet 橱柜execute vt. 执行,实现paycheck n.支付薪金的支票bar chart 直方图joystick 游戏杆encounter vt. 遇到,遇上Mathematical Modelingindustry n. 工业, 产业, 行业, 勤奋commerce n. 商业complexity n. 复杂(性), 复杂的事物, 复杂性career n. (原意:道路, 轨道)事业, 生涯, 速度outset n. 开端, 开始essence n. 基本, [哲]本质, 香精advocation n. (=advocacy)拥护支持provision n. 供应, (一批)供应品, 预备, 防备, 规定publicize v. 宣扬roundabout adj. 迂回的, 转弯抹角的n. 道路交叉处的环形路, 迂回路线, 兜圈子的话trial-error vt. n. 试制, 试生产maneuverability n. 可操作性, 机动性vehicle n. 交通工具, 车辆, 媒介物, 传达手段junction n. 连接, 接合, 交叉点, 汇合处ponder v. 沉思, 考虑contrive v. 发明, 设计, 图谋snooker n. (=snooker pool)彩色台球, 桌球context n. 上下文, 文章的前后关系deviation n. 背离数学专业英语－Groups and RingsDuring the present century modern abstract algebra has become more and more important as a tool for research not only in other branches of mathematics bu t even in other sciences .Many discoveries in abstract algebra itself have been made during the past years and the spirit of algebraic research has definitely t ended toward more abstraction and rigor so as to obtain a theory of greatest p ossible generality. In particular, the concepts of group ,ring,integral domain and field have been emphasized.The notion of an abstract group is fundamental in all sciences ,and it is certai nly proper to begin our subject with this concept. Commutative additive groupsare made into rings by assuming closure with respect to a second operation h aving some of the properties of ordinary multiplication. Integral domains and fi elds are rings restricted in special ways and may be fundamental concepts and their more elementary properties are the basis for modern algebra.GroupsDEFINITION A non-empty set G of elements a,b,…is said to form a group with respect to 0 if:I.G is closed with respect to 0II.The associative law holds in G, that isaо(bоc)=(aоb)оcfor every a, b, c of GⅢ. For every a and b of G there exist solutions χand Уin G of the equ ationsaοχ=b yοa=bA group is thus a system consisting of a set of elements and operation οwit h respect to which G forms a group. We shall generally designate the entire s ystem by the set G of its elements and shall call G a group. The notation use d for the operation is generally unimportant and may be taken in as convenien t a way as possible.DEFINITION A group G is called commutative or abelian ifaοb=bοaFor every a and b of G.An elementary physical example of an abelian group is a certain rotation grou p. We let G consist of the rotations of the spoke of a wheel through multiples of 90ºand aοb be the result of the rotation a followed by the rotation b. T he reader will easily verify that G forms a group with respect to οand that aοb=bοa. There is no loss of generality when restrict our attention to multipl icative groups, that is, write ab in stead of aοb.EQUIVALENCEIn any study of mathematical systems the concept of equivalence of systems of the same kind always arises. Equivalent systems are logically distinct but weusually can replace any one by any other in a mathematical discussion with no loss of generality. For groups this notion is given by the definition: let G an d G´be groups with respective operations o and o´,and let there be a1-1 corr espondenceS : a a´ (a in G and a´in G´)between G and G´such that(aοb)´=a´οb´for all a, b of G. then we call G and G´equivalent(or simply, isomorphic)grou ps.The relation of equivalence is an equivalence relation in the technical sense in the set of all groups. We again emphasize that while equivalent groups may be logically distinct they have identical properties.The groups G and G´of the above definition need not be distinct of course a nd o´may be o. when this is the case the self-equivalence S of G is called a n automorphism.I: a aOf G, but other automorphisms may also exist.RingsA ring is an additive abelian groupB such thatI.the set B is closed with respect to a second operation designated by multiplication; that is , every a and b of B define a unique element ab of B. II.multiplication is associative; that isa (bc) = (ab)cfor every a, b, c of B.Ⅲ. The distributive lawsa (b+c) = ab +ac (b+c) a=ba +cahold for every a, b, c of B.The concept of equivalence again arises. We shall writeB ≌B′to mean that B and B′are equivalent.VocabularyGroup 群rigor 严格ring 环 generalization 推广integral domain 整环Abelian group 阿贝尔群commutative additive group 可交换加法群 rotation 旋转automorphism 自同构数学专业英语－Historical introduction of CalculusThe Two Basic Concepts of CalculusThe remarkable progress that has been made in science and technology during the last century is due in large part to the development of mathematics. That branch of mathematics known as integral and differential calculus serves as a natural and powerful tool for attacking a variety of problems that arise in phys ics,engineering,chemistry,geology,biology, and other fields including,rather recentl y,some of the social sciences.To give the reader an idea of the many different types of problems that can b e treatedby the methods of calculus,we list here a few sample questions.With what speed should a rocket be fired upward so that it never returns to e arth? What is the radius of the smallest circular disk that can cover every isosceles triangle of a given perimeter Ｌ? What volume of material is removed fr om a solid sphere of radius 2 r if a hole of redius r is drilled through the ce nter? If a strain of bacteria grows at a rate proportional to the amount present and if the population doubles in one hour,by how much will it increase at th e end of two hours? If a ten-pound force stretches an elastic spring one inch,h ow much work is required to stretch the spring one foot?These examples,chosen from various fields,illustrate some of the technical quest ions that can be answered by more or less routine applications of calculus.Calculus is more than a technical tool－it is a collection of fascinating and ex eiting idea that have interested thinking men for centuries.These ideas have to do with speed,area,volume,rate of growth,continuity,tangent line,and otherconcept s from a varicty of fields.Calculus forces us to stop and think carefully about the meanings of these concepts. Another remarkable feature of the subject is it s unifying power.Most of these ideas can be formulated so that they revolve a round two rather specialized problems of a geometric nature.We turn now to a brief description of these problems.Consider a cruve C which lies above a horizontal base line such as that show n in Fig.1. We assume this curve has the property that every vertical line inter sects it once at most.The shaded portion of the figure consists of those pointe which lie below the curve C , above the horizontal base,and between two para llel vertical segments joining C to the base.The first fundamental problem of c alculus is this: To assign a number which measures the area of this shaded re gion.Consider next a line drawn tangent to the curve,as shown in Fig.1. The second fundamental problem may be stated as follows:To assign a number which me asures the steepness of this line.Basically,calculus has to do with the precise formulation and solution of these two special problems.It enables us to define the concepts of area and tangent l ine and to calculate the area of a given region or the steepness of a given an gent line. Integral calculus deals with the problem of area while differential cal culus deals with the problem of