数学专业英语-第2章课后答案.docx

2.12.比:ratio 比例:proportion 利率:interest rate 速率:speed 除:divide 除法:division 商:quotient 同类量:like quantity 项:term 线段:line segment 角:angle 长度:length 宽:width高度:height 维数:dimension 单位:unit 分数:fraction 百分数:percentage3.(1)一条线段和一个角的比没有意义,他们不是相同类型的量. (2)比较式通过说明一个量是另一个量的多少倍做出的,并且这两个量必须依据相同的单位. (5)为了解一个方程,我们必须移项,直到未知项独自处在方程的一边,这样就可以使它等于另一边的某量.4.(1)Measuring the length of a desk, is actually comparing the length of the desk to that of a ruler. (3)Ratio is different from the measurement, it has no units. The ratio of the length and the width of the same book does not vary when the measurement unit changes. (5)60 percent of students in a school are female students, which mean that 60 students out of every 100 students are female students.2.22.初等几何:elementary geometry 三角学:trigonometry 余弦定理:Law of cosines 勾股定理/毕达哥拉斯定理:Gou-Gu theorem/Pythagoras theorem 角:angle 锐角:acute angle 直角:right angle 同终边的角:conterminal angles 仰角:angle of elevation 俯角:angle of depression 全等:congruence 夹角:included angle 三角形:triangle 三角函数:trigonometric function直角边:leg 斜边:hypotenuse 对边:opposite side 临边:adjacent side 始边:initial side 解三角形:solve a triangle 互相依赖:mutually dependent 表示成:be denoted as 定义为:be defined as3.(1)Trigonometric function of the acute angle shows the mutually dependent relations between each sides and acute angle of the right triangle. (3)If two sides and the included angle of an oblique triangle are known, then the unknown sides and angles can be found by using the law of cosines. (5)Knowing the length of two sides and the measure of the included angle can determine the shape and size of the triangle. In other words, the two triangles made by these data are congruent.4.(1)如果一个角的顶点在一个笛卡尔坐标系的原点并且它的始边沿着x轴正方向，这个角被称为处于标准位置. (3)仰角和俯角是以一条以水平线为参考位置来测量的,如果正被观测的物体在观测者的上方,那么由水平线和视线所形成的角叫做仰角.如果正被观测的物体在观测者的下方,那么由水平线和视线所形成的的角叫做俯角. (5)如果我们知道一个三角形的两条边的长度和对着其中一条边的角度,我们如何解这个三角形呢?这个问题有一点困难来回答,因为所给的信息可能确定两个三角形,一个三角形或者一个也确定不了.2.32.素数:prime 合数:composite 质因数:prime factor/prime divisor 公倍数:common multiple 正素因子: positive prime divisor 除法算式:division equation 最大公因数:greatest common divisor(G.C.D) 最小公倍数: lowest common multiple(L.C.M) 整除:divide by 整除性:divisibility 过程:process 证明:proof 分类:classification 剩余:remainder辗转相除法:Euclidean algorithm 有限集:finite set 无限的:infinitely 可数的countable 终止:terminate 与矛盾:contrary to3.(1)We need to study by which integers an integer is divisible, that is , what factor it has. Specially, it is sometime required that an integer is expressed as the product of its prime factors. (3)The number 1 is neither a prime nor a composite number;A composite number in addition to being divisible by 1 and itself, can also be divisible by some prime number. (5)The number of the primes bounded above by any given finite integer N can be found by using the method of the sieve Eratosthenes.4.(1)数论中一个重要的问题是哥德巴赫猜想,它是关于偶数作为两个奇素数和的表示. (3)一个数,形如2p-1的素数被称为梅森素数.求出5个这样的数. (5)任意给定的整数m和素数p,p的仅有的正因子是p和1,因此仅有的可能的p和m的正公因子是p和1.因此,我们有结论:如果p是一个素数,m是任意整数,那么p整除m,要么(p,m)=1.2.42.集:set 子集:subset 真子集:proper subset 全集:universe 补集:complement 抽象集:abstract set 并集:union 交集:intersection 元素:element/member 组成:comprise/constitute包含:contain 术语:terminology 概念:concept 上有界:bounded above 上界:upper bound 最小的上界:least upper bound 完备性公理:completeness axiom3.