tangents.Historical BackgroundThe birth of integral calculus occurred more than 2000 years ago when the Gr eeks attempted to determine areas by a procees which they called the method of exhaustion.The essential ideas of this ,method are very simple and can be d escribed briefly as follows:Given a region whose area is to be determined,we inscribe in it a polygonal region which approximates the given region and whos e area we can easily compute.Then we choose another polygonal region which gives a better approximation,and we continue the process,taking polygons with more and more sides in an attempt to exhaust the given region.The method is illustrated for a scmicircular region in Fig.2. It was used successfully by Arch imedes(287－212 B.C.) to find exact formulas for the area of a circle and a fe w other special figures.The development of the method of exhaustion beyond the point to which Ar chimcdcs carried it had to wait nearly eighteen centuries until the use of algeb raic symbols and techniques became a standard part of mathematics. The eleme ntary algebra that is familiar to most high-school students today was completel y unknown in Archimedes’time,and it would have been next to impossible to extend his method to any general class of regions without some convenient w ay of expressing rather lengthy calculations in a compact and simpolified form.A slow but revolutionary change in the development of mathematical notations began in the 16th century A.D. The cumbersome system of Roman numerals was gradually displaced by the Hindu-Arabic characters used today,the symbol s “+”and “-”were introduced for the forst time,and the advantages of the decimal notation began to be recognized.During this same period,the brilliant su ccesse of the Italian mathematicians Tartaglia,Cardano and Ferrari in finding al gebraic solutions of cubic and quadratic equations stimulated a great deal of ac tivity in mathematics and encouraged the growth and acceptance of a new and superior algebraic language. With the wide spread introduction of well-chosen algebraic symbols,interest was revived in the ancient method of exhaustion an d a large number of fragmentary results were discovered in the 16 th century by such pioneers as Cavalieri, Toricelli, Roberval, Fermat, Pascal, and Wallis.Fig.2. The method of exhaustion applied to a semicircular region.Gradually the method of exhaustion was transformed into the subject now calle d integral calculus,a new and powerful discipline with a large variety of applic ations, not only to geometrical problems concerned with areas and volumes but also to jproblems in other sciences. This branch of mathematics, which retaine d some of the original features of the method of exhaustion,received its bigges t impetus in the 17 th century, largely due to the efforts of Isaac Newion (16 42—1727) and Gottfried Leibniz (1646—1716), and its development continued well into the 19 th century before the subject was put on a firm mathematical basis by such men as Augustin-Louis Cauchy (1789-1857) and Bernhard Riem ann (1826-1866).Further refinements and extensions of the theory are still being carried o ut in contemporary mathematicsVocabularygeology 地质学decimal 小数，十进小数biology 生物学discipline 学科social sciences 社会科学 contemporary 现代的disk (disc) 圆盘bacteria 细菌isosceles triangle 等腰三角形 elastic 弹性的perimeter 周长 impetus 动力volume 体积 proportional to 与…成比例center 中心 inscribe 内接steepness 斜度 solid sphere 实心球method of exhaustion 穷举法 refinement 精炼，提炼polygon 多边形，多角形 cumbersome 笨重的，麻烦的polygonal 多角形fragmentary 碎片的，不完全的approximation 近似，逼近 background 背景学专业英语－How to Organize a paper (For Beginers)?The usual journal article is aimed at experts and near-experts, who are the peo ple most likely to read it. Your purpose should be say quickly what you have done is good, and why it works. Avoid lengthy summaries of known results, and minimize the preliminaries to the statements of your main results. There ar e many good ways of organizing a paper which can be learned by studying pa pers of the better expositors. The following suggestions describe a standard acc eptable style.Choose a title which helps the reader place in the body of mathematics. A use less title: Concerning some applications of a theorem of J. Doe. A. good titlecontains several well-known key words, e. g. Algebraic solutions of linear parti al differential equations. Make the title as informative as possible; but avoid re dundancy, and eschew the medieval practice of letting the title serve as an infl ated advertisement. A title of more than ten or twelve words is likely to be m iscopied, misquoted, distorted, and cursed.The first paragraph of the introduction should be comprehensible to any mathe matician, and it should pinpoint the location of the subject matter. The main p urpose of the introduction is to present a rough statement of the principal resul ts; include this statement as soon as it is feasible to do so, although it is som etimes well to set the stage with a preliminary paragraph. The remainder of th e introduction can discuss the connections with other results.It is sometimes useful to follow the introduction with a brief section that estab lishes notation and refers to standard sources for basic concepts and results. N ormally this section should be less than a page in length. Some authors weave this information unobtrusively into their introductions, avoiding thereby a dull section.The section following the introduction should contain the statement of one or more principal results. The rule that the statement of a theorem should precede its proof a triviality. A reader wants to know the objective of the paper, as well as the relevance of each section, as it is being read. In the case of a ma jor theorem whose proof is long, its statement can be followed by an outline of proof with references to subsequent sections for proofs of the various parts.Strive for proofs that are conceptual rather than computational. For an example of the difference, see A Mathematician’s Miscellany by J.E.Littlewood, in wh ich the contrast between barbaric and civilized proofs is beautifully and amusin gly portrayed. To achieve conceptual proofs, it is often helpful for an author t o adopt an initial attitude such as one would take when communicating mathe matics orally (as when walking with a friend). Decide how to state results wit h a minimum of symbols and how to express the ideas of the proof without c omputations. Then add to this framework the details needed to clinch the resul ts.Omit any computation which is routine (i.e. does not depend on unexpected tri cks). Merely indicate the starting point, describe the procedure, and state the o utcome.It is good research practice to analyze an argument by breaking it into a succe ssion of lemmas, each stated with maximum generality. It is usually bad practi ce to try to publish such an analysis, since it is likely to be long and unintere sting. The reader wants to see the path-not examine it with a microscope. A part of the argument is worth isolating as a lemma if it is used at least twice l ater on.The rudiments of grammar are important. The few lines written on the blackbo ard during an hour’s lecture are augmented by spoken commentary, and aat t he end of the day they are washed away by a merciful janitor. Since the publ ished paper will forever speak for its author without benefit of the cleansing s ponge, careful attention to sentence structure is worthwhile. Each author must develop a suitable individual style; a few general suggestions are nevertheless a ppropriate.The barbarism called the dangling participle has recently become more prevalen t, but not less loathsome. “Differentiating both sides with respect to x, the eq uation becomes---”is wrong, because “the equation”cannot be the subject th at does the differentiation. Write instead “differentiating both sides with respec t to x, we get the equation---,”or “Differentiation of both sides with respect to x leads to the equation---”Although the notion has gained some currency, it is absurd to claim that infor mal “we”has no proper place in mathematical exposition. Strict formality is appropriate in the statement of a theorem, and casual chatting should indeed b e banished from those parts of a paper which will be printed in italics. But fif teen consecutive pages of formality are altogether foreign to the spirit of the t wentieth century, and nearly all authors who try to sustain an impersonal digni fied text of such length succeed merely in erecting elaborate monuments to slu msiness.A sentence of the form “if P,Q”can be understood. However “if P,Q,R,S,T”is not so good, even if it can be deduced from the context that the third co mma is the one that serves the role of “then.”The reader is looking at the paper to learn something, not with a desire for mental calisthenics.Vocabularypreliminary 序,小引(名)开端的,最初的(形)eschew 避免medieval 中古的，中世纪的inflated 夸张的comprehensible 可领悟的，可了解的pinpoint 准确指出（位置）weave 插入，嵌入unobtrusivcly 无妨碍地triviality 平凡琐事barbarism 野蛮，未开化portray 写真，描写clinch 使终结rudiment 初步，基础commentary 注解，说明janitor 看守房屋者sponge 海绵dangling participle 不连结分词prevalent 流行的，盛行loathsome 可恶地absurd 荒谬的banish 排除sustain 维持，继续slumsiness 粗俗，笨拙monument 纪念碑calisthenics 柔软体操，健美体操notes1. 本课文选自美国数学会出版的小册子A mamual for authors of mathematical paper的一节，本文对准备投寄英文稿件的读者值得一读。
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
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
数学专业英语第二章精读课文-- 入门必修2.1 数学方程与比例（Mathematics，Equation and Ratio）一、词汇及短语：1. Cha nge the terms about变形2. full of ：有许多的充满的例The StreetS are full of people as on a holiday像假日一样，街上行人川流不息）3. in groups of ten??4. match SOmething against sb. “匹配”例Long ago ,when people had to Count many things ,they matChed them against their fingers. 古时候，当人们必须数东西时，在那些东西和自己的手指之间配对。
5. grow out of 源于由…引起例Many close friendships grew out of common acquaintance6. arrive at 得出（到达抵达达到达成）例We both arrived at the Same COnclusion我们俩个得出了相同的结论）7. stand for “表示，代表”8. in turn “反过来，依次”9. bring about 发生导致造成10. arise out of 引起起源于11. express by “用…表示”12. occur 发生，产生13. come from 来源于，起源于14. resulting method 推论法15. be equal to 等于的相等的例TWiCe two is equal to four（2 乘以 2 等于4）16. no matter 无论不管17. mathematical analysis 数学分析18. differential equation 微分方程19. higher mathematics 高等数学higher algebra 高等代数20. equation of condition 条件等式二句型及典型翻译1. For a long period of the history of mathematics, the centric place of mathematicalmethods was occupied by the logical deductions “在数学史的很长的时期内，是逻辑推理一直占据数学方法的中心地位”2. An equation is a statement of the equality between two equal numbers or numbersymbols.equation ：“方程”“等式” 等式是关于两个数或数的符号相等的一种陈述3. In such an equation either the two members are alike, or become alike onperformance of the indicated operation. 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。
数学专业英语
abelian algebra 阿贝耳代数abelian differential 阿贝耳微分abelian equation 阿贝耳方程abelian extension 阿贝耳扩张abelian function 阿贝耳函数abelian function field 阿贝耳函数域abelian functor 阿贝耳函子abelian group 交换群abelian groupoid 阿贝耳广群abelian integral 阿贝耳积分abelian summation 阿贝耳求和法abelian theorem 阿贝耳定理abelian variety 阿贝耳簇abridge 略abridged notation 简算记号abscissa 横坐标abscissa of absolute convergence 绝对收敛坐标abscissa of summability 可和性坐标abscissa of uniform convergence 一致收敛横坐标absolute convergence 绝对收敛absolute curvature vector 绝对曲率向量absolute deviation 绝对偏差absolute differential calculus 绝对微分学absolute error 绝对误差absolute extremes 绝对极值absolute extremum 绝对极值absolute frequency 绝对频率absolute geometry 绝对几何absolute inequality 绝对不等式absolute instability 绝对不稳定性absolute maximum 绝对极大值absolute minimum 绝对极小值absolute neighborhood 绝对邻域absolute neighborhoodretract 绝对邻域收缩核absolute norm 绝对范数absolute stability 绝对稳定性absolute term 常数项absolute unit 绝对单位absolute value 绝对值absolute value sign 绝对值符号absolute velocity 绝对速度absolutely additivemeasure 完全加性测度absolutely compact set 绝对紧集absolutely completesystem 绝对完备系absolutely continuous 绝对连续的absolutely continuousdistri but ion 绝对连续分布absolutely continuousfunction 绝对连续函数absolutely continuousmeasure 绝对连续测度absolutely continuous part 绝对连续部分absolutely continuoustransformation 绝对连续变换absolutely convergent 绝对收敛的absolutely convergentintegral 绝对收敛积分absolutely convergentseries 绝对收敛级数absolutely convex hull 绝对凸包absolutely discontinuousfunction 绝对不连续函数absolutely integrable 绝对可积的absolutely irreduciblecharacter 绝对不可约特征absolutely irreduciblerepresentation 绝对不可约表示absolutely irreduciblevariety 绝对不可约簇absolutely normal number 绝对范数absolutely prime ideal 绝对素理想absolutely semisimpleabsolutely summablesequence 绝对可和序列absolutely unbiasedestimator 绝对无偏估计量absolutely unramifiedabstract 抽象的abstract algebra 抽象代数abstract algebraicgeometry 抽象代数几何abstract mathematics 抽象数学abstract number 不名数abstract ordered simplicialcomplex 抽象有序单纯复形abstract simplex 抽象单形abstract space 抽象空间abstraction 抽象abstraction operator 抽象算子absurd 谬论的absurdity 谬论abundant number 过剩数accelerated motion 加速运动acceleration 加速度acceleration ofconvergence 收敛性的加速accessibility 可达性accessible boundary point 可达边界点accessible ordinal number 可达序数accessible point 可达点accessible set 可达集accessible vertex 可达顶点accidental 偶然的accidental coincidence 偶然符合accumulating point 聚点accumulation 累积accumulation point 聚点accuracy 准确性accuracy grade 准确度accuracy of measurement 测量精确度accuracy rating 准确度acnod 孤点acount 计算actual 真实的actual infinity 实无穷acute 尖锐的acute angle 锐角acute angled triangle 锐角三角形acute triangle 锐角三角形acuteness 锐度acyclic 非循环的ad infinitum 无穷地adaptability 适应性adaptation 适应add 加addend 加数addition 加法addition formulas 加法公式addition sign 加号addition system 加法系addition table 加法表addition theorem 加法定理addition theorem of probability 概率的加法定理additional 加法的additional code 附加代码additional condition 附加条件additional error 附加误差additive 加法的additive functional 加性泛函数additive functionaltransformation 加性泛函变换additive group 加法群additive interval function 加性区间函数additive operator 加性算子additive process 加性过程additive relation 加性关系additive separable 加法可分的additive valuation 加法赋值additively commutativeordinal numbers 加性交换序数additivity 加法性address 地址adele 阿代尔adele group 阿代尔群adequate 适合的adherent point 触点adhesion 附着adjacency 邻接adjacency matrix 邻接矩阵adjacent angles 邻角adjacent edge 邻棱adjacent side 邻边adjacent supplementaryangles 邻角adjacent vertex 邻顶adjoint determinant 伴随行列式adjoint form 伴随形式adjoint function 伴随函数adjoint functor 伴随函子adjoint hilbert problem 伴随希耳伯特问题adjoint kernel 伴随核adjoint linear map 伴随线性映射adjoint matrix 伴随阵adjoint operator 伴随算子adjoint space 伴随空间adjoint transformation 伴随算子adjoint vector 伴随向量adjunct 代数余子式adjunction 附加adjunction of an identityaffine coordinates 平行坐标affine curvature 仿射曲率affine differentialgeometry 仿射微分几何学affine distance 仿射距离affine figure 仿射图形affine function 仿射函数affine geometry 仿射几何学affine group 仿射群affine group scheme 仿射群概型affine length 仿射长度affine line 仿射直线affine normal 仿射法线affine