(1)Set theory has become one of the common theoretical foundation and the important tools in many branches of mathematics. (3)Set S itself is the improper subset of S; if set T is a subset of S but not S, then T is called a proper subset of S. (5)The subset T of set S can often be denoted by x, that is, T consists of those elements x for which P(x) holds. (7)This example makes the following question become clear, that is, why may two straight lines in the space neither intersect nor parallel.4.(1)设N是所有自然数的集合,如果S是所有偶数的集合,那么它在N中的补集是所有奇数的集合. (3)一个非空集合S称为由上界的,如果存在一个数c具有属性:x=1,这个n元数组称为一个n维点或者一个n维向量，各个数a1，a2，an称为这个向量的坐标或者分量。 (3)向量x1,x2,xn称为线性相关的，如果存在不全为0的标量使得a1x1+a2x2+anxn=0.向量x1，x2xn称为线性无关的，如果x1，x2xn不是线性相关的 。2.72.向量空间：vector space 行向量：row vector 列向量：column vector 线性相关：linearly dependent 线性无关:linearly independent 线性组合:linear combination 数量级:scalar product 矩阵:matrix 方阵:square matrix 行列式:determinant 逆矩阵:inverse matrix 单位矩阵:identity matrix 零矩阵:zero matrix 变换:transformation 到上的:onto 同构:isomorphism 同构的:isomorphic 应用微分方程:applied differential equations 数理经济:mathematical economics 量子力学:quantum mechanics 相容的:consistent 最终的:ultimately3.(1)Linear combination, linear dependence and linear independence are all important concepts of linear spaces. (3)Not only matrix can be used to solve a system of linear equations, but also can be used to judge whether the system of equations have solutions and whether the solution is unique. (5)This conclusion is contradictory to the hypothesis of the problem, so the proposition to be proved is true. (7)let V be an n-dimensional vector space over the field F and W be an m-dimensional vector space over the field F. Let B and B be ordered bases for U and W respectively. Then any linear transformation T from V into W is determined by the matrix of T relative to B and B.4.(1)如果V和W是在域F上的向量空间,任何V到W的到上的一一对应的线性变化T称为一个V到W到上的同构。如果存在一个V到W到上的同构，我们说V同构于W。 (3)当p=1，一个p*q的矩阵只有一行，称为一个行向量。当q=1时，该矩阵只有一列，称为一个列向量。当p和q都是1时，这种情况相当不值一提,这里不需要关注. 一个有p个元素的列向量我们成为一个p向量,所以一个p向量是一个p*1的矩阵。 (5)一个m*n的矩阵有m个行向量和n列向量。设A是一个有实数元素的n阶方阵，那么A的n个行向量是线性无关的当且仅当行列式|A|不等于0.进一步，A的n个列向量是线性相关的当且仅当A的n个列向量是线性相关的。332.8函数关系：function relation 表格：table 反函数：inverse function 简单函数：simple function 特征函数：characteristic function 复合函数：composite function 映射：mapping 定义域：domain 值域：range 像：image与成正比：be directly proportional to 正变：direct variation 反变：inverse variation 性质：property 按推广的定义：in the extended sense 普遍化：generalize/universalize 并入：incorporate 无穷大：infinite 最大值：maximum 可测空间：measurable space3.(1)Because the concept of function originated in physics, in the seventeenth century, people once believed that the function relation was nothing but a formula. (3)If to each value of the variable x， the variable y has a definite value corresponding，then the variable y is the function of variable x. (5)If f is a mapping from a space X to another space Y and g is a mapping from the space Y to the another space Z, then we can define a composite mapping h=g.f, which is a mapping from X to Z.4.(1)函数提供了一个研究一些变量的方法，所研究的变量随着其他的量变化，也就是说，当一个量的变化引起了另一个量相应的变化时。 (3)对一个函数f：D-R.我们定义f（D）=y in R|y=f(x) x属于D，并f(D)是f的像。我们说f：D-R达到一个最大值，如果使像f(D)有一个最大值，即有一个点x0属于D使得f(x)=f(x0). (5)要画出一个新的函数y=f(x)，即一个我们不熟悉的函数图象，我们可以按以下方式使用一些作图建议。(a)确定函数的定义域。(b)确定此方程拥有的任何类型的对称性。如果f(x)=f(-x)，那么函数表现出y轴轴对称性。如果f(-x)=-f(x)，那么函数表现出原点对称性。(c通过x=0处对f(x)取值求y截距，通过求x的值使f(x)=0求x截距。(d)建立一个满足方程y=f(x)的有序对的表格。(e)描画出于有序对有关的点，并用一条平滑的曲线连接它们，然后，如果合适，则根据图像所拥有的任何对称性反射出曲线的这一部分。2.92.