parameter 仿射参数affine space 仿射空间affine sphere 仿射球面affine surface 仿射曲面affine transformation 仿射变换affine variety 仿射簇affinely connectedmanifold 仿射连通廖affinely connected space 仿射连通空间affinity 仿射变换affirmation 肯定affirmative proposition 肯定命题affix 附标aggregate 集aggregation 聚合agreement 一致air coordinates 空间坐标alexander matrix 亚历山大阵alexander polynomial 亚历山大多项式algebra 代数学algebra of events 事件场algebra of logic 逻辑代数algebraic calculus 代数计算algebraic closure 代数闭包algebraic closure operator 代数闭包算子algebraic complement 代数余子式algebraic curve 代数曲线algebraic equation 代数方程algebraic expression 代数式algebraic extension 代数扩张algebraic form 代数形式algebraic fraction 代数分式algebraic function 代数函数algebraic function field 代数函数域algebraic geometry 代数几何学algebraic group 代数群algebraic hull 代数包algebraic hypersurface of the algebraic integer 代数整数algebraic multiplicity 代数重度algebraic number 代数数algebraic number field 代数数域algebraic number theory 代数数论algebraic operation 代数运算algebraic polynomial 代数多项式algebraic singularity 代数奇点algebraic space 代数空间algebraic spiral 代数螺线algebraic structure 代数结构algebraic sum 代数和algebraic system 代数系algebraic variety 代数簇algorithm 算法algorithm of division 辗转相除法algorithm of euclid 欧几里得算法algorithm theory 算法论algorithmic language 算法语言algorithmization 算法化aliquot part 整除部分allowance 允许almost all 几乎处处almost certainconvergence 几乎必然收敛almost everywhere 几乎处处almost impossible event 殆不可能事件alpha limit set 极限集alphabetical 字母的alphanumeric 字母数字式alphanumeric representationof alternating function 反对称函数alternating group 交错群alternating harmonicseries 莱布尼兹级数alternating matrix 交错矩阵alternating method 交错法alternating product 外积alternating sequence 交错序列alternating series 交错级数alternating series test 交错级数检验alternating sum 交错和altitude 高度altitude theorem 高度定理amalgamated product 融合积amalgamated subcategory 融和子范畴amalgamation 合并ambient space 环绕空间ambiguous point 歧点amicable numbers 亲和数amount 量amphicheiral knot 双向纽结amplitude 振幅;角amplitude of a complexnumber 复数角analog computer 模拟计算机analogous 类似的analogue display 相似表示analogue method 相似法analogy 类似analysis 数学分析analysis of time series 时间序列分析analytic 分析的analytic curve 解析曲线analytic expression 解析式analytic function 分析函数analytic function of severalanalytic proof 解析证明analytic proposition 解析命题analytic set 解析集analytic space 解析空间analytic transformation 解析变换analytical differential 解析微分analytical geometry 分析几何学angle 角angle at center 圆心角angle between chord andtangent 弦和切线的角angle function 角函数angle of contact 接触角angle of contingence 切线角angle of declination 俯角angle of diffraction 衍射角angle of incidence 入射角angle of inclination 斜角angle of intersection 相交角angle of lead 超前角angle of reflection 反射角angle of refraction 折射角angle of rotation 旋转角angle of torsion 挠率角angle preserving 保角的angle preserving map 保角映象angular 角的angular acceleration 角加速度angular velocity 角速度anti automorphism 反自同构anti hermitian form 反埃尔米特形式anti isomorphic lattice 反同构格anti isomorphism 反同构anti position 反位置anti reflexiveness 反自反性anti semiinvariant 反半不变量antianalytic function 反解析函数antichain 反链anticlockwise 逆时针的anticlockwise revolution 逆时针回转anticlockwise rotation 逆时针回转anticoincidence 反重合anticoincidence method 反重法anticommutation 反交换anticommutative 反交换的anticommutativity 反交换性anticommutator 反换位子antiderivative 不定积分的antiholomorphic 反全纯的antihomomorphism 反同态antiisomorphy 反同构antilinear 反线性的antilinear mapping 反线性映射antilinear transformation 反线性变换antiorder homomorphism operator 反序同态算子antiordered set 反有序集合antiparallel 逆平行的antiplane 反平面antipodal map 对映映射antipodal point 对映点antipodal set 对映集antipode 对映点antipoints 反点antisymmetric 反对称的antisymmetric function 反对称函数antisymmetric matrix 反对称矩阵antisymmetric relation 反对称关系antitone mapping 反序映射antitone sequence 反序列antitonic function 反序函数antitonicity 反序性antitony 反序性antitrigonometric function 反三角函数antiunitary 反酉的apex 顶点apex angle 顶角apical angle 顶角application 应用applied mathematics 应用数学approach 接近approach infinity 接近无穷大approximability 可逼近性approximable 可逼近的approximate 近似的;使近似approximate calculation 近似计算approximate construction 近似准approximate continuity 近似连续性approximate derivative 近似导数approximatedifferentiability 近似可微性approximate differential 近似微分approximate formula 近似公式approximate integration 近似积分approximate limit 近似极限approximate lower semicontinuity 近似下半连续性approximate method 近似法approximate number 近似数approximate partialderivative 近似偏导函数approximate partial derivedfunction 近似偏导函数approximate partialdifferential 近似偏微分approximate solution 近似解approximate totaldifferentiability 近似全可微性approximate totaldifferential 近似全微分approximate upper semicontinuity 近似上半连续性approximate value 近似值approximate value inexcess 过剩近似值approximately equal 近似等于approximatelysemicontinuous 近似半连续的approximation 逼近approximation by excess 过剩逼近approximation calculus 近似计算approximation error 近似误差approximation function 逼近函数approximation in excess 过剩逼近approximation method 近似法approximation methods in physics 物理学中的逼近法approximation theorem 逼近定理approximation theory 逼近理论arbitrarily small 任意小arbitrary 任意的arbitrary constant 任意常数arbitrary element 任意元素arbitrary parameter 任意参数arbitrary small number 任意小数arc 弧arc tangent 反正切arch 拱形archimedean 阿基米德性的archimedean group 阿基米德群archimedean semigroup 阿基米德半群archimedean space 阿基米德空间archimedean total order 阿基米德全序archimedean valuation 阿基米德赋值archimedes axiom 阿基米德公理archimedically ordered field 阿基米德有序域archimedically ordered number field 阿基米德有序数域arcwise connected set 弧连通集arcwise connected space 弧连通空间arcwise connectedness 弧连通性are 公亩area 面积area function 面积函数area of a circle 圆面积area preserving mapping 保面积映射areal coordinates 重心坐标argand plane 复数平面argument 自变数;辐角argument function 辐角函数argument of a function 函数的自变数argument principle 辐角原理argumentation 论证aristotelian logic 亚里斯多德逻辑学arithmetic 算术arithmetic al function 数论函数arithmetic difference 算术差arithmetic division 算术除法arithmetics 算术arithmetization 算术化arithmometer 四则计算机arrangement 排列array 排列arrow 射artificial variable 人工变量artificial variable method 人工变量法associate equation 相伴方程associated equation 相伴方程associated homogeneoussystem 相伴齐次组associated legendrefunction 相伴勒让德函数associated legendrepolynomial 连带的勒让德多项式associated minimalsurface 相伴极小曲面associated power series 相伴幂级数associated prime ideal 相伴素理想associated radius ofconvergence 相伴收敛半径associated space 相伴空间associated system 相伴系associated undirectedgraph 相伴无向图association 结合associative algebra 结合代数associative law 结合律associative law for series 级数的结合律associativity 结合性assume 假定assumption 假定assumption formula 假定公式asterisk 星号asteroid 星形线asymmetric 非对称的asymmetric relation 非对称关系asymmetric variety 非对称簇asymmetrical 非对称的asymmetry 非对称性asymptote 渐近线asymptote of curve 曲线的渐近线asymptotic 浙近的asymptotic behavior 渐近状态asymptotic circle 渐近圆asymptotic cone 渐近锥面asymptotic convergence 渐近收敛asymptotic curvature 渐近曲率asymptotic curve 渐近曲线asymptotic density 渐近密度asymptotic direction 渐近方向asymptotic efficiency 渐近效率asymptotic expansion 渐近展开asymptotic formula 渐近公式asymptotic line 渐近线asymptotic mean value 渐近平均值asymptotic minimal basis 渐近极小基asymptotic order 渐近阶asymptotic path 渐近路线asymptotic plane 渐近平面asymptotic point 渐近点asymptotic rate of convergence 渐近收敛速度asymptotic series 渐近级数asymptotic solution 渐近解asymptotic stability 渐近稳定性asymptotic surface 渐近曲面asymptotic unbiased estimator 渐近无偏估计量asymptotic value 渐近值asymptotically efficient estimator 渐近有效估计量asymptotically equal 渐近相等asymptotically equal sequence 渐近相等序列asymptotically equivalent function 渐近等价函数asymptotically normal distri but ion 渐近正态分布asymptotically normally distri but ed 渐近正规分布的asymptotically stable 渐近稳定的asymptotically