变量：variable 极限：limit 左极限：left limit 右极限：right limit 绝对值：absolute value 序列：sequence 柯西列：Cauchy sequence 级数：series 递增的：increasing 递减的;decreasing 单调的：monotonic 有界的：bounded 无界的：unbounded 上界：upper bound 最小上界：least upper bound 最大下界：greatest lower bound 有限区间：finite interval 无穷区间：infinite interval 有界集：bounded set 实数系：real-number system 趋近：approach 收敛：converge 一致收敛：converge uniformly 判别法：criterion 等价：be equivalent to 逐次的：successive 与矛盾：contradict3.(1)A sequence is called monotonic if it is increasing or decreasing. (3)If a sequence an converges, then for any positive number c, there exists a positive integer N such that |an|=N. (5)Suppose the function f(x) is defined on the interval(a,infinite). If any c0, there exists a positive integer K such that |f(x)-A|=K(where A is a constant), then f(x) is said to converge to A as X-+infinite. If g(x)=-f(x)converges to A as xinfinite, then f(x) is said to converge to A as xinfinite. If f(x) converges to A both as x-+infinite and as xinfinite, then f(x) is said to converge to A as x-infinite.4.(1)一个序列an被称为有极限L如果对于任意的正整数c，存在另一个正整数N（N可能与c有关），使得|an-L|=N成立。一个不收敛的数列就被称为发散的。 (3)短语“收敛序列”只适用于极限是有限的序列。一个序列具有无穷极限被称为发散的。当然，发散序列并不具有无穷极限，例子由下面公式定义： (5)引理 假设序列dn收敛于数d ，并且dn=0对于每一个自然数n成立，那么我们有d0。上一个引理断言，一个非负数收敛序列也有一个非负的极限。一个正数的收敛序列也有一个正的极限，这不总是真实的。例如1/n是一个收敛于0的正数列。2.102.导数：derivation 微分：differential 切线：tangent line 即时速度：instantaneous velocity 差商：difference quotient 可视化的解释：visual interpretation 子区域：subdomain 逼近：approximation 原函数：primitive function 反导数：antiderivative 不定积分：indefinite integral 被积函数：integrand 任意常数：arbitrary constant 积分常数：integral constant 积分公式：integral formula 逆过程：reverse process 替换法积分：integration by substitution 分部积分：integration by parts 连续函数：continuous function 可微函数：differentiable function3.(1)Peoples researches on the tangent line to a curve and instantaneous velocity of moving body leads to the generation of the concept of a derivative. (3)The tangent problem provides a visual interpretation for the derivative, which leads people to define the derivative as the limit of a difference quotient. (5)If f(x) has a first derivative f(x) in an interval, then, it can be used to define the differential of f(x).4.(1)导数的例子例1 一个常函数的导数 设f是一个常值函数，比如说f（x）=c对于所有的x成立，差商是 由于对有所有h！=0。此商为0，它的极限f(x)对于每一个x也是0.换句话说，一个常数函数在各处有零导数。例2 一个线性函数的导数 设f是一个线性函数，比如说f(x)=mx+b对于所有的实数x成立。如果h！=0，我们有由于当h趋近于0时，差商不改变，我们得出结论f(x)=m对于每一个x成立。因此一个线性函数的导数是一个常函数。（3）方向导数 我们现在介绍一个偏导数的自然推广。在定义中，差商的分子用于涉及到f(x,y)在和两点的值。当趋近于0时，第一个点沿着直线y=y0趋近于后面的点。对于一个点沿直线x=x0趋近于。（5）相差一个常数的反导数 如果F是连续函数f的反导数，那么任何其他的反导数一定有形式 G(x)=F(x)+C，这就说明同一个函数的两个反导数一个相差积分常数值，此值可以为0.2.112. 定积分：definite integral 不变性：invariance 可加性：additivity 唯一性：uniqueness 阶梯函数：step function 非负函数：nonnegative function 放大：expansion 压缩：contraction 建立：establish 比较：comparison 成立：hold 严格不等式：strict inequality 直立柱子：right cylindrical solid 子区间:subinterval 旋转体:solid of revolution 圆柱形的:cylindrical 单调的:monotonic 可测的:measurable 可积的:integrable 全等的:congruent 截面的:cross-sectional 凸的:convex 有效的:valid3.(1)The definite integral satisfies the linearity with respect to the integrand，the additivity with respect to the interval of integration and invariance under translation. (3)Step functions or other simple functions can be used to approximate integrand above and below to find the approximate value of the integral of the function discussed. (5)Let a function y=f(x) be nonnegative on the intervala,b. Then the solid obtained by revolving ordinate set of this function about the x-axis, is a solid of revolution, whose volume can be computed by the integral of the function over the intervala,b.4.