stable solution 渐近稳定解atom 原子atomic element 原子元素atomic formula 原子公式atomic lattice 原子格atomic proposition 原子命题atomicity 原子性augmented matrix 增广矩阵austausch 交换autocorrelation 自相关automorphism 自同构automorphism group 自同构群autonomous system 自治系统autoregressivetransformation 自回归变换auxiliary 辅助的auxiliary angle 辅助角auxiliary circle 辅助圆auxiliary equation 相伴齐次方程auxiliary function 辅助函数auxiliary line 辅助线auxiliary variable 辅助变数average 平均值average deviation 平均偏差average error 平均误差axial 轴的axial symmetry 轴对称axial vector 轴向量axiom 公理axiom of accessibility 可达性公理axiom of addition 加法公理axiom of choice 选择公理axiom of completeness 完备性公理axiom of comprehension 概括公理axiom of constructibility 可构成性公理axiom of constructivity 可构成性公理axiom of continuity 连续公理axiom of regularity 正则性公理axiom of subsets 子集公理axiom of substitution 替换公理axiom of sum set 并集公理axiom of the empty set 空集公理axiom of union 并集公理axiom scheme 公理格式axiomatic 公理的axiomatic method 公理法axiomatic set theory 公理论的集论;公理集合论axiomatic system 公理系统axiomatics 公理学axiomatization 公理化axiomatize 公理化axioms of continuity 连续公理axioms of denumerability 可数公理axioms of incidence 关联公理axioms of order 次序公理axis 轴axis of a cone 锥轴axis of abscissas 横坐标轴axis of absoluteconvergence 绝对收敛轴axis of affinity 仿射轴axis of convergence 收敛轴axis of coordinate 坐标轴axis of curvature 曲率轴axis of ordinates 纵坐标轴axis of projection 射影轴axis of reals 实轴axis of revolution 回转轴axis of rotation 回转轴axis of symmetry 对称轴axisymmetric 轴对称的axonometric perspective 轴测投影法axonometric projection 轴测投影法b measurability b可测性b measurable function 波莱尔可测函数babylonian numerals 巴比伦数字back substitution 逆计算backward difference 后向差分backward difference operator 后向差分算子backward difference quotient 后向差商backward solution 后向解法baire function 贝利函数baire measure 贝利测度baire set 贝利集baire space 贝利空间baire theorem 贝利定理balance 平衡数banach lie group 巴拿赫李群banach space 巴拿赫空间band 带band chart 带状图band matrix 带状矩阵barycentric 重心的barycentric complex 重心复形barycentric coordinates 重心坐标barycentric mapping 重心映射barycentric subdivision 重心重分base 底base angle 底角base line 底线base number 底数base of logarithms 对数的底base point 基点base space 底空间base vector 基向量basic 基础的basic block 基本块basic field 基域basic form 基本形式basic point 基础点basic representation 基本表示basic ring 基环basic solution 基本解basic symbol 基本符号basic variable 基本变量basis 基basis of linear space 线性空间的基basis of vector space 向量空间的基basis replacementprocedure 基替换过程basis theorem of hilbert 希耳伯特基定理basis vector 基本向量bayes decision function 贝叶斯判定函数bayes formula 贝叶斯公式bayes postulate 贝叶斯公设bayes solution 贝叶斯解behavior 行为bellman principle 贝尔曼原理beltrami equation 贝尔特拉米方程bending point 转向点bergman metric 伯格曼度量bernoulli equation 伯努利方程bernoulli inequality 伯努利不等式bernoulli method 伯努利法bernoulli number 伯努利数bernoulli polynomial 伯努利多项式bernoulli trials 伯努利试验bernoullian polynomial 伯努利多项式bernstein inequality 伯思斯坦不等式bernstein polynomial 伯思斯坦多项式bertrand curves 柏特龙曲线bertrand paradox 柏特龙悖论bessel equation 贝塞耳方程bessel function 贝塞耳函数bessel inequality 贝塞耳不等式bessel integral 贝塞耳积分best approximation 最佳逼近best uniformapproximation 最佳一致逼近beta distri but ion 分布beta function 函数bbicompact 紧bicompact set 紧集bicompact space 列紧空间bicompact transformationgroup 列紧变换群bicompactification 紧化biconditional 等价biconnected space 双连通空间bicontinuous function 双连续函数bicontinuouslydifferentiable 双连续可微bicylinder 双圆柱bidimensional 二维的bidimensionality 二维性bidual banach space 双对偶巴拿赫空间bifunctor 二变项函子bijection 双射bijective mapping 双射bijectivity 双射性bilateral 两面的bilateral derivative 双侧导数bilateral laplacetransform 双侧拉普拉斯变换bilaterally boundedsequence 双侧有界序列bilinear 双线性的bilinear form 双线性形式bilinear functional 双线性泛函binary arithmetic 二进制算术binary code 二进制吗binomial distri but ion 二项分布binomial equation 二项方程binomial expansion 二项展开式binomial integral 二项式积分binomial probability paper 二项式概率纸bisect 平分bisecting point 平分点bisection 平分bisector 平分线bisector of angle 角的平分线border 边缘border element 边缘元素bornological set 有界型集bornological space 有界型空间bornological topology 有界型拓扑学bornology 有界型性bound 界bound decision variable 约束决策变量boundary 边界;边缘boundary cell 边界胞腔boundary condition 边界条件boundary correspondence 边界对应boundary value problem 边值问题bounded 有界的bounded above 上有界的bounded above sequence 上有界序列bounded below 下有界的bounded below sequence 下有界序列bounded closed set 有界闭集bounded domain 有界域bounded function 有界函数bounded matrix 有界矩阵bounded minimization 有界最小化bounded operator 有界算子bounded point sequence 有界点序列bounded quantification 有界量词限制bounded quantifier 有界量词bounded sequence 有界序列bounded set 有界集合bounded to thedownwards 下有界的bounded to the upwards 上有界的bounded variation 有界变分boundedly convergentseries 有界收敛级数boundedness 有界性boundless 无限的box 框brace 大括号brachistochrone 最速降线bracket 括号break point 断点;分割点briggs' logarithm 常用对数briggsian logarithm 常用对数broken diagonal 折对角线broken line 折线broken number 分数bus 母线byte 字节calculability 可计算性calculable mapping 可计算映射calculable relation 可计算关系calculate 计算calculating automaton 计算自动机calculation 计算calculation of areas 面积计算calculator 计算机calculus 演算calculus ofapproximations 近似计算calculus of probability 概率calculus of variations 变分法calibration 校准cancel 消去cancellation 消去cancellation law 消去律cancellation property 消去性质cancelling of significantfigures 有效数字消去canonical coordinates 标准坐标canonical correlationcoefficient 典型相关系数canonical decomposition 标准分解canonical form 标准型canonical function 标准函数canonical fundamentalsystem 标准基本系统canonical hyperbolicsystem 典型双曲线系canonical mapping 标准映射canonical representation 典型表示canonical sequence 标准序列canonical solution 标准解canonical system ofdifferential equations 标准微分方程组canonical variable 典型变量canonical variationalequations 标准变分方程canonical variationalproblem 标准变分问题cap 交capacity 容量card 卡片cardinal number 基数cardinal product 基数积cardioid 心脏线carrier 支柱cartesian coordinatesystem 笛卡儿坐标系cartesian coordinates 笛卡尔座标cartesian equation 笛卡儿方程cartesian folium 笛卡儿叶形线cartesian product 笛卡儿积cartesian space 笛卡儿空间category 范畴category of groups 群范畴category of modules 模的范畴category of sets 集的范畴category of topological spaces 拓扑空间的范畴catenary 悬链线catenary curve 悬链线catenoid 悬链曲面cauchy condensation test 柯微项收敛检验法center of projection 射影中心center of symmetry 对称中心centered process 中心化过程centered system of sets 中心集系centi 厘centigram 厘克centimetre 厘米central angle 圆心角central confidence interval 中心置信区间central conic 有心圆锥曲线central derivative 中心导数central difference operator 中心差分算子central divided difference 中心均差central element 中心元central extension 中心扩张central extension field 中心扩张域central limit theorem 中心极限定理central line 中线central moment 中心矩central point 中心点central processing unit 中央处理器central projection 中心射影central quadric 有心二次曲面central series 中心群列central symmetric vectorfield 中心对称向量场central symmetry 中心对称centralizer 中心化子centre 中心centre of a circle 圆心centre of gyration 旋转中心centre of projection 射影中心centre of similarity 相似中心centre of similitude 相似中心centrifugal force 离心力centripetal acceleration 向心加速度centroid 形心certain event 必然事件certainty 必然cesaro mean 纬洛平均cesaro method ofsummation 纬洛总求法chain 链chainette 悬链线chamber complex 箱盒复形chance 偶然性;偶然的chance event 随机事件chance move 随机步chance quantity 随机量chance variable 机会变量change 变化change of metrics 度量的变换change of the base 基的变换change of the variable 变量的更换channel 信道channel width 信道宽度character 符号circulant 循环行列式circulant matrix 轮换矩阵circular 圆的circular arc 圆弧circular cone 圆锥circular correlation 循环相关circular cylinder 圆柱circular disk 圆盘circular domain 圆形域circular frequency 角频率circular functions 圆函数circular helix 圆柱螺旋线circular measure 弧度circular motion 圆运动circular neighborhood 圆邻域circular orbit 圆轨道circular pendulum 圆摆circular permutation 循环排列circular ring 圆环circular section 圆截面circular sector 圆扇形circular segment 圆弓形circular slit domain 圆形裂纹域circular symmetry 圆对称circular transformation 圆变换circulation 循环circulation index 环粮数circulation of vector field 向量场的循环circulatory integral 围道积分circumcenter 外心circumcentre 外心circumcircle 外接圆circumcone 外切圆锥circumference 圆周circumscribe 外接circumscribed circle 外接圆circumscribed figure 外切形circumscribed polygon 外切多边形circumscribedquadrilateral 外切四边形circumscribed triangle 外切三角形circumsphere 外接球cissoid 蔓叶类曲线cissoidal curve 蔓叶类曲线cissoidal function 蔓叶类函数clairaut equation 克莱罗方程class 类class bound 组界class field 类域class field tower 类域塔class frequency 组频率class function 类函数class interval 组距class mean 组平均class number 类数class of conjugate elements 共轭元素类classical groups 典型群classical lie algebras 典型李代数classical mechanics 经典力学classical sentential calculus 经典语句演算classical set theory 经典集论classical statistical mechanics 经典统计力学classical theory of probability 经典概率论classification 分类classification statistic 分类统计classification theorem 分类定理classify 分类classifying map 分类映射classifying space 分类空间clear 擦去clifford group 克里福特群clifford number 克里福特数clockwise 顺时针的clockwise direction 顺时针方向clockwise rotation 顺时针旋转clopen set 闭开集closable linear operator 可闭线性算子closable operator 可闭算子closed ball 闭球closed circuit 闭合电路closed complex 闭复形closed convex curve 卵形线closed convex hull 闭凸包closed cover 闭覆盖closed curve 闭曲线closed disk 闭圆盘closed domain 闭域closed equivalencerelation 闭等价关系closed extension 闭扩张closed filter 闭滤子closed form 闭型closed formula 闭公式closed geodesic 闭测地线closed graph 闭图closed graph theorem 闭图定理closed group 闭群closed half plane 闭半平面closed half space 闭半空间closed hull 闭包closed interval 闭区间closed kernel 闭核closed linear manifold 闭线性廖closed loop system 闭圈系closed manifold 闭廖closed map 闭映射closed neighborhood 闭邻域closed number plane 闭实数平面closed path 闭路closed range theorem 闭值域定理closed region 闭域closed riemann surface 闭黎曼面closed set 闭集closed shell 闭壳层closed simplex 闭单形closed solid sphere 闭实心球closed sphere 闭球closed star 闭星形closed subgroup 闭子群closed subroutine 闭型子程序closed surface 闭曲面closed symmetricextension 闭对称扩张closed system 闭系统closed term 闭项closeness 附近closure 闭包closure operation 闭包运算closure operator 闭包算子closure property 闭包性质clothoid 回旋曲线cluster point 聚点cluster sampling 分组抽样cluster set 聚值集coadjoint functor 余伴随函子coalgebra 上代数coalition 联合coanalytic set 上解析集coarser partition 较粗划分coaxial circles 共轴圆cobase 共基cobordant manifolds 配边廖cobordism 配边cobordism class 配边类cobordism group 配边群cobordism ring 配边环coboundary 上边缘coboundaryhomomorphism 上边缘同态coboundary operator 上边缘算子cocategory 上范畴cochain 上链cochain complex 上链复形cochain homotopy 上链同伦cochain map 上链映射cocircuit 上环道cocommutative 上交换的cocomplete category 上完全范畴cocycle 上闭键code 代吗coded decimal notation 二进制编的十进制记数法codenumerable set 余可数集coder 编器codiagonal morphism 余对角射codifferential 上微分codimension 余维数coding 编码coding theorem 编码定理coding theory 编码理论codomain 上域coefficient 系数coefficient domain 系数域coefficient function 系数函数coefficient functional 系数泛函coefficient group 系数群coefficient of alienation 不相关系数coefficient of association 相伴系数coefficient of covariation 共变系数coefficient of cubical expansion 体积膨胀系数coefficient of determination 可决系数coefficient of diffusion 扩散系数coefficient of excess 超出系数coefficient of friction 摩擦系数coefficient of nondetermination 不可决系数coefficient of rankcorrelation 等级相关系数coefficient of regression 回归系数coefficient of theexpansion 展开系数coefficient of thermalexpansion 热膨胀系数coefficient of variation 变差系数coefficient of viscosity 粘性系数coefficient problem 系数问题coefficient ring 系数环coercive operator 强制算子cofactor 代数余子式cofiber 上纤维cofibering 上纤维化cofibration 上纤维化cofilter 余滤子cofinal set 共尾集cofinal subset 共尾子集cofinality 共尾性cofinite subset 上有限子集cofunction 余函数cogenerator 上生成元cogredient automorphism 内自同构coherence 凝聚coherence condition 凝聚条件coherent module 凝聚摸coherent ring 凝聚环coherent set 凝聚集coherent sheaf 凝聚层coherent stack 凝聚层coherent topology 凝聚拓扑coherently orientedsimplex 协同定向单形cohomological dimension 上同惮数cohomological invariant 上同祷变量cohomology 上同调cohomology algebra 上同碟数cohomology class 上同掂cohomology functor 上同弹子cohomology group 上同岛cohomology group withcoefficients g 有系数g的上同岛cohomology module 上同担cohomology operation 上同邓算cohomology ring 上同捣cohomology sequence 上同凋列cohomology spectralsequence 上同底序列cohomology theory 上同帝cohomotopy 上同伦cohomotopy group 上同伦群coideal 上理想coimage 余象coincidence 一致coincidence number 叠合数coincidence point 叠合点coincident 重合的coinduced topology 余导出拓扑cokernel 上核collect 收集collectionwise normalspace 成集体正规空间collective 集体collinear diagram 列线图collinear points 共线点collinear vectors 共线向量collinearity 共线性collineation 直射变换collineation group 直射群collineatorytransformation 直射变换collocation method 配置法collocation of boundary 边界配置collocation point 配置点colocally small category 上局部小范畴cologarithm 余对数colorable 可着色的column 列column finite matrix 列有限矩阵column matrix 列阵column rank 列秩column space 列空间column vector 列向量combination 组合combination principle 结合原理combination with repetitions 有复组合combination without repetition 无复组合combinatorial analysis 组合分析combinatorial closure 组合闭包combinatorial dimension 组合维数combinatorial geometry 组合几何学combinatorial manifold 组合廖combinatorial method 组合方法combinatorial optimization problem 组合最优化问题combinatorial path 组合道路combinatorial problem 组合最优化问题combinatorial sphere 组合球面combinatorial sum 组合和combinatorial theory of probabilities 概率组合理论combinatorial topology 组合拓朴学combinatorially equivalentcomplex 组合等价复形combinatories 组合分析combinatory logic 组合逻辑combinatory topology 组合拓朴学combined matrix 组合矩阵comma 逗点command 命令commensurability 可通约性commensurable 可通约的commensurable quantities 可公度量common denominator 公分母common difference 公差common divisor 公约数common factor 公因子common factor theory 公因子论common fraction 普通分数common logarithm 常用对数common measure 公测度common multiple 公倍元common perpendicular 公有垂线common point 公共点common ratio 公比common tangent of twocircles 二圆公切线communality 公因子方差communication channel 通讯通道commutant 换位commutation law 交换律commutation relation 交换关系commutative 可换的commutative diagram 交换图表commutative group 交换群commutative groupoid 阿贝耳广群commutative law 交换律commutative lie ring 交换李环commutative ordinalnumbers 交换序数commutative ring 交换环commutativity 交换性commutator 换位子commutator group 换位子群commute 交换compact 紧的compact convergence 紧收敛compact group 紧群compact open topology 紧收敛拓扑compact operator 紧算子compact set 紧集compact space 紧空间compact subgroup 紧子群compact support 紧支柱compactification 紧化compactification theorem 紧化定理compactness 紧性compactness theorem 紧性定理compactum 紧统comparability of cardinals 基数的可比较性comparable curve 可比曲线comparable function 可比的函数comparable topology 可比拓扑comparable uniformity 可比一致性comparison function 比较函数comparison method 比较法comparison series 比较用级数comparison test 比较检验comparison theorem 比较定理。
数学专业英语翻译
第一段翻译(2)：what is the exact valueo f the number pai?a mathematici an made an experi mentin orderto find his own estima tionof the number pai.in his experi ment,he used an old bicycl e wheelof diamet er 63.7cm.he marked the pointon the tire wherethe wheelwas touchi ng the ground and he rolled the wheelstraig ht aheadby turnin g it 20 times.next,he measur ed the distan ce travel ed by the wheel,whichwas 39.69 meters.he divide d the number3969 by 20*63.7 and obtain ed 3.115384615 as an approx imati on of the number pai.of course,this was just his estima te of the number pai and he was awarethat it was not very accura te.数π的精确值是什么？一位数学家做了实验以便找到他自己对数π的估计。