(1)微积分学的基本定理详细阐述了两种核心运算之间的关系微分和积分。定理的第一部分，有时被称为微积分第一基本定理,表明一个不定积分可以由一个微分法反转而来,第一部分也很重要因为它保证了对于连续函数反导数的存在性.第二部分,有时被称为微积分第二基本定理,允许人们通过使用它的无穷多个反导数的任意一个来计算一个函数的定积分.订立的这一部分有着不可估量的实际应用,因为它显著的简化定积分的计算. (3)推论 基本定理经常被用于计算一个函数f的定积分,它的一个反导数F已知.详细的说,如果f是一个a,b上的实值连续函数,F是f在a,b上的一个反导数,那么。推论假定整个区间上的连续性，这个定理在下面的定理中被稍微加强了。2.122.点状收敛：converge pointwise 绝对收敛：converge absolutely 一致收敛的：uniformly convergent 一致收敛：converge uniformly WM判别法：weierstrass M-test 比较判别法： geometric series 幂级数:power series 收敛圆:circle of convergence 收敛半径:radius of convergence 上界:upper bound 上有界:bounded above 最小上界:least upper bound 相邻项:consecutive terms 空集:empty set 在内部:interior to 控制:dominate 操作:manipulate3.(1)A sequence of functions which converges uniformly on an open interval(a,b) must converge at every point in the interval, but a sequence of functions which converges pointwise on an inteval(a,b) may not converge uniformly on the interval. (3)Any power series converge at least at the point z=0, if it converges in the entire complex plane, then its radius of convergence is positive infinite. (5)The general term of a convergent series always approaches zero, from which follows a necessary condition for a series to converge. But it is not sufficient because a series whose general term approaches zero may not converge.4.(1)这个特殊幂级数在收敛圆的每个边界点处都发散，因为如果|z|=r，则通项有绝对值n。 (3)一个级数被称为绝对收敛的如果收敛，如果收敛而发散，则它被称为条件收敛的。 （5）现在我们考虑各项在实直线或在复平面上具有共同定义域S的实值或复值函数的序列fn，假定fn在S上一致收敛于一个函数f，如果一个fn在S上的p点处连续，则极限函数f也在p上连续。 2.132.方程：equation 齐次方程：homogeneous equation 齐次的：homogeneous 非齐次的：non-homogeneous 存在性：existence 唯一性：uniqueness 唯一的：unique 惟一地：uniquely 微分方程的阶：order of a differential equation 初始条件：initial condition 通解：general solution 特解：particular solution 补解：complementary solution 产生：yield 变化：range 输入：input 输出：output 先验的：a priori3.(1)The order of a differential equation is defined as the order of the highest derivative that appears in the equation. (3)The study of existence and uniqueness of solutions of differential equation is very important because the practical problems require that we answer under what conditions a differential equation in question has solutions and has a unique solution. (5)The general solution of a nonhomogeneous equation is of the following form ，where yc（x）is the general solution of the corresponding homogeneous equation and yp（x) is a particular solution of the nonhomogeneous equation.4.(1)一般的二阶微分方程是具有形式F(x,y,y,y)=0的一个方程,与这种方程相联系的理论相当复杂,因此我们以把我们的注意力限制在那些可以就y接触的方程开始,即可以写成形如y=f(x,y,y)的方程. (3)如果函数p和q在开区间(a,b)上连续,那么存在微分方程在区间axb上的一个基础解组，集y1，y2是该方程的一个基础解组，当且仅当y1和y2都是该方程的解，且该方程的每一个解都可以表示成y1和y2的一个线性组合。 (5)很多研究已被致力于常微分方程的求解方法，在方程为线性的情形，它可以通过解析方法求解，不幸地是，大多数有趣的微分方程是非线性的,除了少数例外情况以外,它们不能被精确求解,其近似解可以通过计算机逼近求得.2.14随机变量：random variable 密度函数：density function 方差：variance 偏差：deviation 标准差：standard deviation 带权平均：weighted average 平方根：square root 均方根 root mean square 正态分布：normal distribution 斯蒂尔切丝积分：Stieltjes integral 大数定律：law of large numbers 质点：mass point 质心：center of mass 二阶矩：second moment惯性矩：moment of inertia 旋转半径：radius of gyration 散步：spread 使散开：disperse 离散的：discrete 连续的：continuous 理论上：theoretically3.（1）概率的一种字典定义为：一个事件可能发生的程度，通过喜好的情形与可能出现的情形的总数之比来衡量。 （3）为了推广这种想法，以至于它可以被应用于任何数据集，这些数据集由对许多个体中一个变量取值的标本组成，下面的统计记号法被发现是很有用的。 （5）在很多概率论的应用中，感兴趣的变量是其他随机变量的和，例如，在一种赌博中在若干局之后的获利结果是每次比赛获利的和。4.（1）The mathematical expectation E（X） and the variance Var(X) are both theoretically computable values associated with th