在试验中，他用了一直径63.1厘米的旧自行车轮。
他在车轮接触地面的轮胎上做了标记，而且将车轮向前转动20次。
接下来，他测量了车轮经过的距离，是39.69米。
他用3969除20*63.7得到了数π的近似值3.115384615。
(抱佛脚必备)数学专业英语重点词汇
【Mathematical analysis】1 、上确界(supremum value) 下确界(infimum value)2 、极限(limit) 导数(derivative) 一阶导数(firstderivative) 二阶导数(second derivate)偏导数(partialderivative)方向导数(directional derivative)3 、泰勒展式(Taylor’s expansion) 洛必达法则(L’Hopital’srule) 链式法则(chain rule)连续可微(continuouslydifferentiable)4、微积分 (calculus)微分(differential)级数(series)积分(integral)原函数 (antiderivative)不定积分(indefinite integral)定积分(definite integral)5 、调和级数(harmonic series)马克劳林级数(Maclaurin series)交错级数 (alternating series)傅里叶级数(Fourier series)6 、二重积分(double integral)三重积分(triple integral)多重积分(multiple integral) 格林公式(Green’s formula)斯托克斯定理(Stokes’ theorem)7、收敛(convergence)发散(divergence)一致收敛(uniformly convergent)绝对收敛(absolutely convergent)条件收敛(conditionally convergent) 连续(continuity) 一致连续(uniformly continuous)8 、指数函数(exponential function)对数函数(logarithmicfunction)幂函数(power function)初等函数(elementary function)三角函数(trigonometric function) 调和函数(harmonic function) 周期函数(periodic function) 可微函数(differential function)单调函数(monotonefunction)9 、素数(prime number) 正数(positive number) 负数(negative number) 相反数(opposite number) 自然数(natural number) 有理数(rational number)无理数(irrational number)实数(real number)虚数(imaginarynumber)复数(complex number) 10、等式(equality)不等式(inequality)三角不等式(triangle inequality)勾股定理(Pythagorean theorem)无穷大(infinity)无穷小(infinitesimal)【Linear algebra】1、向量(vector)秩(rank)行列式(determinant)线性方程(linear equation)2 、奇异矩阵(singular matrix)可逆矩阵(invertible matrix)逆矩阵(inverse matrix) 转置矩阵(transposed matrix)酉矩阵(unitary matrix)对称矩阵(symmetric matrix)正定矩阵(positive definite matrix)负定矩阵(negative definitematrix) 雅克比矩阵(Jacobian matrix)3 、迹(trace)主子式(principle minor)代数余子式(algebraic cofactor)4 、二次型(quadratic form)标准型(normal form)最小多项式(minimal polynomial) 特征多项式(characteristicpolynomial)约当块(Jordan block)5 、特征值(eigenvalue)特征向量(eigenvector)约当标准型(Jordan canonical form)6 、几何重数(geometric multiplicity)可对角矩阵(diagonalizable matrix) 【Analytic Geometry】1 、平面几何(plane geometry) 立体几何(solid geometry)射影几何(projective geometry)代数几何(algebraicgeometry)黎曼几何(Riemann geometry)微分几何(differential geometry)几何分析(geometryanalysis)分形几何(fractal geometry)2 、平行线(parallel line) 中线(median line) 直线(straightline) 垂直线(vertical line)水平线(horizontal line)切线(tangent line)法线(normal line)3 、抛物线(parabola)椭圆(ellipse)双曲线(hyperbola)斜率(slope) 【Complex analysis】1 、解析函数(holomorphic function) 留数定理(residualtheorem) 柯西积分公式 (Cauchy’s integral formula)2 、定义域(domain) 值域(range) 像(image) 中值定理(mean value theorem)3 、凸函数(convex function) 凸集(convex set) 变分不等式(variational inequality) 变分法(calculus of variation)线性规划(linear programming)【Real analysis】1 、测度(measure) 可测函数(measure function) 可积函数(integrable function) 平方可积函数(square integrablefunction)黎曼积分(Riemann integral) 勒贝格积分(Lebesgue integral)【Functional analysis】1 、内积(inner product) 向量积(cross product)范数(norm)2 、向量空间(vector space)距离空间(distance space)拓扑空间(topological space) 测度空间(measure space)线性空间(linear space)内积空间(inner product space) 希尔伯特空间(Hilbert space) 赋范空间(normed space) 巴拿赫空间(Banach space)完备空间(complete space)可分空间(separable space) 概率空间(probability space)3 、有穷维的(finite dimensional)无穷维的(infinitedimensional)基底(basis) 线性相关(linear dependence) 线性无关(linear independence) 最佳逼近(bestapproximation ) 最大值原理( maximum principle) 比较原理( comparison principle)最小二乘(least squares)4、泛函(functional)线性算子(linear operator)有界线性算子(bounded linear operator) 有界线性泛函(boundedlinear functional) 闭图像定理(closed graph theorem)一致有界定理(uniform boundedness principle)不动点定理(fixed point theorem)压缩映照定理(contraction mapping theorem)【Supplements】1 、引理(lemma)推论(corollary)公理(axiom)命题(proposition)猜想(conjecture) 数学归纳法(mathematical induction) 充分性(sufficiency) 必要性(necessity) 反例(counterexample)2、加法(addition)减法(subtraction)乘法(multiplication)除法(division)3、理想(ideal)环(ring)单位(unit)陪集(coset)群(group)域(field)置换群(permutation group)有限群(finite group)同态(homomorphism) 同构(isomorphism)维数(dimension)4 、抽象代数(abstract algebra) 广义函数论(theory of distribution) 弦理论(stringtheory) 随机变量(random variable) 动力系统(dynamical system) 偏微分方程(partial differentialequation)。
数学专业英语
数学专业英语（1）、Given ε>0,there exists a number N >0 such that ε<-a a n for all N n ≥ 译文：对给定的ε>0，存在一个数N >0使得不等式ε<-a a n 对所有的N n ≥都成立。
（2）、The function )(x f approaches infinity as x tends to zero.译文：当x 趋于0时，函数)(x f 趋于∞。
（3）、Suppose D is an open set with its closure in G .译文：假定D 是一个开区间，且其闭包在G 中。
（4）、Suppose )(x f is a function on domain D such that M x f <)( for all x ∈D ,where M is a constant .译文：假定)(x f 是区域D 上的一个函数，使得对所有x ∈D ，不等式M x f <)(成立，其中M 是一个常数。
（可用“satisfying ”代替上述“such that ”。
）（5）、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 ≥.译文：根据引理2可推出x y ≥。
（6）、有时用现在分词表示“经过……而得到……”（推论）。
Integrating the above inequality twice,we see that . log )(0t t c t y ≥译文：将上一不等式两次积分得到. log )(0t t c t y ≥。
（7）、表示充分必要条件The sufficient and necessary condition for the equality isα>0 and ≥p 3. 译文：该等式成立的充分必要条件是α>0 且≥p 3。
数学专业英语名词_部分6
sequencing problem 日程计划问题sequent 串联的sequential analysis 序贯分析sequential compactness 列紧性sequential control 顺序控制sequential estimation 序列估计sequential likelihood ratio test 序贯似然比值检验sequential programming 顺序规划sequential sampling 序贯抽样sequential sampling plan 序列抽样法sequential switching circuit 依次转接电路sequential test 序贯检定sequential word function 序贯字函数sequentially compact set 列紧集sequentially complete space 序列完备空间serial correlation 自相关serial correlation coefficient 序列相关系数series 级数series development 级数展开series expansion 级数展开series of functions 函数级数series solution 级数解serpentine 蛇形线service model 服务模型sesquilinear form 半双线性形式set 集set algebra 集代数set function 集函数set of condensation points 凝聚点集set of continuum power 连续统势的集set of limit 极限集合set of lower bounds 下界集set of measure zero 零测度集set of numbers 数集set of points 点集set of sets 集的集set of strategies 策略集set of ternary numbers 三进制数集set of upper bounds 上界集set theoretic addition 集论的加法set theoretic image 集合论的象set theoretic intersection 集论的交set theoretic operation 集论的运算set theoretic proof 集论的证set theoretic union 集论的并集set theoretical 集论的set theory 集论set topology 集论拓扑set valued functor 集值函子set valued mapping 集值映射sexadecimal digit 十六进制数字sexadecimal notation 十六进记法sexadecimal number system 十六进制数系sexagesimal arithmetic 六十进算术sexagesimal system 六十进制sextant 六分仪sextic 六次曲线sextic equation 六次方程。
数学专业英语2-6
For a given real number x>0, it is possible to count the number of primes less than or equal to x. This number is said to be a function of x even though no simple algebraic formula is known for computing it (without counting) when x is known. 对于一个给定的实数x>0，数出小于或者等于x的素数 的个数是有可能的。这个数称为x的函数，尽管还没 有一个简单代数式可以由已知的x计算(不通过计数求) 出它的值。
These are merely devices for describing special relations in a quantitative fashion. Mathematicians refer to certain types of these relations as functions. 这些只是以定量的方式描述特定关系的方法。数学家 将这些关系中的某些类型视作函数。 In this section, we give an informal description of the function concept. A formal definition is given in Section 3. 在本节中，我们给出一个非正式的描述函数的概念。 在第3节给出一个正式的定义 。
EXAMPLE 1. The force F necessary to stretch a steel spring a distance x beyond its natural length is proportional to x. That is, F=cx, where c is a number independent of x called the spring constant. 把一个钢制的弹簧拉伸到超过其自然长度的距离为x 时所需要的力F与x成正比。即，F=cx,这里c是不依赖 与x的数，叫做弹性系数。 This formula, discovered by Robert Hooke in the mid17th century, is called Hooke’s law, and it is said to express the force as a function of the displacement.
数学专业英语(吴炯圻)
New Words & Expressions:algebra 代数学geometrical 几何的algebraic 代数的identity 恒等式arithmetic 算术, 算术的measure 测量，测度axiom 公理numerical 数值的, 数字的conception 概念，观点operation 运算constant 常数postulate 公设logical deduction 逻辑推理proposition 命题division 除，除法subtraction 减，减法formula 公式term 项，术语trigonometry 三角学variable 变化的，变量2.1 数学、方程与比例Mathematics, Equation and Ratio4Mathematics comes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches.1－A What is mathematics数学来源于人类的社会实践，比如工农业生产，商业活动，军事行动和科学技术研究。
And in turn, mathematics serves the practice and plays a great role in all fields. No modern scientific and technological branches could be regularly developed without the application of mathematics. 反过来，数学服务于实践，并在各个领域中起着非常重要的作用。
数学专业英语词汇(大学)
数学专业英语词汇（大学）数学专业英语词汇（大学）a priori bound 先验界限a priori distribution 先验分布a priori probability 先验概率a summable a可和的abacus 算盘abbreviate 略abbreviation 简化abel equation 阿贝耳方程abel identity 阿贝耳恒等式abel inequality 阿贝耳不等式abel summation method 阿贝耳求和法abelian algebra 阿贝耳代数abelian differential 阿贝耳微分abelian equation 阿贝耳方程abelian extension 阿贝耳扩张abelian function 阿贝耳函数abelian function field 阿贝耳函数域abelian functor 阿贝耳函子abelian group 交换群abelian groupoid 阿贝耳广群abelian integral 阿贝耳积分abelian summation 阿贝耳求和法abelian theorem 阿贝耳定理abelian variety 阿贝耳簇abridge 略abridged notation 简算记号abscissa 横坐标abscissa of absolute convergence 绝对收敛坐标abscissa of summability 可和性坐标abscissa of uniform convergence 一致收敛横坐标absolute 绝对形absolute address 绝对地址absolute class field 绝对类域absolute coding 绝对编码absolute cohomology 绝对上同调absolute conic 绝对二次曲线absolute convergence 绝对收敛absolute curvature vector 绝对曲率向量absolute deviation 绝对偏差absolute differential calculus 绝对微分学absolute error 绝对误差absolute extremes 绝对极值absolute extremum 绝对极值absolute frequency 绝对频率absolute geometry 绝对几何absolute homology group 绝对同岛absolute homotopy group 绝对同伦群absolute inequality 绝对不等式absolute instability 绝对不稳定性absolute maximum 绝对极大值absolute minimum 绝对极小值absolute moment 绝对矩absolute neighborhood 绝对邻域absolute neighborhood retract 绝对邻域收缩核absolute norm 绝对范数absolute number 不名数absolute parallelism 绝对平行性absolute quadric 绝对二次曲面absolute ramification index 绝对分歧指数absolute rotation 绝对旋转absolute singular homology group 绝对奇异同岛absolute space 绝对空间absolute space time 绝对时空absolute stability 绝对稳定性absolute term 常数项absolute unit 绝对单位absolute value 绝对值absolute value sign 绝对值符号absolute velocity 绝对速度absolutely additive measure 完全加性测度absolutely compact set 绝对紧集absolutely complete system 绝对完备系absolutely continuous 绝对连续的absolutely continuous distribution 绝对连续分布absolutely continuous function 绝对连续函数absolutely continuous measure 绝对连续测度absolutely continuous part 绝对连续部分absolutely continuous transformation 绝对连续变换absolutely convergent 绝对收敛的absolutely convergent integral 绝对收敛积分absolutely convergent series 绝对收敛级数absolutely convex hull 绝对凸包absolutely discontinuous function 绝对不连续函数absolutely integrable 绝对可积的absolutely irreducible character 绝对不可约特征absolutely irreducible representation 绝对不可约表示absolutely irreducible variety 绝对不可约簇absolutely normal number 绝对范数absolutely prime ideal 绝对素理想absolutely semisimple algebra 绝对半单代数absolutely simple group 绝对单群absolutely summable sequence 绝对可和序列absolutely unbiased estimator 绝对无偏估计量absolutely unramified extension 绝对非分歧扩张absorbing barrier 吸收障碍absorbing medium 吸收媒体absorbing set 吸收集absorbing state 吸收态absorption 吸收absorption coefficient 吸收系数absorption curve 吸收曲线absorption factor 吸收因数absorption index 吸收指数absorption law 吸收律absorption probability 吸收概率abstract 抽象的abstract algebra 抽象代数abstract algebraic geometry 抽象代数几何abstract automaton 抽象自动机abstract category 抽象范畴abstract code 理想码abstract complex 抽象复形abstract group 抽象群abstract interval function 抽象区间函数abstract mathematics 抽象数学abstract number 不名数abstract ordered simplicial complex 抽象有序单纯复形abstract simplex 抽象单形abstract simplicial subcomplex 抽象单子复形abstract space 抽象空间abstraction 抽象abstraction operator 抽象算子absurd 谬论的absurdity 谬论abundance ratio 丰度比abundant number 过剩数accelerated motion 加速运动acceleration 加速度acceleration of convergence 收敛性的加速acceleration of gravity 重力加速度acceptable quality level 合格质量水平acceptance 肯定acceptance inspection 接受检查acceptance limit 接受界限acceptance line 接受线acceptance number 接受数acceptance probability 接受概率acceptance region 接受区域acceptance zone 接受带access 存取access speed 存取速度access time 存取时间accessibility 可达性accessible boundary point 可达边界点accessible ordinal number 可达序数accessible point 可达点accessible set 可达集accessible vertex 可达顶点accessory extremal 配连极值accidental 偶然的accidental coincidence 偶然符合accidental error 随机误差accommodation 第accumulated error 累积误差accumulating point 聚点accumulation 累积accumulation point 聚点accumulator 累加器存储器accuracy 准确性accuracy grade 准确度accuracy of measurement 测量精确度accuracy rating 准确度acnod 孤点acount 计算action integral 酌积分action variable 酌变量active restriction 有效限制actual 真实的actual infinity 实无穷acute 尖锐的acute angle 锐角acute angled triangle 锐角三角形acute triangle 锐角三角形acuteness 锐度acyclic 非循环的acyclic complex 非循环复形acyclic graph 环道自由图acyclic model theorem 非循环模型定理ad infinitum 无穷地adams circle 阿达姆斯圆adams extrapolation method 阿达姆斯外插法adaptability 适应性adaptation 适应adapted basis 适应基add 加added circuit 加法电路addend 加数adder 加法器addition 加法addition formulas 加法公式addition sign 加号addition system 加法系addition table 加法表addition theorem 加法定理addition theorem of probability 概率的加法定理additional 加法的additional code 附加代码additional condition 附加条件additional error 附加误差additive 加法的additive category 加性范畴additive class 加性类additive functional 加性泛函数additive functional transformation 加性泛函变换additive group 加法群additive interval function 加性区间函数additive inverse element 加性逆元素additive number theory 堆垒数论additive operator 加性算子additive process 加性过程additive relation 加性关系additive separable 加法可分的additive theory of numbers 堆垒数论additive valuation 加法赋值additively commutative ordinal numbers 加性交换序数additivity 加法性address 地址address part 地址部分address register 地址寄存器。
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Mathematical English 6: Function concept and function idea
undergone many stages of generalization. Today, the meaning of function is essentially this: Given two sets, say X and Y, a function is a correspondence which associates with each element of X one and only one element of Y. The set X is called the domain of the function. Those elements of Y associated with the elements in X form a set called the range of the function. (This may be all of Y, but it need not be.) Letters of the English and Greek alphabets are often used to denote functions. The particular letters f, g, h, F, G, H, and ϕ are frequently used for this purpose, If f is a given function and if x is an object of its domain, the notation f(x) is used to designate that object in the range which is associated to x by the function f; and it is called the value of f at x or the image of x under f. The symbol f(x) is read as “f of x.”
Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University
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Mathematical English 6: Function concept and function idea
The totality of such points (x, y) is called the graph of the function. Now we consider some more examples of real functions. EXMAPLE 4. The identity function. Suppose that f(x) =x for all real x. This function is often called the identity function. Its domain is the real line, that is, the set of all real numbers. Here x=y for each point (x, y) on the graph of f. The graph is a straight line making equal angles with the coordinates axes (see Figure 2-6-2). The range of f is the set of all real numbers. EXMAPLE 5. The absolute-value function. Consider the function which assigns to each real number x the nonnegative number |x|. A portion of its graph is shown in Figure 2-6-3. Denoting this function by ϕ, we have ϕ(x)=|x| for all real x. For example, ϕ(0)=0, ϕ(2)=2, ϕ(-3)=3. We list here some properties of absolute values expressed in function notation. (a) ϕ(-x)=ϕ(x) (b) ϕ(x2)=x2 (c) ϕ(x+y)≤ϕ(x)+ϕ(y) (the triangle inequality)
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Mathematical English 6: Function concept and function idea
§2.6 Function concept and function idea
TEXT A Informal description of functions
Various fields of human have to do with relationships that exist between one collection of objects and another. Graphs, charts, curves, tables, formulas, and Gallup polls are familiar to everyone who reads the newspapers. These are merely devices for describing special relations in a quantitative fashion. Mathematicians refer to certain types of these relations as functions. In this section, we give an informal description of the function concept. A formal definition is given in Section 3. EXAMPLE 1. The force F necessary to stretch a steel spring a distance x beyond its natural length is proportional to x. That is, F=cx, where c is a number independent of x called the spring constant. This formula, discovered by Robert Hooke in the mid-17th century, is called Hooke sncept and function idea
Mathematical English
Dr. Xiaomin Zhang Email: zhangxiaomin@
Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University
Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University
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Mathematical English 6: Function concept and function idea
TEXT B The function idea
Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University
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Mathematical English 6: Function concept and function idea
law, and it is said to express the force as a function of the displacement. EXAMPLE 2. The volume of a cube is a function of its edge-length. If the edges have length x, the volume V is given by the formula V=x3. EXAMPLE 3. A prime is any integer n>1 that cannot be expressed in the form n=ab, where a and b are positive integers, both less than n. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19. For a given real number x>0, it is possible to count the number of primes less than or equal to x. This number is said to be a function of x even though no simple algebraic formula is known for computing it (without counting) when x is known. The word “function” was introduced into mathematics by Leibniz, who used the term primarily to refer to certain kinds of mathematical formulas. It was later realized that Leibniz’s idea of function was much too limited in its scope, and the meaning of the word has since